LONG-RANGE INTERATOMIC INTERACTIONS

LONG-RANGE INTERATOMIC INTERACTIONS: OSCILLATORY TAILS AND HYPERFINE PERTURBATIONS

by

CHANDRA MANI ADHIKARI

A DISSERTATION Presented to the Faculty of the Graduate School of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in PHYSICS 2017 Approved by

Dr. Ulrich D. Jentschura, Advisor Dr. Don H. Madison Dr. Michael Schulz Dr. Daniel Fischer Dr. Peter J. Mohr

Copyright 2017 CHANDRA MANI ADHIKARI All Rights Reserved

iii ABSTRACT

We study the long-range interaction between two hydrogen atoms, in both the van der Waals and Casimir-Polder regimes. The retardation regime is reached when the finiteness of the speed of light becomes relevant. Provided that both atoms are in the ground states, the retardation regime is achieved when the interatomic distance, R, is larger than 137 a0 , where a0 is the Bohr radius. To study the interaction between two hydrogen atoms in 1S and 2S states, we differentiate three different ranges for the interatomic distance: van der Waals range (a0  R  a0 /α, where α is the fine structure constant), the intermediate or Casimir-Polder range (a0 /α  R  ~c/L, where L is the Lamb shift energy), and the very long or Lamb shift range (R  ~c/L). We also study the Dirac-δ perturbation potential acting on the metastable excited states in the context of hyperfine splitting. The |2P1/2 i levels, which are displaced from the reference 2S-levels just by the Lamb shift, make the study of hyperfine resolved 2S-2S system very interesting. Each S and P state have a hyperfine singlet and a triplet. Thus, there are 8-hyperfine states per hydrogen atom and 8 × 8 = 64 states in the two atom system. The Hamiltonian matrix of the quasi-degenerate 2S-2S system is thus a (64 × 64)-matrix. Our treatment, which profits from adjacency graphs, allows us to do the hyperfineresolved calculation. We examine the evolution of the energy levels in the hyperfine subspaces. We notice that there is a possibility of level crossings in higher dimensional quantum mechanical systems, which is a breakdown of the non-crossing theorem. For higher excited reference states, we match the scattering amplitude and effective Hamiltonian of the system. In the Lamb-shift range, we find an oscillatory term whose magnitude falls off as R−2 and dominates the Wick-rotated term, which otherwise has a retarded Casimir-Polder type of interaction.

iv ACKNOWLEDGMENTS

I have been supported by many people to achieve the success that I have today. It is a great pleasure to acknowledge all of them. First and foremost, I would like to express my sincere gratitude to my dissertation advisor, Dr. Ulrich D. Jentschura, for bringing me into the very interesting field of research. His constant motivation, encouragement, and support is the key to my success. I would forever remain indebted to him. I would like to thank Dr. Vincent Debierre for his immeasurable help. I enjoyed all the scientific and non-scientific discussions with him. I am very grateful to all the members of my family for their everlasting love, support, and understanding. I do not have any word to express my acknowledgment to my late mother Sushila Adhikari, who used to wait for my phone call every evening just to hear that it was a good day for me. I would like to acknowledge Dr. George Wadill and Dr. Jerry Peacher for their help. I also acknowledge Pamela Crabtree and Janice Gragus for their administrative support. I acknowledge the financial support provided by National Science Foundation with grant PHY-1403973 through the principal investigator Dr. Ulrich D. Jentschura. I would also like to extend my acknowledgment to the dissertation evaluation committee, Dr. Don H. Madison, Dr. Michael Schulz, Dr. Daniel Fischer, and Dr. Peter J. Mohr.

v TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii SECTION 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

ORGANIZATION OF THE DISSERTATION . . . . . . . . . . . . .

2

2 DERIVATION OF LONG-RANGE INTERACTIONS . . . . . . . . . . . . . . . . . . . . . .

4

2.1

ORIENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

DERIVATION OF THE vdW AND CP ENERGIES

. . . . . . . . .

4

2.2.1

Derivation Based on Expansion of Electrostatic Interaction . .

4

2.2.2

Derivation Using Time-Ordered Perturbation Theory . . . . .

9

2.3

CHIBISOV’S APPROACH . . . . . . . . . . . . . . . . . . . . . . . .

27

2.4

ASYMPTOTIC REGIMES . . . . . . . . . . . . . . . . . . . . . . .

34

2.5

LONG-RANGE TAILS IN THE vdW INTERACTION . . . . . . . .

41

2.5.1

S -matrix in the Interaction Picture . . . . . . . . . . . . . . .

42

2.5.2

Interaction Energy for nS-1S Systems . . . . . . . . . . . . .

47

2.5.3

Close-Range Limit, a0  R  a0 /α . . . . . . . . . . . . . . .

53

2.5.4

Intermediate Range, a0 /α  R  ~c/L . . . . . . . . . . . . .

54

2.5.5

Very Long-Range Limit, ~c/L  R . . . . . . . . . . . . . . .

54

vi 3 MATRIX ELEMENTS OF THE PROPAGATOR. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1

STURMIAN DECOMPOSITION OF THE GREEN FUNCTION . .

55

3.2

ENERGY ARGUMENT OF THE GREEN FUNCTION . . . . . . .

56

3.3

ANGULAR ALGEBRA (CLEBSCH-GORDAN COEFFICIENTS) . .

58

3.4

1S, 2S, 3S, 4S, AND 5S MATRIX ELEMENTS . . . . . . . . . . . .

61

3.4.1

1S Matrix Element . . . . . . . . . . . . . . . . . . . . . . . .

62

3.4.2

2S Matrix Element . . . . . . . . . . . . . . . . . . . . . . . .

67

3.4.3

3S, 4S, and 5S Matrix Elements . . . . . . . . . . . . . . . .

69

4 DIRAC-DELTA PERTURBATION OF THE vdW ENERGY . . . . . . . . . . . . . . . 73 4.1

HYPERFINE HAMILTONIAN AND DIRAC-DELTA POTENTIAL

73

4.2

WAVE FUNCTION PERTURBATION . . . . . . . . . . . . . . . . .

76

4.3

CALCULATION OF THE DIRAC-DELTA PERTURBATION TO EvdW 81

5 LONG-RANGE INTERACTION IN THE 1S-1S SYSTEM . . . . . . . . . . . . . . . . . 86 5.1

CALCULATION OF C6 (1S; 1S) IN THE vdW RANGE . . . . . . .

86

5.2

CALCULATION OF C7 (1S; 1S) in the LAMB SHIFT RANGE . . .

87

5.3

CALCULATION OF THE 1S-1S DIRAC-δ PERTURBATION EvdW

88

5.4

5.3.1

δD6ψ (1S; 1S) Coefficient . . . . . . . . . . . . . . . . . . . . .

89

5.3.2

δD6E (1S; 1S) Coefficient . . . . . . . . . . . . . . . . . . . . .

95

CALCULATION OF δC7 (1S; 1S) IN THE LAMB SHIRT RANGE .

96

6 LONG-RANGE INTERACTION IN THE 2S-1S SYSTEM . . . . . . . . . . . . . . . . . 99 6.1

2S-1S SYSTEM IN THE vdW RANGE . . . . . . . . . . . . . . . .

99

6.1.1

Calculation of the 2S-1S Direct vdW Coefficient . . . . . . . .

99

6.1.2

Calculation of the 2S-1S vdW Mixing Coefficient . . . . . . . 108

6.2

2S-1S SYSTEM IN THE INTERMEDIATE RANGE . . . . . . . . . 114

6.3

2S-1S SYSTEM IN THE LAMB SHIFT RANGE . . . . . . . . . . . 117

vii 6.4

2S-1S-DIRAC-δ PERTURBATION TO EvdW . . . . . . . . . . . . . 121 6.4.1

δD6E (2S; 1S) Coefficient . . . . . . . . . . . . . . . . . . . . . 122

6.4.2

δD6ψ (2S; 1S) Coefficient . . . . . . . . . . . . . . . . . . . . . 124

6.5

2S-1S-DIRAC-δ MIXING PERTURBATION TO EvdW . . . . . . . . 137

6.6

DIRAC-δ INTERACTION FOR 2S-1S SYSTEM IN THE CP RANGE142

6.7

6.6.1

Wave Function Contribution . . . . . . . . . . . . . . . . . . . 142

6.6.2

Energy Contribution . . . . . . . . . . . . . . . . . . . . . . . 144

δE2S,1S (R) IN THE LAMB SHIFT RANGE . . . . . . . . . . . . . . 153

7 HYPERFINE-RESOLVED 2S-2S SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1

ORIENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.2

CONSERVED QUANTITY . . . . . . . . . . . . . . . . . . . . . . . 158

7.3

HYPERFINE-RESOLVED BASIS STATES . . . . . . . . . . . . . . 161

7.4

MATRIX ELEMENTS OF ELECTRONIC POSITION OPERATORS 168

7.5

SCALING PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . 170

7.6

GRAPH THEORY (ADJACENCY GRAPH) . . . . . . . . . . . . . 172

7.7

HAMILTONIAN MATRICES IN THE HYPERFINE SUBSPACES . 173 7.7.1

Manifold Fz = +2 . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.7.2

Manifold Fz = +1 . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.7.3

Manifold Fz = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.7.4

Manifold Fz = −1 . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.7.5

Manifold Fz = −2 . . . . . . . . . . . . . . . . . . . . . . . . . 206

7.8

REPUDIATION OF NON-CROSSING RULE . . . . . . . . . . . . . 209

7.9

HYPERFINE SHIFT IN SPECIFIC SPECTATOR STATES . . . . . 211 7.9.1

Manifold Fz = +1 . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.9.2

Manifold Fz = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.9.3

Manifold Fz = −1 . . . . . . . . . . . . . . . . . . . . . . . . . 223

viii 8 LONG-RANGE INTERACTION IN nS-1S SYSTEMS . . . . . . . . . . . . . . . . . . . . . 227 8.1

8.2

DIRECT INTERACTION ENERGY IN THE vdW RANGE . . . . . 227 8.1.1

3S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

8.1.2

4S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.1.3

5S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

MIXING INTERACTION ENERGY IN THE vdW RANGE . . . . . 234 8.2.1

3S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

8.2.2

4S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

8.2.3

5S-1S System . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

8.3

DIRECT INTERACTION ENERGY IN THE CP RANGE . . . . . . 239

8.4

MIXING INTERACTION ENERGY IN THE CP RANGE . . . . . . 247

8.5

OSCILLATORY TAILS IN THE DIRECT TERM IN THE LAMB SHIFT RANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

8.6

OSCILLATORY TAILS IN THE MIXING TERM IN THE LAMB SHIFT RANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

9 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 APPENDICES A DISCRETE GROUND STATE POLARIZABILITY . . . . . . . . . . . . . . . . . . . . . . . . 265 B MAGIC WAVELENGTHS OF nS-1S SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

ix LIST OF FIGURES Figure

Page

2.1

The vdW interaction of two neutral hydrogen atoms A and B. . . . .

6

2.2

Diagram showing the CP interaction between two atoms A and B. The ρ and σ lines are the virtual states associated with the atom A and the atom B. The k1 is the magnitude of the momentum of the photon to the left, and the k2 is the magnitude of the momentum of the photon to the right of the line. . . . . . . . . . . . . . . . . . . .

13

The contours to compute integrals in Eq. (2.70). We close the contour in the upper half plane to evaluate the integral containing the exponential factor eix . As the pole x = −x1 align along the real axis, the integral has a value 12 (2πi) times the residue at the pole. The contour is closed in the lower half plane to calculate the integral containing e−ix . In such a case, the integral has a value 12 (−2πi) times the residue at the pole enclosed by the contour. The negative sign is because the contour is negatively oriented. . . . . . . . . . . . . . . . . . . . . . .

23

Figure showing a numerical model for the interaction energy as a function of interatomic distance in three different range. The interaction energy shows 1/R7 asymptotic for R  a0 /α. . . . . . . . . . . . . .

39

Figure showing a numerical model for the interaction energy as a function of interatomic distance in three different range. In the presence of quasi-degenerate states, the 1/R6 range extends much farther out up to ~c/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

The figure shows an integration contour in the complex ω-plane when we carry out the Wick rotation. In the Wick rotation, the ω ∈ (0, ∞) axis is rotated by 900 in a counter clockwise direction to an imaginary E axis. The counter picks up only the poles at ω = − m,A + i. Thus, ~ the contribution of the integration is 2πi times the sum of residues at the poles enclosed by the contour. . . . . . . . . . . . . . . . . . . . .

49

2.3

2.4

2.5

2.6

6.1

Energy levels of the hydrogen atom for n=1 and n=2. L2 and F2 stand for the Lamb shift energy and the fine structure respectively. The Dirac fine structure lowers the ground state energy and resolves the degeneracy corresponding to the first excited state. The degenerate 2S1/2 and 2P1/2 level is a low-lying energy level than 2P3/2 [1]. The degeneracy of the 2S1/2 and 2P1/2 levels is resolved by the Lamb shift, which is in the order of α5 [2; 3]. . . . . . . . . . . . . . . . . . . . . . 100

x 6.2

Asymptotics of the modification of the interaction energy due to the energy type correction in all three ranges. The interaction energy follows the 1/R6 power law in the vdW range, and the 1/R7 power law in the Lamb shift range. However, it follows the peculiar 1/R5 power law in the CP range. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.1

Fine and hyperfine levels of the hydrogen atom for n=1, 2. Here, L and F represent the Lamb shift and fine structure, F stands for the hyperfine quantum number and Fz indicates the z-component of the hyperfine quantum number, where z-axis is the axis of quantization. The numerical values presented in this figure are taken from Refs. [4; 5; 6; 7; 8; 9]. The spacing between the levels is not well scaled. In other words, some closed levels are also spaced widely for better visibility.

162

7.2

An adjacency graph of the matrix A(Fz =+2) . The first diagonal entry, i.e., first vertex is adjacent to the fourth diagonal entry, i.e., fourth vertex and vice versa. The second diagonal element, i.e., the second vertex is adjacent to the third diagonal element, i.e., third vertex and vice versa. However, the two pieces of the graph do not share any edges between the vertices. . . . . . . . . . . . . . . . . . . . . . . . . 176

7.3

Energy levels as a function of interatomic separation R in the Fz = +2 hyperfine manifold. The horizontal axis which represents the interatomic distance is expressed in the unit of Bohr’s radius, a0 , and the vertical axis, which is the energy divided by the plank constant, is in hertz. The energy levels in the subspace (I) deviate heavily from their unperturbed values 21 H and 32 H + L for R < 500a0 . The doubly degenerate energy level L + H splits up into two levels, which repel each other as the interatomic distance decreases. . . . . . . . . . . . . . . . 179

7.4

An adjacency graph of the matrix A(Fz =+1) . The graph for A(Fz =+1) is (I) (II) disconnected having two components G(Fz =+1) and G(Fz =+1) which do not share any edges between the vertices. . . . . . . . . . . . . . . . . 182

7.5

Evolution of the energy levels as a function of interatomic separation R in the subspace (I) of the Fz = +1 hyperfine manifold. For infinitely long interatomic distance, we observe three distinct energy levels same as in the unperturbed case. However, for small interatomic separation, the energy levels split and deviate from the unperturbed energies and become separate and readable. . . . . . . . . . . . . . . . . . . . . . . 186

xi 7.6

Evolution of the energy levels as a function of interatomic separation R in the subspace (II) of the Fz = +1 hyperfine manifold. For infinitely long interatomic distance, we observe four distinct energy levels same as in the unperturbed case. However, for small interatomic separation, the energy levels split and deviate from the unperturbed energies and become distinct and readable. . . . . . . . . . . . . . . . . . . . . . . 188

7.7

An adjacency graph of the matrix A(Fz =0) . The graph for A(Fz =0) has (I) (II) two disconnected components G(Fz =0) and G(Fz =0) which do not share any edges between the vertices. . . . . . . . . . . . . . . . . . . . . . 191

7.8

Evolution of the energy levels as a function of interatomic separation R in the subspace (I) of Fz = 0 hyperfine manifold. The energy levels are asymptotic for large interatomic separation. Although at the large separation, there are six unperturbed energy levels, the degeneracy is removed in small separation and hence, the energy levels spread widely. The small figure inserted on the right top of the main figure is the magnified version of a small portion as indicated in the figure. The figure shows several level crossings. . . . . . . . . . . . . . . . . . 194

7.9

Evolution of the energy levels as a function of interatomic separation R in the subspace (II) of Fz = 0 hyperfine manifold. The vertical axis is the energy divided by the plank constant, and the horizontal axis is the interatomic distance in the unit of Bohr’s radius a0 . The energy levels are asymptotic for large interatomic separation. Although at the large separation, there are six unperturbed energy levels, the degeneracy is removed in small separation and hence, the energy levels spread widely. We observe two level crossings for small atomic separation. The arrow, ‘ ↑ 0 , shows the location of crossings. . . . . . . . . . . . . . . . . . . 199

7.10 An adjacency graph of the matrix A(Fz =−1) . The graph for A(Fz =−1) is (II) (I) disconnected having two components G(Fz =−1) and G(Fz =−1) which do not share any edges between the vertices. . . . . . . . . . . . . . . . . 201 7.11 Energy levels as a function of interatomic separation R in the subspace (I) of the Fz = −1 hyperfine manifold. For infinitely long interatomic separation, there are three distinct energy levels, as expected from the (I) unperturbed energy values of the Hamiltonian matrix, H(Fz =−1) , given by Eq. (7.131). However, for small interatomic separation, the energy levels split. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

xii 7.12 Energy levels as a function of interatomic separation R in the subspace (II) of the Fz = −1 hyperfine manifold. For infinitely long interatomic separation, there are four distinct energy levels, as expected from the (II) unperturbed energy values of the Hamiltonian matrix, H(Fz =−1) , given by Eq. (7.142). However, for small interatomic separation, the energy levels split and deviate from the unperturbed values. . . . . . . . . . 207 7.13 Energy levels as a function of interatomic separation R in the Fz = −2 hyperfine manifold. For large interatomic separation, there are three distinct energy levels. However, for small interatomic separation, the degenerate energy level L + H splits into two, and the level repulsion occurs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.1

Distance dependent direct-type interaction energy in the 3S-1S system in the CP range. The vertical axis is an absolute value of the interaction energy divided by the Plank constant. We have used the logarithmic scale on the vertical axis. The horizontal axis is the interatomic separation in units of Bohr’s radius. The arrow indicates the minimum distance where the pole term and the Wick-rotated term are equal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

8.2

Distance dependent direct-type interaction energy in the 4S-1S system in the CP range. The vertical axis is an absolute value of the interaction energy divided by the Plank constant. We have used the logarithmic scale on the vertical axis. The horizontal axis is the interatomic separation in units of Bohr’s radius. The arrow indicates the point where the pole term becomes comparable to the Wick-rotated term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.3

Distance dependent direct-type interaction energy in the 5S-1S system in the CP range. The vertical axis is an absolute value of the interaction energy divided by the Plank constant. We have used the logarithmic scale on the vertical axis. The horizontal axis is the interatomic separation in units of Bohr’s radius. The arrow indicates the point where the pole term becomes comparable to the Wick-rotated term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8.4

Distance dependent direct-type interaction energy in the 3S-1S system in the very long range. This is a semi-log plot. The vertical axis is an absolute value of the interaction energy divided by the Plank constant. We have used the logarithmic scale on the vertical axis. The pole-type contribution approaches to −∞ upon the change of sign of the pole term contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

xiii LIST OF TABLES

Table

Page (+)

7.1

The energy differences between the symmetric superposition ∆EII (−) and the antisymmetric superposition ∆EII in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2S1/2 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.2

The energy differences between the symmetric superposition ∆EI (−) and the antisymmetric superposition ∆EI in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2P1/2 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.3

The energy differences between the symmetric superposition ∆EI (−) and the antisymmetric superposition ∆EI in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2S1/2 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.4

The energy differences between the symmetric superposition ∆EII (−) and the antisymmetric superposition ∆EII in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2P1/2 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

(+)

(+)

(+)

1. INTRODUCTION

1.1. BACKGROUND The motion of electrons in their orbitals around the atomic nucleus makes an atom polarized to some extent. If two atoms or molecules are brought near each to other, then quantum fluctuations mutually induce dipole moments. The weak interaction which links the dipoles is the so-called van dar Waals (vdW) interaction. Study of the vdW interaction is now popular not only among the physicists but also among the vast majority of researchers from biochemistry, the pharmaceutical industry, nanotechnology, chemistry, biology, etc. For a biochemist, it is the vdW force which determines the interaction of enzymes with biomolecules [10]. For a pharmacist, the binding nature of a drug molecule to the target molecule is determined by the vdW force [11]. In nanoscience, by virtue of origin, the interaction between polarizable nano structures which have wavelike charge density fluctuation is the vdW interaction [12]. The vdW interaction between two atomic states is proportional to R−6 , where R is the interatomic separation. The vdW interaction is a weak interaction, however, if two interacting objects have a significant number of such interactions, the net vdW interaction of the system can be significantly strong [13]. In 1948, the Dutch physicist H. Casimir found that two perfectly conducting parallel plates placed in a vacuum attract each other [14]. This force of attraction is related to the vdW interaction in the retardation regime [15]. In the same year, Casimir and Polder showed that if the distance between the atoms is much larger than the distance related to the retardation time, the interaction potential will be proportional to R−1 times the potential in the non-retardation regime. Thus in the dispersive retardation regime, the van der Waals interaction changes the power law

2 from R−6 to R−7 . This power law modification has been verified experimentally by D. Tabor and R. H. S. Winterton in 1968 [16]. The Russian physicist E. M. Lifshitz developed a more general theory of vdW interactions about ten years after Casimir and Polder proposed Casimir−Polder (CP) forces [17]. In 1997, S. K. Lamoreaux of Los Alamos National Laboratory measured the Casimir force between a plate and a spherical lens with good accuracy [18]. The Casimir effect received, even more, attention of the scientific world when U. Mohideen and A. Roy of the University of California measured the Casimir force between a plate and a sphere even more accurately in 1998 [19]. Recent experimental work includes measurement of the Casimir force between parallel metallic surfaces of silicon cantilever coated with chromium in the 500 −3000 nm range [20], measurement of the Casimir force between dissimilar metals [21], and the Casimir force measurements in a sphere-plate configuration [22].

1.2. ORGANIZATION OF THE DISSERTATION This dissertation provides a detailed analysis of the long-range interaction between two electrically neutral hydrogen atoms. Based on the interatomic distances and nature of the state of the atoms of the system, we apply three different approaches to study the long-range interaction. Every approach has its pros and cons. The first approach is to make a Taylor series expansion of the electrostatic interaction. This approach is valid in a short range regime. However, it does not talk anything about the retardation effect. The other approach is a calculation based on a fourthorder time-ordered perturbation theory. This approach is valid for a wide range of interatomic distances ranging from a0 to ∞, but it suffers from a limitation that both the interacting atoms must be in the ground state. If an atom interacting with the ground state atom is in the excited reference states we match the effective perturbative Hamiltonian with the scattering matrix amplitude.

3 This dissertation is organized as follows. In Section 2, we discuss the basic mathematical formulation. We present derivations based on the expansion of electrostatic interaction and time-ordered perturbation theory. We will realize that the interatomic distance has to be distinguished into three regimes. The last subsection of Section 2 focuses on the long-range tails to the vdW interaction. This subsection shows how an oscillatory dependence of the interaction energy naturally arises due to the presence of quasi-degenerate states. In Section 3, we introduce the Sturmian decomposition of the Green function and determine direct and mixing matrix elements for the first few nS-states of hydrogen. Section 4 highlights what is a Dirac-δ perturbation of the vdW energy, why we care it, and how we determine it. Section 5 is devoted only to the 1S-1S system. We calculate the vdW coefficient for the 1S-1S system. We also evaluate the δ-modification to the vdW interaction energy for the 1S-1S system. In Section 6, we extend our study to the 2S-1S system. In the 2S-1S system, an atom in the ground state now interacts with the other atom in the n = 2 excited states. This causes many complications. We will see how important a role the quasi-degenerate levels play in the interaction energy. We also study the modification of the interaction energy due to the δ-type potential. We make use of our model parameters to verify that our expressions of the interaction energy in the three different regimes are optimal. Section 7 is all about the hyperfine-resolved 2S-2S system. We make use of an applied graph theory to solve the Hamiltonian matrix of the 2S-2S system. We extend our analysis to the vdW energy to the nS-1S system, for 3 ≤ n ≤ 5, in Section 8. Conclusions are drawn in Section 9. Appendix A is about discrete part of ground state static polarizability. We show that the contribution of continuum wave functions to the ground state static polarizability can not be neglected. Appendix B contains an analysis of the magic wavelengths to the nS-1S systems for 2 ≤ n ≤ 6.

4 2. DERIVATION OF LONG-RANGE INTERACTIONS

2.1. ORIENTATION Whenever I look into the internet for some quotes, my eyes pause for a moment on the following quote of a famous physicist Galileo Galilei, “The laws of nature are written by the hand of God in the language of mathematics”. This quote speaks the importance of mathematical formulation in any scientific work very loud and clear. We devote this Section to develop some mathematical formulations which we later use to calculate many quantities in this project.

2.2. DERIVATION OF THE vdW AND CP ENERGIES In what follows, we present a detailed derivation of the vdW and the CP interaction energies. We here discuss two approaches to deduce interaction energies, namely, derivation based on an expansion of electrostatic interaction and derivation based on a non-relativistic quantum electrodynamics using time-ordered perturbation theory. 2.2.1. Derivation Based on Expansion of Electrostatic Interaction. ~ A and ρ~a are the position Let us consider two neutral hydrogen atoms A and B. Let R ~ B and ρ~b are the vectors of the nucleus and the electron of the atom A. Similarly, R position vectors of the nucleus and the electron of the atom B as shown in Figure 2.1. The Hamiltonian of the system can be written as

ˆ =H ˆA + H ˆB + H ˆ AB , H

(2.1)

5 ˆ A and H ˆ B are Hamiltonians of the atoms A and B respectively, which read where H e2 1 p~a2 ˆ − HA = ~ A| 2m 4π0 |~ ρa − R

e2 1 p~b2 ˆ − and HB = , ~ B| 2m 4π0 |~ ρb − R

(2.2)

ˆ AB represents where p~a and p~b are momenta of the atoms A and B respectively. The H the perturbation Hamiltonian of the system. Let us first consider the electrostatic interaction between the atoms A and B.

Velec = −

1 1 1 e2 e2 e2 − + + ~ A | 4π0 |~ ~ B | 4π0 |R ~A − R ~ B| 4π0 |~ ρa − R ρb − R

1 e2 1 e2 1 e2 − − . ~ B | 4π0 |~ ~ A| 4π0 |~ ρa − ρ~b | 4π0 |~ ρa − R ρb − R

(2.3)

The first and the second terms on the right-hand side of Eq. (2.3) are the electrostatic potentials of atoms A and B respectively. Thus, the remaining terms can be treated as the perturbation on the electrostatic interaction. With this, the perturbation ˆ AB can be written as: Hamiltonian H ˆ AB H

e2 =− 4π0

( −

1 ~A − R ~ B| |R

−

1

~ A − ρ~b + R ~B + R ~A − R ~ B| |~ ρa − R ) 1 1 + . ~A + R ~A − R ~ B | |~ ~B + R ~B − R ~ A| |~ ρa − R ρb − R

+ (2.4)

~A −R ~ B = R, ~ ρ~a − R ~ A = ~r(A) , and ρ~b − R ~B = For the sake of simplicity, let us denote R ~r(B) . We have,

ˆ AB H

e2 =− 4π0

(

) 1 1 1 1 − − + + . ~ ~ ~ ~ |R| |~r(A) − ~r(B) + R| |~r(A) + R| |~r(B) − R|

(2.5)

The distance between the proton of an atom and its electron is much smaller than ~ and |~r(B) |  |R|. ~ This allows us the distance between two protons i.e. |~r(A) |  |R| to expand Eq. (2.5) into a series. The Taylor series expansion of Eq. (2.5) about ~r(A)

6

~ra A ρ~a

~ R ~rb

~A R

ρ~b

B

~B R

Figure 2.1: The vdW interaction of two neutral hydrogen atoms A and B.

and/or ~r(B) , to second order, is given by 2

(

X X 1 1 ~ −1 − + (r(A) − r(B) )i νi (R) (r(A) − r(B) )i ~ ~ 2 ij |R| |R| i X (A) (A) X (A) ~ ~ +1 ~ + 1 − ri rj νij (R) ri νi (R) × (r(A) − r(B) )j νij (R) ~ 2 |R| ij i ) X (B) X (B) (B) 1 ~ +1 ~ , + + ri νi (R) ri rj νij (R) (2.6) ~ 2 |R| i ij

ˆ AB ≈ − e H 4π0

−

where ~ = − Ri νi (R) R3

2 ~ = 3Ri Rj − δij R and νij (R) R5

(2.7)

7 correspond to the dipole and the quadrupole contributions of the interaction poten~ as tial. We can rewrite νij (R) ~ = − βij νij (R) R3

such that

βij = δij −

3Ri Rj . R2

(2.8)

After some algebra, Eq. (2.6) leads to (A) (B) e2 X ri rj ˆ HAB ≈ βij . 4π0 ij R3

(2.9)

The first order energy shift for a pair of hydrogen atoms in their ground state is given by (a) (b) ˆ (a) (b) ∆E (1) = hψ100 ψ100 |H AB |ψ100 ψ100 i.

(2.10)

Due to the configurational symmetry of the ground state of hydrogen atoms, we have,

~ A i = h~rb − R ~ B i = 0. h~ra − R

(2.11)

Consequently, the first order energy shift is zero, i.e., ∆E (1) = 0. The first non-vanishing energy shift comes from the second order correction. To second order in perturbation, the energy shift is

∆E

(2)

(A) (B) X hψ (A) ψ (B) |H ˆ AB |ψ (A) ψ (B) ihψ A) ψ (B) |H ˆ AB |ψ100 ψ100 i 100 100 n`m n`m n`m n`m = A B A B E0 − En + E0 − En n6=1

e4 2 2 ~A − R ~ B |6 (4π0 ) |R X X hψ (A) |xi |ψ (A) ihψ (B) |xj |ψ (B) ihψ (A) |xi |ψ (A) ihψ (B) |xj |ψ (B) i

=− ×

100

n6=1 i,j

n`m

n`m E0A

−

100 EnA +

100 E0B −

n`m B En

n`m

100

. (2.12)

8 This is in the form

∆E (2) = −

C , ~ ~ B |6 |RA − R

(2.13)

where C is the vdW coefficient and given by

C=

(A) (A) (B) (B) (A) (A) (B) (B) 2 e4 X X hψ100 |xi |ψn`m ihψn`m |xj |ψ100 ihψ100 |xi |ψn`m ihψn`m |xj |ψ100 i . (4π0 )2 n6=1 i,j E0A − EnA + E0B − EnB

(2.14) Making use of the identity X δ ij X hψ100 |xi |ψn`m ihψn`m |xj |ψ100 i = hψ100 |xs |ψn`m ihψn`m |xs |ψ100 i, 3 s i,j

(2.15)

which is valid for any S state, the vdW coefficient given in Eq. (2.14) yields 2 e4 X X X δ ij δ ij C= (4π0 )2 n6=1 s k 3 3 (A)

(A)

(A)

(A)

(B)

(B)

(B)

(B)

hψ |xs |ψn`m ihψn`m |xs |ψ100 ihψ100 |xk |ψn`m ihψn`m |xk |ψ100 i × 100 E0A − EnA + E0B − EnB (B) (B) (B) (B) (A) (A) (A) (A) 2 e4 X X δ ii hψ100 |xs |ψn`m ihψn`m |xs |ψ100 ihψ100 |xk |ψn`m ihψn`m |xk |ψ100 i = (4π0 )2 n6=1 s,k 9 E0A − EnA + E0B − EnB (A) (A) (B) (B) 2 e4 X X X |hψ100 |xs |ψn`m i|2 |hψ100 |xk |ψn`m i|2 . = 3 (4π0 )2 n6=1 s k E0A − EnA + E0B − EnB

(2.16)

With the following integral identity 2ab π

Z 0

∞

Z dx ab ∞ dx = 2 2 2 2 (a + x )(b + x ) π −∞ (i|a| + x)(−i|a| + x)(i|b| + x)(−i|b| + x)   ab 1 1 1 1 = 2πi + π 2|a|i |b|2 − |a|2 2|b|i |a|2 − |b|2 h i |a| |b| = sgn(a)sgn(b) − |a|2 − |b|2 |a|2 − |b|2

9 =

sgn(a)sgn(b) , |a| + |b|

(2.17)

where sgn(a) and sgn(b) are sign functions, Eq. (2.16) can be written as Z 4e4 ~ X X ∞ dω (E0A − EnA )(E0B − EnB ) C= 3π(4π0 )2 n6=1 j,k 0 ×

|hψ100 |xj |ψn`m i|2 |hψn`m |xk |ψ100 i|2 .  B A B 2 2 A 2 2 (E0 − En ) + (~ω) (E0 − En ) + (~ω)

(2.18)

The sign function sgn(a) of the real number a is +1 if a > 0 and −1 if a < 0 and similarly for sgn(b). The quantity 2e2 X A |hψ100 |xj |ψn`m i|2  = α1S (iω, A), (E0 − EnA )  3 j (E A − E A )2 + (~ω)2 0

(2.19)

n

is the dipole polarizability of the hydrogen atom A in its ground state. We have a similar expression for the atom B. The polarizability of an atom measures the distortion of the charge distribution of the atom in the presence of the electric field. An atom having high polarizability has large fluctuations in local charge distribution [23]. Thus, from Eq. (2.18), the vdW coefficient can be expressed as 3~ C= π(4π0 )2

Z

∞

dω α1S (iω, A) α1S (iω, B).

(2.20)

0

The important feature of expression (2.20) is the dependence of vdW coefficient on the polarizabilities of the atoms. 2.2.2. Derivation Using Time-Ordered Perturbation Theory. The unperturbed Hamiltonian for a system of two neutral hydrogen atoms A and B is 2 2 ˆ 0 = p~a + V (~ra ) + p~b + V (~rb ) + H ˆF , H 2ma 2mb

(2.21)

10 where (ma , mb ), (~ra , ~rb ), and (~pa , p~b ) are masses, coordinates and momenta of electrons in atoms A and B. And

ˆF = H

2 Z X

d3 k k a†λ (~k) aλ (~k)

(2.22)

λ=1

is the electromagnetic field Hamiltonian where a†λ and aλ are the usual creation and ~ A |  |~ra −R ~ B| annihilation operators. If the two atoms are far enough such that |~ra −R ~ B |  |~rb − R ~ A |, where R ~ A and R ~ B are the coordinates of the nuclei, the and |~rb − R potential V (~ra ) and V (~rb ) can be approximated as

V (~ra ) = −

e2 1 , ~ A| 4π0 |~ra − R

and

V (~rb ) = −

1 e2 . ~ B| 4π0 |~rb − R

(2.23)

Substituting V (~ra ) and V (~rb ) in Eq. (2.21), the unperturbed Hamiltonian of the system yields 2 2 p~2 1 e2 1 ˆ 0 = p~a − e ˆF . H + b − +H ~ A | 2mb 4π0 |~rb − R ~ B| 2ma 4π0 |~ra − R

(2.24)

ˆ A , the sum of The first two terms stand for the Schr¨odinger-Coulomb Hamiltonian H ˆ B , and the third and the fourth terms are the Schr¨odinger-Coulomb Hamiltonian H ˆ F is the field Hamiltonian. Along with the dipole approximation, the interaction the H Hamiltonian in the so-called length gauges formulation of quantum electrodynamics (QED) reads

ˆ AB = −e ~ra · E( ~ R ~ A ) − e ~rb · E( ~ R ~ B ), H

(2.25)

~ R ~ A ) and E( ~ R ~ B ) are the electric field operators given as where E(

~ R ~ A) = E(

r

2

~c X 0 λ=1

Z

d3 k (2π)3/2

r

h i k ~ ~ ~ ~ ˆλ (~k) i aλ (~k)eik·RA − ia†λ (~k)e−ik.RA , 2

(2.26)

11 and

~ R ~ B) = E(

r

2

~c X 0 λ=1

Z

d3 k (2π)3/2

r

h i k ~B ~B † ~ −i~k·R i~k·R ~ ~ ˆλ (k) i aλ (k)e − iaλ (k)e . 2

(2.27)

In terms of the creation, annihilation operators of the field, the interaction Hamiltonian becomes

ˆ AB = − H

r



2 Z d3 k ~c X e 0 λ=1 (2π)3/2

~ ~ + i aλ (~k)ˆλ (~k)eik·RB

r

  k  ~ ~ ~ ~ i aλ (~k)ˆλ (~k)eik·RA − ia†λ (~k)ˆλ (~k)e−ik.RA · ~ra 2   ~ ~ † ~ −i k· R B · ~rb . (2.28) − iaλ (k)ˆλ (~k)e

We take the state with zero photons |φ0 i as the reference state and calculate the perturbation effect of the interaction Hamiltonian. The creation operator increases the number of particles in a given state |ni by one and brings the system to the state |n + 1i while the annihilation operator decreases the number of particles by one and brings the system into the new state |n − 1i. In the first order perturbation, the annihilation operator kills the state as our system is already in the ground state and the creation operators bring the system into its first excited state. The orthonormality condition,

hn|mi = δnm =

   1, if n = m,

(2.29)

  0, if n 6= m, requires that the first order contribution should vanish. In the similar fashion, no odd order perturbation contributes to the interaction energy. The second order terms are the self-energy terms and do not contribute to the CP interaction. Thus, we look

12 into the fourth order perturbation which reads

ˆ AB ∆E (4) = hφ0 |H

1 ˆ 0 )0 (E0 − H

The prime in the operator

ˆ AB H

1 ˆ 0 )0 (E0 −H

1 ˆ 0 )0 (E0 − H

ˆ AB H

1 ˆ 0 )0 (E0 − H

ˆ AB |φ0 i. H

(2.30)

indicates that the reference state is excluded from

the spectral decomposition of the operator. Consider a CP interaction between two atoms A and B involving two virtual photons. A time-ordered sequence results four different types of intermediate states [24; 25], namely, (1) Both atoms are in ground states, and two virtual photons are present, (2) Only one atom is in the excited state, and only one virtual photon is exchanged, (3) Both atoms are excited state, but no photon is present, and (4) Both atoms are excited state, and two photons are present. Thus, the electrons and photons can couple in 4 × 3 × 2 × 1 = 12 distinct ways. Figure 2.2 represents all these 12 possible interactions. Let us first investigate the first diagram of the Figure (2.2). There are four factors which give contributions to the interaction energy, namely, emission of ~k2 at RB , emission of ~k1 at RB , absorption of ~k2 at RA , and absorption of ~k1 at RA . The corresponding fourth order energy shift reads "  3 X X k k d k ~ ~ 2 1 2 (4) e4 hφ | − i aλ1 (~k1 )e−ik1 ·RA + ∆E1 = 0 3 (2π) λ ,λ ρ,σ 2 2 1 2    ~ ~ ~ ~ ~ ~ ia†λ1 (~k1 )eik1 ·RA ˆλ1 (~k1 ) · ~ra |ρihρ| + i aλ1 (~k1 )eik1 ·RB − ia†λ1 (~k1 )e−ik1 ·RB #"   ~A ~A † ~ −i~k2 ·R i~k2 ·R ~ ~ × ˆλ1 (k1 ) · ~rb |σihσ| − i aλ2 (k2 )e + iaλ2 (k2 )e ˆλ2 (~k2 ) · ~ra +

2

d3 k1 (2π)3



~c 0



  ~B ~B † ~ i~k2 ·R −i~k2 ·R ~ ~ i aλ2 (k2 )e − iaλ2 (k2 )e ˆλ2 (k2 ) · ~rb |φ0 i

Z

Z

1 1 1 . E1S,a − Eρ − ~ck1 −~ck1 − ~ck2 E1S,b − Eσ − ~ck2

(2.31)

13 ρ

ρ

ρ

k2 k1

k2

k1 σ

k2

k1 σ

(I)

ρ

k1

k2

k1 k2

σ (V)

σ (VI)

ρ

σ (VII)

ρ

k1

k2

k1

σ (X)

σ (IX)

σ (IV) ρ

k1

k2

σ (VIII)

ρ

k1

k2

k2

ρ

k1

k2

k1

σ (III)

(II)

ρ

ρ

ρ

k2 σ (XI)

k1

k2 σ (XII)

Figure 2.2: Diagram showing the CP interaction between two atoms A and B. The ρ and σ lines are the virtual states associated with the atom A and the atom B. The k1 is the magnitude of the momentum of the photon to the left, and the k2 is the magnitude of the momentum of the photon to the right of the line.

The annihilation operator kills the ground state however the creation operator can raise a particular state to the corresponding excited state. Thus, Eq. (2.31) yields (4) ∆E1

 =

~c 0

2

4

e

Z

d3 k1 (2π)3

Z

d3 k2 X X k1 k2 ~ ~ (i)hφ1S,a |ˆλ1 (~k1 ) · ~ra |ρieik1 ·RA (−i) 3 (2π) λ ,λ ρ,σ 4 1

~B −i~k1 ·R

× hφ1S,b |ˆλ1 (~k1 ) · ~rb |σie ~

~

× e−ik2 ·RB

2

~ ~ (i)hρ|ˆλ2 (~k2 ) · ~ra |φ1S,a ieik2 ·RA (−i)hσ|ˆλ2 (~k2 ) · ~rb |φ1S,b i

1 1 1 . E1S,a − Eρ − ~ck1 −~ck1 − ~ck2 E1S,b − Eσ − ~ck2

(2.32)

14 The polarization vectors ˆλi (~ki ), i = 1, 2 satisfy ˆλi (~k) · ˆλj (~k) = δλi λj ,

(2.33)

~k · ˆλ (~k) = 0, i

(2.34)

2 X λi =1

ˆpλi (k~r )ˆqλi (k~r ) = δ pq −

krp krq . ~k 2

(2.35)

r

Thus, the contribution to the interaction energy from the first diagram reads (4) ∆E1

2 Z 3 Z 3   d k1 ~c d k2 k1 k2 k1m k1r 4 mr = e δ − 2 0 (2π)3 (2π)3 4 k1   n s k k ~ ~ ~ ~ × δ ns − 2 2 2 ei(k1 +k2 )·(RA −RB ) k2 X hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i × . (E1S,a − Eρ − ~ck1 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ) ρ,σ 

(2.36)

In the diagram (II), the four factors which contribute to the interaction energy are emission of ~k2 at RB , emission of ~k2 at RA , absorption of ~k1 at RB , and absorption of ~k1 at RA . This leads to the following contributions to the interaction energy (4) ∆E2

2 Z 3 Z 3   ~c k1m k1r d k1 d k2 k1 k2 4 mr = e δ − 2 0 (2π)3 (2π)3 4 k1   n s k k ~ ~ ~ ~ × δ ns − 2 2 2 ei(k1 +k2 )·(RA −RB ) k2 X hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i × . (E1S,a − Eρ − ~ck2 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ) ρ,σ 

(2.37)

If we denote a propagator denominator by D, then for diagrams (I) and (II), we have,

DI = (E1S,a − Eρ − ~ck1 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ),

(2.38)

DII = (E1S,a − Eρ − ~ck2 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ).

(2.39)

15 The diagram (III) in Figure (2.2) involves the emission of ~k2 at RB , the emission of ~k1 at RA and the excitation of both atoms. Thus the propagator denominator (D ) III corresponding to the diagram (III) reads

DIII = (E1S,a − Eρ − ~ck1 )(E1S,a − Eσ + E1S,b − Eρ )(E1S,b − Eσ − ~ck2 ).

(2.40)

The corresponding energy shift is (4) ∆E3



~c 0

2

4

Z

d3 k1 (2π)3

Z

d3 k2 k1 k2 (2π)3 4



mr

kmkr − 121 k1



= δ e   k2n k2s i(~k1 +~k2 )·(R~ A −R~ B ) ns e × δ − 2 k2 X hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i × . (2.41) (E1S,a − Eρ − ~ck1 )(E1S,a − Eσ + E1S,b − Eρ )(E1S,b − Eσ − ~ck2 ) ρ,σ

Let us investigate diagram (IV) in Figure (2.2). The contribution to the interaction energy from the diagram (IV) reads (4) ∆E4



~c 0

2

4

Z

d3 k1 (2π)3

Z

d3 k2 k1 k2 (2π)3 4

  k1m k1r mr δ − 2 k1

e =   k2n k2s i(−~k1 +~k2 )·(R~ A −R~ B ) ns × δ − 2 e k2 X hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i × . (2.42) (E − E − ~ck )(E − E + E − E )(E − E − ~ck ) 1S,b σ 1 1S,a ρ 1S,b σ 1S,b σ 2 ρ,σ

We change the sign of k1 under the integral sign to get the same exponential for all diagrams in Figure (2.2). Diagrams (V) and (VI) involve the emission of photon, excitation of both atoms and the absorption of photons. The propagator denominators for the diagrams (V) and (VI) are

DV = (E1S,a − Eρ − ~ck2 )(E1S,a − Eρ + E1S,b − Eσ − ~ck1 − ~ck2 ) × (E1S,b − Eσ − ~ck2 ),

(2.43)

16 DV I = (E1S,b − Eσ − ~ck1 )(E1S,a − Eρ + E1S,b − Eσ − ~ck1 − ~ck2 ) × (E1S,b − Eσ − ~ck2 ).

(2.44)

Under the exchange of the ρ line and the σ line, the six diagrams (I) to (VI) correspond to the other six diagrams (VII) to (XII). The corresponding propagator denominators for the diagrams (VII) to (XII) are

DV II = (E1S,b − Eσ − ~ck1 )(−~ck1 − ~ck2 )(E1S,a − Eρ − ~ck2 ),

(2.45)

DV III = (E1S,b − Eσ − ~ck2 )(−~ck1 − ~ck2 )(E1S,a − Eρ − ~ck2 ),

(2.46)

DIX = (E1S,b − Eσ − ~ck1 )(E1S,a − Eρ + E1S,b − Eσ )(E1S,a − Eρ − ~ck2 ),

(2.47)

DX = (E1S,a − Eρ − ~ck1 )(E1S,a − Eρ + E1S,b − Eσ )(E1S,a − Eρ − ~ck2 ),

(2.48)

DXI = (E1S,b − Eσ − ~ck2 )(E1S,a − Eσ + E1S,b − Eρ − ~ck1 − ~ck2 ) × (E1S,a − Eρ − ~ck2 ),

(2.49)

DXII = (E1S,a − Eρ − ~ck1 )(E1S,a − Eρ + E1S,b − Eσ − ~ck1 − ~ck2 ) × (E1S,a − Eρ − ~ck2 ).

(2.50)

The net fourth order energy shift is the sum of the contributions of all the 12 diagrams. Explicitly,

∆E

(4)

 ~c 2 Z d3 k Z d3 k k k  k1m k1r  ns k2n k2s  i(~k1 +~k2 )·(R~ A −R~ B ) 1 2 1 2 mr e4 δ − δ − 2 e = 0 (2π)3 (2π)3 4 k12 k2 XII X X × hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i Dj−1 . (2.51) ρ,σ

j=I

The propagator denominators corresponding to the diagrams (I), (II) and (IV) are the denominators of the summands of Eqs. (2.36), (2.37) and (2.42). Namely,

DI = (E1S,a − Eρ − ~ck1 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ),

(2.52)

17 DII = (E1S,a − Eρ − ~ck2 )(−~ck1 − ~ck2 )(E1S,b − Eσ − ~ck2 ),

(2.53)

DIV = (E1S,b − Eσ − ~ck1 )(E1S,a − Eρ + E1S,b − Eσ )(E1S,b − Eσ − ~ck2 ).

(2.54)

while remaining D0 s are given by Eq. (2.40) and Eqs. (2.43) to (2.50). We first compute the sum of D’s. For simplicity, let us denote E1S,a − Eρ = Eaρ and E1S,b − Eσ = Ebσ . Let us now group, simplify, and then assemble all the terms as below. −(Eaρ − ~ck1 ) − (Ebσ − ~ck2 ) (Eaρ + Ebσ )(Eaρ − ~ck1 )(Ebσ − ~ck2 )(~ck1 + ~ck2 ) −1 = + (Eaρ + Ebσ )(Ebσ − ~ck2 )(~ck1 + ~ck2 ) −1 , (2.55a) (Eaρ + Ebσ )(Eaρ − ~ck1 )(~ck1 + ~ck2 ) 1 = (Ebσ − ~ck1 )(Eaρ + Ebσ )(Ebσ − ~ck2 )   1 1 1 1 = − , (2.55b) (Eaρ + Ebσ ) (Ebσ − ~ck1 ) (Ebσ − ~ck2 ) (~ck1 − ~ck2 ) 1 + = (Ebσ − ~ck2 )(−~ck1 − ~ck2 )(Eaρ − ~ck2 ) 1 (Ebσ − ~ck1 )(Eaρ + Ebσ )(Eaρ − ~ck2 ) −(Eaρ − ~ck2 ) − (Ebσ − ~ck1 ) = (Eaρ + Ebσ )(Eaρ − ~ck2 )(Ebσ − ~ck1 )(~ck1 + ~ck2 ) −1 = + (Eaρ + Ebσ )(Ebσ − ~ck1 )(~ck1 + ~ck2 ) −1 , (2.55c) (Eaρ + Ebσ )(Eaρ − ~ck2 )(~ck1 + ~ck2 ) 1 = (Eaρ − ~ck1 )(Eaρ + Ebσ )(Eaρ − ~ck2 )   1 1 1 1 = − , (2.55d) (Eaρ + Ebσ ) (Eaρ − ~ck1 ) (Eaρ − ~ck2 ) (~ck1 − ~ck2 ) 1 = (Eaρ − ~ck2 )(Eaρ + Ebσ − ~ck1 − ~ck2 )(Ebσ − ~ck2 ) 1 + (Ebσ − ~ck1 )(Eaρ + Ebσ − ~ck1 − ~ck2 )(Ebσ − ~ck2 ) 1 = , (2.55e) (Ebσ − ~ck1 )(Ebσ − ~ck2 )(Eaρ − ~ck2 )

−1 DI−1 + DIII =

−1 DIV

−1 DV−1II + DIX

DX−1

DV−1 + DV−1I

18 1 (Ebσ − ~ck2 )(Ebσ + Eaρ − ~ck1 − ~ck2 )(Eaρ − ~ck2 ) 1 + (Eaρ − ~ck1 )(Eaρ + Ebσ − ~ck1 − ~ck2 )(Eaρ − ~ck2 ) 1 = . (Eaρ − ~ck1 )(Eaρ − ~ck2 )(Ebσ − ~ck2 )

−1 −1 DXI + DXII =

(2.55f)

−1 −1 −1 The six propagator denominators DI−1 , DIII , DIV , DV−1II , DIX , and DX−1 can be grouped

as  1 1 − DI + DIII + DIV + DV II + DIX + DX = (Eaρ + Ebσ ) (Eaρ − ~ck1 )    1 1 1 1 − − × (~ck1 + ~ck2 ) (~ck1 − ~ck2 ) (Ebσ − ~ck1 ) (~ck1 + ~ck2 )     1 1 1 1 + − + − (~ck1 − ~ck2 ) (Eaρ + Ebσ ) (Eaρ − ~ck2 ) (~ck1 + ~ck2 )    1 1 1 1 − + (~ck1 − ~ck2 ) (Ebσ − ~ck2 ) (~ck1 + ~ck2 ) (~ck1 − ~ck2 )    1 1 1 1 1 =− + − (Eaρ + Ebσ ) Eaρ − ~ck1 Ebσ − ~ck1 ~ck1 + ~ck2 ~ck1 − ~ck2    1 1 1 1 1 − . (2.56) + + (Eaρ + Ebσ ) Eaρ − ~ck2 Ebσ − ~ck2 ~ck1 + ~ck2 ~ck1 − ~ck2 −1

−1

−1

−1

−1

−1

Interchanging k1 and k2 in the second term of Eq. (2.56) we get 2 (Eaρ + Ebσ )    1 1 1 1 × + − . (2.57) (Eaρ − ~ck1 ) (Ebσ − ~ck1 ) (~ck1 + ~ck2 ) (~ck1 − ~ck2 )

−1 −1 −1 DI−1 +DIII + DIV + DV−1II + DIX + DX−1 = −

−1 Let us group the three D’s DII , DV−1 and DV−1I .

−1

DII

  1 1 1 − + + DV +DV I = (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (Ebσ − ~ck1 )  1 1 1 = − + − (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (Ebσ − ~ck1 )  1 1 + (~ck1 − ~ck2 ) (~ck1 − ~ck2 ) −1

−1

19   1 1 1 =− + + (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (~ck1 − ~ck2 ) 1 . (2.58) (Eaρ − ~ck2 )(Ebσ − ~ck1 )(~ck1 − ~ck2 ) −1 −1 The three D’s, namely DV−1III , DXI and DXII can be grouped as

−1 + (Ebσ − ~ck2 )(~ck1 + ~ck2 )(Eaρ − ~ck2 ) 1 (Eaρ − ~ck1 )(Eaρ − ~ck2 )(Ebσ − ~ck2 )   1 1 1 − + = (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (Eaρ − ~ck1 )  1 −1 = + (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 )  1 1 1 − + (~ck1 − ~ck2 ) (~ck1 − ~ck2 ) (Eaρ − ~ck1 )   1 1 1 =− + + (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (~ck1 − ~ck2 ) 1 . (2.59) (Ebσ − ~ck2 )(Eaρ − ~ck1 )(~ck1 − ~ck2 )

−1 −1 DV−1III + DXI + DXII =

Adding Eqs. (2.58) and (2.59) we get −2 −1 −1 −1 DII + DV−1 + DV−1I + DV−1III + DXI + DXII = (Ebσ − ~ck2 )(Eaρ − ~ck2 )   1 1 1 × + + ~ck1 + ~ck2 ~ck1 − ~ck2 (Eaρ − ~ck2 )(Ebσ − ~ck1 )(~ck1 − ~ck2 ) 1 + . (2.60) (Ebσ − ~ck2 )(Eaρ − ~ck1 )(~ck1 − ~ck2 ) Under the interchange of k1 and k2 , the second term in the right hand side of the Eq. (2.60) is equal in magnitude but opposite in sign with the third term. Thus,

−1 −1 −1 DII + DV−1 + DV−1I + DV−1III + DXI + DXII   2 1 1 + . (2.61) =− (Ebσ − ~ck2 )(Eaρ − ~ck2 ) (~ck1 + ~ck2 ) (~ck1 − ~ck2 )

20 Interchanging k1 and k2 in Eq. (2.61) and adding the result to Eq. (2.57), the sum of the reciprocal of all the twelve propagator denominators evaluates to XII X

  1 1 1 2 + − =− (Eaρ + Ebσ ) Eaρ − ~ck1 Ebσ − ~ck1 ~ck1 + ~ck2 j=I    1 2 1 1 − − ~ck1 − ~ck2 (Ebσ − ~ck1 )(Eaρ − ~ck1 ) ~ck1 + ~ck2 ~ck1 − ~ck2   −4(Eaρ + Ebσ − ~ck1 ) 1 1 = − . (2.62) (Eaρ + Ebσ )(Ebσ − ~ck1 )(Eaρ − ~ck1 ) ~ck1 + ~ck2 ~ck1 − ~ck2 Dj−1

The fourth order energy shift is now simplified to 2

Z 3 d k2 k1 k2  mr k1m k1r  ns k2n k2s  d3 k1 ∆E = − δ − 2 δ − 2 e (2π)3 (2π)3 4 k1 k2 XII X X ~ A −R ~B) i(~k1 +~k2 )·(R m n r s ×e hφ1S,a |x |ρihρ|x |φ1S,a ihφ1S,b |x |σihσ|x |φ1S,b i Dj−1 

(4)

~c 0

4

Z

ρ,σ

 =−

~c 0 ~

2

e4

~

Z

~

d3 k1 (2π)3 X ~

Z

× ei(k1 +k2 ).·(RA −RB )

j=I

d3 k2 k1 k2  mr k1m k1r  ns k2n k2s  δ − 2 δ − 2 (2π)3 4 k1 k2

hφ1S,a |xm |ρihρ|xn |φ1S,a ihφ1S,b |xr |σihσ|xs |φ1S,b i

ρ,σ

  4(Eaρ + Ebσ − ~ck1 ) 1 1 × − . (Eaρ + Ebσ )(Ebσ − ~ck1 )(Eaρ − ~ck1 ) ~ck1 + ~ck2 ~ck1 − ~ck2

(2.63)

Let us use the identity (2.15) in Eq. (2.63). We get, 2 Z Z  ~c e4 k1m k1r  ns k2n k2s  3 3 mn rs mr ∆E = − d k d k k k δ δ δ − δ − 2 1 2 1 2 0 576π 6 k12 k2 X X X ~ ~ ~ × ei(k1 +k2 )·R hφ1S,a |xj |ρihρ|xj |φ1S,a ihφ1S,b |x` |σihσ|x` |φ1S,b i (4)



ρ,σ

j

`

  (Eaρ + Ebσ − ~ck1 ) 1 1 × − , (Eaρ + Ebσ )(Ebσ − ~ck1 )(Eaρ − ~ck1 ) ~ck1 + ~ck2 ~ck1 − ~ck2

(2.64)

21 ~ A −R ~ B = R. ~ Now substitute where, R

R

d3 k =

R∞ 0

k 2 dk

Rπ 0

sin θdθ

R 2π 0

dφ in Eq. (2.64)

and carry out the integration of the angular part first. Note that Z

π

2π

Z sinθdθ 0

0

Z π   k1m k1r  ik1 Rcosθ k1m k1r  i~k1 ·R~ mr mr sinθdθ δ − 2 e dφ δ − 2 e = 2π k1 k1 0 Z 1   1 ∂ ∂ = 2π δ mr + 2 du eik1 Ru m k1 ∂R ∂Rr −1   1 ∂ ∂ 2 sink1 R mr = 2π δ + 2 m r k1 ∂R ∂R k1 R     m r Rm Rr cosk1 R R R sink1 R mr mr + δ −3 = 4π δ − R2 k1 R R2 (k1 R)2    Rm Rr sink1 R − δ mr − 3 . (2.65) R2 (k1 R)3

With the help of Eq. (2.65), Eq. (2.64) can be re-expressed as  ~c 2 e4 X X hφ |xj |ρihρ|xj |φ ihφ |x` |σihσ|x` |φ i Z ∞ 1S,a 1S,a 1S,b 1S,b dk1 ∆E = − 0 36π 4 ρ,σ j,` (Eaρ + Ebσ ) 0   Z ∞ (Eaρ + Ebσ − ~ck1 ) 1 1 3 3 dk2 k1 k2 × − δ mn δ rs (E − ~ck )(E − ~ck ) (~ck + ~ck ) (~ck − ~ck ) bσ 1 aρ 1 1 2 1 2  0 Rm Rr  cosk1 R  mr Rm Rr  sink1 R Rm Rr  sink1 R  mr mr + δ −3 − δ −3 δ − R2 k1 R R2 (k1 R)2 R2 (k1 R)3   Rn Rs  sink2 R  ns Rn Rs  cosk2 R  ns Rn Rs  sink2 R ns δ − + δ −3 2 − δ −3 2 R2 k2 R R (k2 R)2 R (k2 R)3 ~c e4 X X hφ1S,a |xj |ρihρ|xj |φ1S,a i hφ1S,b |x` |σihσ|x` |φ1S,b i = − 36π 4 20 ρ,σ j,` (Eaρ + Ebσ ) Z ∞ (Eaρ + Ebσ − ~ck1 ) dk1 k13 × Amr (k1 R) δ mn δ rs (E − ~ck )(E − ~ck ) bσ 1 aρ 1 0Z ∞  Z ∞ ns ns A (k R) 2 3 A (k2 R) 3 − dk2 k2 , (2.66) × dk2 k2 (k1 + k2 ) (k1 − k2 ) 0 0 (4)

where

ns

A (x) =



δ

ns

 Rn Rs  sinx  ns Rn Rs  cosx  ns Rn Rs  sinx − + δ −3 2 − δ −3 2 . R2 x R x2 R x3 (2.67)

22 The Ans (x) is an even function of x. Thus Eq. (2.66) can be equivalently written as below extending the integration limit from −∞ to +∞:

∆E

(4)

Z ~c e4 X X hφ1S,a |xj |ρihρ|xj |φ1S,a i hφ1S,b |x` |σihσ|x` |φ1S,b i ∞ =− dk1 k13 2 4 36π 0 ρ,σ j,` (Eaρ + Ebσ ) 0 Z ∞ Ans (k2 R) (Eaρ + Ebσ − ~ck1 ) dk2 k23 Amr (k1 R) δ mn δ rs . (2.68) × (Ebσ − ~ck1 )(Eaρ − ~ck1 ) (k1 + k2 ) −∞

Let us evaluate the k2 -integral first. The k2 -integral has a pole of order one at k2 = −k1 . Let k2 R = x and k1 R = x1 . Then the k2 -integral can be written as Z

∞

Z ∞ Ans (k2 R) 1 Ans (x) = 3 dx x3 (k1 + k2 ) R −∞ (x1 + x) (Z )   Z ∞ ∞ 2 2 n s ix −ix x x 1 R R e e dx dx − = 3 δ ns − R R2 x + x1 2i x + x1 2i −∞ −∞ ) (   Z ∞ Z ∞ ix −ix n s x x e e 1 R R dx dx + + 3 δ ns − 3 2 R R x + x1 2 x + x1 2 −∞ −∞ ) (   Z ∞ Z ∞ ix −ix n s 1 1 1 e e R R dx + 3 δ ns − 3 2 dx − . (2.69) R R x + x1 2i x + x1 2i −∞ −∞

dk2 k23

−∞

All the first integrals under curly brackets in Eq. (2.69) diverge as x → ∞ while all the second integrals in the same equation diverge as x → −∞. Let us introduce a convergence factor e−η|x| to make our integrands divergence-free. We have, Z

∞

dk2 k23

−∞

1 = 3 R +

1 R3

+

1 R3

Ans (k2 R) (k1 + k2 )

(Z )   Z ∞ ∞ n s 2 ix−η|x| 2 −ix−η|x| R R x e x e δ ns − lim dx − dx η→0 R2 x + x1 2i x + x1 2i −∞ −∞ ( )   Z ∞ Z ∞ n s ix−η|x| −ix−η|x| R R x e x e δ ns − 3 2 lim dx + dx η→0 R x + x1 2 x + x1 2 −∞ −∞ ( )   Z ∞ Z ∞ n s ix−η|x| −ix−η|x| 1 R R e 1 e dx δ ns − 3 2 lim − dx . η→0 R x + x1 2i x + x1 2i −∞ −∞ (2.70)

23 We evaluate integrals in Eq. (2.70) with the help of contours as shown in Figure 2.3 and perform the integration. We finally take the limit η → 0 which yields )  ( −ix1 ix1 n s 1 e 1 e A (k2 R) R R 1 = 3 δ ns − (2πi)x21 − (−2πi) x21 dk2 k23 2 (k1 + k2 ) R R 2 2i 2 2i −∞ ) (   1 e−ix1 1 eix1 Rn Rs 1 + 3 δ ns − 3 2 (2πi)(−x1 ) + (−2πi) (−x1 ) R R 2 2 2 2 ( )   1 Rn Rs 1 e−ix1 1 eix1 + 3 δ ns − 3 2 (2πi) − (−2πi) R R 2 2i 2 2i     n s n s 1 R R R R 1 πx21 cosx1 − 3 δ ns − 3 2 πx1 sinx1 = 3 δ ns − 2 R R R R   1 Rn Rs + 3 δ ns − 3 2 π cosx1 . (2.71) R R

Z

∞

ns

e−ix

eix

Figure 2.3: The contours to compute integrals in Eq. (2.70). We close the contour in the upper half plane to evaluate the integral containing the exponential factor eix . As the pole x = −x1 align along the real axis, the integral has a value 21 (2πi) times the residue at the pole. The contour is closed in the lower half plane to calculate the integral containing e−ix . In such a case, the integral has a value 12 (−2πi) times the residue at the pole enclosed by the contour. The negative sign is because the contour is negatively oriented. Here we have used the following well known Euler’s formula,

e±iθ = cos θ ± i sin θ ,

(2.72)

24 to express complex exponential functions into trigonometric functions. Rearranging Eq. (2.71), we have ∞

ns 3 A (k2 R) dk2 k2 (k1 + k2 ) −∞

Z

  π x31 Rn Rs cosx1 ns = 3 δ − R R2 x1    n s sinx1 cosx1 R R ns . + − δ −3 2 R x21 x31

(2.73)

Replacing the assumed variable x1 by its value x1 = k1 R, we get Z

∞

dk2 k23

−∞

"  Rn Rs cosk1 R A(k2 R) 3 ns =πk1 δ − (k1 + k2 ) R2 k1 R    # n s R R sink1 R cosk1 R + . − δ ns − 3 2 R (k1 R)2 (k1 R)3

(2.74)

Substituting the value of the integral (2.74) in Eq. (2.68), we have

∆E (4) = −

~c e4 X X X hφ1S,a |xj |ρihρ|xj |φ1S,a i hφ1S,b |x` |σihσ|x` |φ1S,b i 36π 4 20 ρ,σ j ` (Eaρ + Ebσ ) ∞

(Eaρ + Ebσ − ~ck1 ) δ mn δ rs (E − ~ck )(E − ~ck ) bσ 1 aρ 1 0     m r m r R R sink R R R cosk1 R sink1 R 1 mr mr × δ − + δ −3 − R2 k1 R R2 (k1 R)2 (k1 R)3    Rn Rs  cosk1 R  ns Rn Rs  sink1 R cosk1 R ns × δ − + δ −3 2 + . (2.75) R2 k1 R R (k1 R)2 (k1 R)3 Z

×

dk1 πk16

The Kronecker delta satisfies the following relations:

δ ij δ jk = δ ik ,

δ ii = 3,

(2.76)

as a result, Eq. (2.75) gets simplified to

∆E (4) = −

~c e4 X X X hφ1S,a |xj |ρihρ|xj |φ1S,a i hφ1S,b |x` |σihσ|x` |φ1S,b i 36π 4 20 ρ,σ j ` (Eaρ + Ebσ )

25 ∞

 (Eaρ + Ebσ − ~ck1 ) sin2k1 R sin2 k1 R sin2k1 R × − 2 − + (Ebσ − ~ck1 )(Eaρ − ~ck1 ) (k1 R)2 (k1 R)3 (k1 R)4 0  sin2k1 R sin2k1 R cos2 k1 R sin2 k1 R cos2 k1 R sin2k1 R − −3 +3 −6 +6 2 (k1 R)3 (k1 R)4 (k1 R)4 (k1 R)6 (k1 R)5 (k1 R)5 ~c e4 X X X hφ1S,a |xj |ρihρ|xj |φ1S,a ihφ1S,b |x` |σihσ|x` |φ1S,b i =− 36π 4 20 ρ,σ j ` (Eaρ + Ebσ )  Z ∞ (Eaρ + Ebσ − ~ck1 ) sin2k1 R sin2 k1 R cos2 k1 R 6 dk1 πk1 × − 2 + 2 (Ebσ − ~ck1 )(Eaρ − ~ck1 ) (k1 R)2 (k1 R)3 (k1 R)3 0  sin2k1 R cos2 k1 R sin2 k1 R sin2k1 R . (2.77) −5 − 6 + 6 +3 4 5 5 (k1 R) (k1 R) (k1 R) (k1 R)6 Z

dk1 πk16

We make use of the following identities

coskR =

eikR + e−ikR 2

and

sinkR =

eikR − e−ikR , 2i

(2.78)

such that the Eq. (2.77) can be expressed in the following form

∆E

(4)

~c e4 X X X hφ1S,a |xj |ρihρ|xj |φ1S,a i hφ1S,b |x` |σihσ|x` |φ1S,b i =− 36π 3 20 ρ,σ j ` (Eaρ + Ebσ ) " Z 1 ∞ (Eaρ + Ebσ − ~ck1 )e2ik1 R × dk1 k16 2i 0 (Ebσ − ~ck1 )(Eaρ − ~ck1 )   2i 5 6i 3 1 + − − + × (k1 R)2 (k1 R)3 (k1 R)4 (k1 R)5 (k1 R)6 Z (Eaρ + Ebσ − ~ck1 )e−2ik1 R 1 ∞ dk1 k16 − 2i 0 (Ebσ − ~ck1 )(Eaρ − ~ck1 )  # 1 2i 5 6i 3 . (2.79) × − − + + (k1 R)2 (k1 R)3 (k1 R)4 (k1 R)5 (k1 R)6

Now, let us introduce a new variable u which has values u = i k1 c in the first hi k1 -integral and u = −i k1 c in the second k1 -integral inside the square bracket in Eq. (2.79). Consequently, We get

∆E

(4)

XXXZ ∞ ~c e4 hφ1S,a |xj |ρihρ|xj |φ1S,a i 4 =− 5 du u E Ebσ aρ 2 + ~2 u2 ) c 36π 3 R2 20 ρ,σ j ` 0 (Eaρ

26 2c 5c2 hφ1S,b |x` |σihσ|x` |φ1S,b i −2uR/c h 6c3 3c4 i e 1 + + + + 2 (Ebσ + ~2 u2 )) uR (uR)2 (uR)3 (uR)4 Z ∞ ~ u4 e−2uR/c =− 4 du α(a, iu) α(b, iu) πc (4π0 )2 0 R2 i h 5c2 2c 6c2 3c4 + (2.80) 1+ + + uR (uR)2 (uR)3 (uR)4 Z ∞ ~ ω 4 e−2ωR/c =− 4 dω α(a, iω) α(b, iω) πc (4π0 )2 0 R2  c 2  c 3  c 4 i h  c  +5 +6 +3 , (2.81) 1+2 ωR ωR ωR ωR ×

where the quantities α(a, iu) and α(b, iu) are the dynamic polarizabilities of atoms A and B respectively and given by

α(a, iu) =

2e2 X X |ρihρ| xj |φ1S,a i Eaρ hφ1S,a |xj 2 3 ρ j (Eaρ + ~2 u2 )

2e2 X Eaρ = xj |φ1S,a i, hφ1S,a |xj 2 2 2 3 j (Eaρ + ~ u ) 2e2 X X |σihσ| Ebσ hφ1S,b |x` 2 x` |φ1S,b i 3 σ ` (Ebσ + ~2 u2 ) 2e2 X X Ebσ = hφ1S,b |x` 2 x` |φ1S,b i. 3 σ ` (Ebσ + ~2 u2 )

(2.82)

α(b, iu) =

(2.83)

Making use of u = ±ik1 c = ±ω, the dynamic polarizabilities can be rewritten as

α(a, iω) =

2e2 X H − E1S,a hφ1S,a |xj xj |φ1S,a i 3 j (H − E1S,a )2 + ~2 ω 2

=

e2 X X 1 hφ1S,a |xj xj |φ1S,a i, 3 ± j (H − E1S,a ) ± i~ω

(2.84)

α(b, iω) =

e2 X X 1 hφ1S,b |x` x` |φ1S,b i. 3 ± ` (H − E1S,b ) ± i~ω

(2.85)

27 It is obvious to state from Eqs. (2.84) and (2.85) that the dynamic polarizability of an atom is the sum of two matrix elements of the Sch¨odinger-Coulomb propagator.

α(a, ω) = P (a, ω) + P (a, −ω)

(2.86)

where,

P (a, ±ω) =

3 e2 X 1 xj |na i. hna |xj 3 j=1 En − Ea ± ~ω − i

(2.87)

For large ω, the polarizability shows ω −2 behavior. The expression for the CP interaction between any two atoms A and B given by the Eq. (2.81) is valid for any interatomic separation R provided their wave functions do not overlap.

2.3. CHIBISOV’S APPROACH Let us consider two neutral hydrogen atoms in which one atom is in the ground state 1S and the other in the excited nS state. Consider the case in which the wave function of the system is in the state of quantum entanglement. The wave function of the system can be expressed as

Ψ = K1n |1SiA |nSiB + Kn1 |nSiA |1SiB .

(2.88)

ˆ is Total Hamiltonian of the system H

ˆ =H ˆA + H ˆB + H ˆ AB = H ˆS + H ˆ AB , H

(2.89)

28 ˆ S stands for the Schr¨odinger Hamiltonian. H ˆ A and H ˆ B are Hamiltonians of where H the atom A and the atom B which are respectively e2 p~a2 ˆ HA = − ~ A| 2m 4π0 |~ra − R

e2 p~b2 ˆ − and HB = . ~ B| 2m 4π0 |~rb − R

(2.90)

ˆ AB is given by As derived in section 2.2, the interaction Hamiltonian H (A) (B) e2 X ri rj ˆ βij HAB ≈ . 4π0 ij R3

(2.91)

Taking the entangled state |Ψi given by Eq. (2.88) as the eigenstate, the eigenvalue ˆ is equation of the system with the Hamiltonian H   ˆ ˆA + H ˆB + H ˆ AB |Ψi = E|Ψi. H|Ψi = H

(2.92)

The total wave function of the system can be expressed as sum of all possible products

|Ψi =

X

Kpq |pSiA |qSiB ,

(2.93)

pq

where Kpq is the expansion coefficient. In the first order perturbation approximation, the expansion coefficients Kpq are approximated as (0) (1) Kpq = Kpq + Kpq ,

(2.94)

(0)

(1)

where Kpq are the unperturbed coefficients of expansion and the Kpq are the first (1)

order corrections to the expansion coefficients. The first order correction Kpq is given as ˆ

(0) h1SnS|HAB |pqi (0) (0) E1n − Epq

(1) Kpq = K1n

ˆ

(0) hnS1S|HAB |pqi . (0) (0) En1 − Epq

+ Kn1

(2.95)

29 ˆA + H ˆB + H ˆ AB , the eigenvalue In the eigen basis of the sum of the Hamiltonian H Eq. (2.92) can be expressed as 

ˆA + H ˆB + H ˆ AB H

X

Kpq |pqi = E

pq

Or,

X

Kpq

pq

X

X

Kpq |pqi,

pq

ˆ AB |pqi = |rsihrs|H

rs

X

(0) )|pqi. Kpq (E − Epq

(2.96)

pq

Here we have used the completeness relation X

|rsihrs| = 1.

(2.97)

rs

Eq. (2.96) can be re-expressed as )

( X

(0) Kpq (E − Epq )−

X

ˆ AB |pqi |pqi = 0, Krs hrs|H

(2.98)

ˆ AB |pqi = 0. Krs hrs|H

(2.99)

rs

pq

which implies

(0) Kpq (Epq − E) +

X rs

(0)

Using Eq. (2.95) in Eq. (2.99), the two equations with the expansion coefficients K1n (0)

and Kn1 are (0) K1n

(0) E1n

ˆ AB |1SnSi + − E + h1SnS|H

X h1SnS|H ˆ AB |pqihpq|H ˆ AB |1SnSi (0)

(0)

ˆ AB |1SnSi + hnS1S|H

(0)

E1n − Epq

pq6=1n

+Kn1

X hnS1S|H ˆ AB |pqihpq|H ˆ AB |1SnSi (0)

pq6=1n

!

(0)

E1n − Epq

! = 0, (2.100)

30 and (0)

(0) ˆ AB |nS1Si + En1 − E + hnS1S|H

Kn1

X hnS1S|H ˆ AB |pqihpq|H ˆ AB |nS1Si (0)

ˆ AB |nS1Si + h1SnS|H

(0)

En1 − Epq

pq6=n1 (0) +K1n

!

X h1SnS|H ˆ AB |pqihpq|H ˆ AB |nS1Si (0)

(0)

En1 − Epq

pq6=n1

! = 0. (2.101)

The homogeneous linear Eqs. (2.100) and (2.101) can also be written as the homogeneous matrix equation 



(0) K1n







  0  W X    =  ,   (0) Kn1 0 Y Z

(2.102)

where (0)

ˆ AB |1SnSi + W =E1n − E + h1SnS|H

X |h1SnS|H ˆ AB |pqi|2 (0)

pq6=1n

ˆ AB |1SnSi + X =hnS1S|H

(0)

Z

(0)

E1n − Epq

X h1SnS|H ˆ AB |pqihpq|H ˆ AB |nS1Si (0)

pq6=n1 (0) =En1

,

X hnS1S|H ˆ AB |pqihpq|H ˆ AB |1SnSi pq6=1n

ˆ AB |nS1Si + Y =h1SnS|H

(0)

E1n − Epq

ˆ AB |nS1Si + − E + hnS1S|H

(0)

En1 − Epq

X |hnS1S|H ˆ AB |pqi|2 (0)

pq6=n1

(0)

En1 − Epq

(2.103a) ,

(2.103b)

,

(2.103c)

.

(2.103d)

ˆ AB is symmetric with respect to the order of the The interaction Hamiltonian H selection of 1S and nS in the eigenstates |1SnSi and |nS1Si. Namely,

ˆ AB |1SnSi = hnS1S|H ˆ AB |nS1Si. h1SnS|H

(2.104)

31 ˆ AB |1SnSi = hnS1S|H ˆ AB |nS1Si = 0 as the interaction Hamiltonian, Indeed, h1SnS|H as required by the selection rule, does not couple S states. In addition to this, with the interchange of 1 and n, the following two quantities are equal. X |h1SnS|H ˆ AB |pqi|2 (0)

(0)

E1n − Epq

pq6=1n

X |hnS1S|H ˆ AB |pqi|2

=

(0)

(0)

En1 − Epq

pq6=n1

.

(2.105)

As a result, the diagonal elements of the 2 × 2 matrix in (2.102) are equal. And for the same reasons the off-diagonal elements in (2.102) are also equal. Thus, the matrix in (2.102) is a 2 × 2 symmetric Toeplitz matrix [26]. To have non-trivial solutions, (0)

(0)

we require the determinant of the matrix to be zero which implies K1n = ±Kn1 . Provided the determinant of the matrix vanishes, the 2 × 2 matrix in Eq. (2.102) gives

(0)

E1n − E +

X |h1SnS|H ˆ AB |pqi|2 (0)

pq6=1n

=

!2

(0)

E1n − Epq

X hnS1S|H ˆ AB |pqihpq|H ˆ AB |1SnSi (0)

!2

(0)

E1n − Epq

pq6=1n

.

(2.106)

Solving energy E from Eq. (2.106), we get (0)

E =E1n −

X |h1SnS|H ˆ AB |pqi|2 (0)

pq6=1n (0)

=E1n −

(0)

E1n − Epq

±

X hnS1S|H ˆ AB |pqihpq|H ˆ AB |1SnSi (0)

X 2e4 ~A − R ~ B |6 3(4π0 )2 |R pq6=1n

(0)

E1n − Epq

pq6=1n

X |h1S|xi |pihnS|xj |qi|2 (0)

(0)

E1n − Epq ij ! X (hnS|xi |pih1S|xj |qi)∗ (h1S|xi |pihnS|xj |qi) (0)

(2.107)

(0)

E1n − Epq

ij (0)

=E1n − ±

±

X 2e4 ~A − R ~ B |6 3(4π0 )2 |R pq6=1n

X h1S|xr |pihp|xr |1SihnS|xs |qihq|xs |nSi (0)

X (h1S|xr |pihp|xr |nSi) (hnS|xs |qihq|xs |1Si) (0)

rs

(0)

E1n − Epq

(0)

E1n − Epq

rs

! .

(2.108)

32 In the second line of equation (2.107) we have used the following identities X i,j

δ ij X hψ100 |xs |ψn`m ihψn`m |xs |ψ100 i hψ100 |x |ψn`m ihψn`m |x |ψ100 i = 3 s i

j

(2.109)

and

δ ij δ ij = δ ii = 3.

(2.110)

Let us define

D6 (nS; 1S) =

X X h1S|xr |pihp|xr |1SihnS|xs |qihq|xs |nSi 2e4 , (0) (0) 3(4π0 )2 pq6=1n rs E1n − Epq

(2.111)

and

M6 (nS; 1S) =

X X (h1S|xr |pihp|xr |nSi) (hnS|xs |qihq|xs |1Si) 2e4 , (2.112) (0) (0) 3(4π0 )2 pq6=1n rs E1n − Epq

such that (0)

E = E1n −

D6 (nS; 1S) ± M6 (nS; 1S) . ~A − R ~ B |6 |R

(2.113)

By the notations D6 (nS; 1S) and M6 (nS; 1S), we are referring to the direct and the mixing term contributions to the vdW C6 (nS; 1S) coefficient such that

C6 (nS; 1S) = D6 (nS; 1S) ± M6 (nS; 1S).

(2.114)

The ± sign depends on the symmetry of the wave function of the two-atom state. Making use of the standard integral identity (2.17), we can express Eqs. (2.111) and

33 (2.112) in terms of integrals over ω as  X  (0)
4e4 (0) (0) (0) − E E D6 (nS; 1S) = E − E 1 q n p 3(4π0 )2 pq6=1n Z∞ dω 0

X  rs

h1S|xr |pihp|xr |1SihnS|xs |qihq|xs |nSi  , (0) (0) 2 (0) (0) 2 2 2 2 2 (E1 − Ep ) + ~ ω (En − Eq ) + ~ ω

(2.115)

and ∞

 Z X  1 (0) 4e4 1 (0) (0) (0) (0) (0) M6 (nS; 1S) = (E + En ) − Ep (E + En ) − Eq dω 3(4π0 )2 pq6=1n 2 1 2 1 0

×

r

X rs



r

s

s

(h1S|x |pihp|x |nSi) (hnS|x |qihq|x |1Si)   . 2 2 (0) (0) (0) (0) (0) (0) 1 1 2 2 2 2 (E1 + En ) − Ep +~ ω ( 2 (E1 + En ) − Eq +~ ω 2 (2.116)

Identifying
hnS|xs |qihq|xs |nSi 2 2 X X (0)  = αnS (iω), e En − Eq(0)  (0) (0) 3 pq6=1n s (En − Eq )2 + ~2 ω 2

(2.117)

and   2 2 X X 1 (0) (0) (0)  e (E + En ) − Eq 3 pq6=1n s 2 1

hnS|xs |qihq|xs |1Si  2 (0) (0) (0) 1 2 2 ( 2 (E1 + En ) − Eq +~ ω = αnS1S (iω), (2.118)

the direct and the mixing vdW coefficients can be rewritten as Z ∞ 4e4 dω αnS (iω)α1S (iω), D6 (nS; 1S) = 3(4π0 )2 0 Z ∞ 4e4 M6 (nS; 1S) = dω α1SnS (iω)αnS1S (iω). 3(4π0 )2 0

(2.119) (2.120)

34 There are some advantages of using the average energy corresponding to the states of interest as the reference energy. First, there is no state which is degenerate with the reference state. Next, the issue of the contributions arising from the intermediate P -states is resolved. However, it requires calculating the quantum number associated with the reference state. Taking 1S state as one state of interest and nS as another, the energy associated with the reference state is

Eref =

E1 + En α2 mc2 =− . 2 2n2ref

(2.121)

Solving for nref , Eq. (2.121) yields 1 1 1 = + 2. 2 nref 2 2n It is obvious to note that nref = 1 when n = 1 and nref =

(2.122) √ 2 when n = ∞.

Thus the quantum number corresponding to the reference state always lies in the √ range 1 ≤ nref < 2. A downside of this approach is that this is valid only in the short range. Here, by the short range of the interatomic distance, we mean that the interatomic distance must be less than the wavelength corresponding to a typical atomic transition. To put it another way, R must satisfy

a0  R  a0 /α,

(2.123)

where a0 is the Bohr radius and α = 1/137.035 999 139, is the fine-structure constant. This is so-called vdW range.

2.4. ASYMPTOTIC REGIMES To study the interaction between two atoms in S-states, we differentiate three different ranges for the interatomic distance: van der Waals range (a0  R  a0 /α),

35 CP range (a0 /α  R  1/L, where L is lamb shift energy [27]), and Lamb shift range (R  1/L). Equivalently, we will call them the short range, the intermediate range, and the long range of the interatomic distances. In this work, we consider the interaction between two atoms in which one atom sits in the ground state while the other one can be either in the ground state or one of the excited nS-states. The ground state is nondegenerate. However, the excited state has quasi-degenerate neighbors. If an atom is in an excited state, the polarizability of the atom αnS (ω) is the sum

αnS (ω) = α enS (ω) + αnS (ω),

(2.124)

where α enS (ω) is the nondegenerate contribution to the nS polarizability while αnS (ω) represents the contribution of the quasi-degenerate nP levels. The dipole polarizability αnS (ω) is computed by a sum over all states. The degenerate polarizability αnS (ω) is a sum over quasi-degenerate neighbors. Mathematically, 1 e2 X X X X |hnS|xj |nP (m = µ)i|2 . αnS (ω) = 3 ± j µ=−1 EnPj − EnS ± ~ω − i

(2.125)

1 3

nPj = , 2 2

The symbol nPj indicates the total angular quantum number j of the quasi-degenerate P -states which are resonant . The total orbital angular quantum number l has the value 1 for P -state. Thus, the total angular quantum number j, and hence and 23 . The energy difference between the quasi-degenerate     levels with the principal quantum number n, EnP1/2 − EnS and EnP3/2 − EnS are nPj , can have values

1 2

respectively the Lamb shift Ln and the fine structure splitting Fn . Mathematically,

EnP1/2 − EnS1/2 ≡Ln ,

(2.126a)

EnS1/2 − EnP3/2 ≡Fn .

(2.126b)

36 The nondegenerate polarizability α enS (ω) is the sum over all states excluding degenerate levels. It is the measurement of the polarizability due to states having the different principal quantum number. More explicitly, 1 e2 X X X X |hnS|xj |nP (m = µ)i|2 α enS (ω) = . 3 ± j µ=−1 k>n Ek − EnS ± ~ω − i

(2.127)

The sum over k in the non degenerate polarizability indicates that we include all the possible states whose principal quantum number is greater than the reference state. The total interaction energy can be written as the sum (4) e (4) + ∆E (4) + PnS;1S , ∆EnS;1S = ∆E nS;1S nS;1S

(2.128)

e (4) and ∆E (4) where ∆E nS;1S are the nondegenerate and the degenerate contributions nS;1S to the interaction energy respectively. The PnS;1S is the pole term contribution, which arises as the integration contour picks up a number of poles under the Wickrotation. Detailed discussion of the pole term is presented in Sec.2.5. Being the e (4) and ∆E (4) f Wick-rotated contribution, the ∆E nS;1S can be renamed as WnS;1S and nS;1S W nS;1S respectively, such that the total Wick-rotated contribution reads

fnS;1S + W nS;1S , WnS;1S = W

(2.129)

fnS;1S + W nS;1S + PnS;1S = WnS;1S + PnS;1S . ∆EnS;1S =W

(2.130)

which allows us to write (4)

37 fnS;1S and the degenerate W nS;1S contributions The Wick-rotated nondegenerate W are given by Z∞ ~ ω 4 e−2ωR/c dω α(1S, iω) α e (nS, iω) πc4 (4π0 )2 R2 0   c   c 2  c 3  c 4  × 1+2 +5 , +6 +3 ωR ωR ωR ωR Z∞ ~ ω 4 e−2ωR/c =− 4 α(nS, iω) dω α(1S, iω) πc (4π0 )2 R2 0   c   c 2  c 3  c 4  × 1+2 +5 . +6 +3 ωR ωR ωR ωR

fnS;1S = − W

W nS;1S

(2.131a)

(2.131b)

For the short range (a0  R  a0 /α) of interatomic distances, there is no oscillatory suppression in the interaction energy and the first four terms under the hi bracket in both Eqs. (2.131a) and (2.131b) are negligible in comparison to the fifth term. Furthermore, the exponential can be approximated to unity. Thus we can fnS;1S and W nS;1S as approximate the Wick-rotated contributions W Z ∞ ω 4 3 c4 ~ dω α(1S, iω) α e (nS, iω) πc4 (4π0 )2 0 R2 (ωR)4 Z ∞ 3~ =− dω α(1S, iω) α e(nS, iω); a0  R  a0 /α, π(4π0 )2 R6 0 Z ∞ ~ ω 4 3 c4 dω α(1S, iω) ≈− 4 α(nS, iω) πc (4π0 )2 0 R2 (ωR)4 Z ∞ 3~ =− dω α(1S, iω) α(nS, iω); a0  R  a0 /α. π(4π0 )2 R6 0

fnS;1S ≈ − W

W nS;1S

(2.132a)

(2.132b)

Both the nondegenerate and the degenerate contributions to the energy follow the R−6 power law in the short range. Let us examine the behavior of the interaction for very large interatomic distances (R  ~c/L). As the interatomic distance is very large, the exponential term and the negative powers of R vary very fast but not the polarizabilities

38 [28].

Specifically, we can approximate the dynamic polarizabilities of atoms by

their static values. i.e. α(1S, iω) = α(1S, ω = 0), α e(nS, iω) = α e(nS, ω = 0), and α(nS, iω) = α(nS, ω = 0). Consequently, we have

fnS;1S W

W nS;1S

Z∞ ~ ω 4 e−2ωR ≈− 4 α(1S, ω = 0) α e (nS, ω = 0) dω πc (4π0 )2 R2 0   c 2  c 3  c 4   c  +5 × 1+2 +6 +3 ωR ωR ωR ωR 23 ~c α(1S, ω = 0) α e(nS, ω = 0) =− , R  ~c/L. 4π(4π0 )2 R7 Z∞ ω 4 e−2ωR ~ dω ≈− 4 α(1S, ω = 0) α(nS, ω = 0) πc (4π0 )2 R2 0   c   c 2  c 3  c 4  × 1+2 +5 +6 +3 ωR ωR ωR ωR 23 ~c α(1S, ω = 0) α(nS, ω = 0) , R  ~c/L. =− 4π(4π0 )2 R7

(2.133a)

(2.133b)

Hence, in the long range of interatomic distances, both the nondegenerate and the degenerate contributions has R−7 dependence. We recovered the famous CP result. Let us now investigate the interaction energy in the intermediate interatomic distances (a0 /α  R  ~c/F < ~c/L). The transition energies, in the nondegenerate cases, are in the order of the Hartree energy and the polarizabilities due to the nondegenerate states can be approximated by their static values. Thus, we still get a R−7 power law dependence of the interaction energy. To illustrate the analytic considerations of power law behavior of the interaction energy, we consider model integrals. In the nondegenerate case, the model integral can be expressed as Z∞ I(a, b, R) = dω

a b ω 4 e−2ωR (a − i)2 + ω 2 (b − i)2 + ω 2 R2 0   5 6 3 2 × 1+ + + + , ωR (ωR)2 (ωR)3 (ωR)4

(2.134)

39 where a and b are the energy parameters. Let us choose the parameters:

a = 1,

b = 1/4,

and

 = 10−6 .

(2.135)

For small interatomic distance, the curve for a model integral with no approximation (blue curve), matches with a 1/R6 asymptotic (red-dashed) curve while for large interatomic distance, the model curve matches with 1/R7 asymptotic (greendashed) curve (see Figure 2.4). a0 /α ≈ 137.036 a0 is the transition from 1/R6 to 1/R7 asymptotic.

Figure 2.4: Figure showing a numerical model for the interaction energy as a function of interatomic distance in three different range. The interaction energy shows 1/R7 asymptotic for R  a0 /α.

40 In the presence of the quasi-degenerate states, the model integral can be written as Z∞ J(a, b, R) = dω

a (−η) ω 4 e−2ωR (a − i)2 + ω 2 (−η − i)2 + ω 2 R2 0   5 2 6 3 + × 1+ + + , ωR (ωR)2 (ωR)3 (ωR)4

(2.136)

where η is the energy shift of the degenerate levels which represents the Lamb shift or fine structure. One good choice of the numerical values of the parameters are

a = 1,

η = 10−3 ,

and  = 10−6 .

(2.137)

Figure 2.5 shows an exact, and approximate 1/R6 and 1/R7 asymptotic for interaction energy. For small interatomic distance, the curve for a model integral with no approximation (blue curve), matches with a 1/R6 asymptotic (red-dashed) curve while for large interatomic distance, the model curve matches with 1/R7 asymptotic (green-dashed) curve. ~c/L is the transition from 1/R6 to 1/R7 asymptotic. Now, it is time to clarify why we choose R  ~c/L. As the long range of the interatomic distances instead of R  ~c/F. As F ≈ 10L, the interatomic distances ~c/F and ~c/L differ by an order of magnitude. One might argue that there is a window ~c ~c 100a0 . However, the fine structure splitting energy is much larger than them for R > 100a0 . Assuming that fine structure levels are sufficiently apart, we do not take the fine structure splitting Hamiltonian into account. Thus, the total

156 Hamiltonian of the system is the sum of the vdW, the Lamb shift, and the hyperfine splitting Hamiltonians. More explicitly,

H = HLS + HHFS + HvdW ,

(7.1)

where HLS is the Lamb shift, HHFS is the hyperfine splitting, and HvdW is the vdW Hamiltonians. If A and B are the two hydrogen atoms, at the first excited states, interacting with each other, the Lamb shift Hamiltonian is given as

HLS = HLS,A + HLS,B

4 = α2 mc2 3



~ mc

3

 ln

1 α2

 X

δ 3 (~rj ),

(7.2)

j=A,B

where α is the fine-structure constant, m is the mass of an electron, and ~rj is the relative distance of an electron in an atom with respect to its nucleus. The Lamb shift energy ELS shifts the nS1/2 state upwards relative to the Dirac position for the corresponding j = 1/2 level, thereby splitting the nS1/2 and the nP1/2 states, which are otherwise degenerate according to the Dirac theory of the hydrogen atom. It is believed that the origin of the Lamb shift is the interactions of the electron and the quantum vacuum fluctuations of the electromagnetic field within the atom [63]. The HHFS in Eq. (7.1) represents the hyperfine splitting Hamiltonian given by

HHFS = HHFS,A + HHFS,B =

i ~pj · L ~j ~αgp X S ~αgp X 1 h ~ ~ ~ ~ + 3( S · r ˆ )(ˆ r · S ) − S · S ej j j pj ej pj 4mM c j=A,B rj3 2mM c j=A,B rj3 4 X ~ ~ej ) π~α δ 3 (~rj ), + gp (Spj · S (7.3) 3 j=A,B mM c

~ej and S ~pj are the spin angular momenta of the electron and the proton of where S the atoms A or B. M and gp = 5.585694702 are the mass and the g-factor of the ~ j is the orbital angular momentum of the electron. The first term on the proton. L

157 right-hand side of Eq. (7.3) has a zero contribution for S states as l = 0 for S-states and the second term is zero for S-states as 

+  * ~ ~ 3 Sej . Spj ~ ~ nS 3 (Sej . rˆj )(ˆ rj . Spj ) nS = nS nS . rj3 rj

(7.4)

Thus for S-states, the hyperfine splitting Hamiltonian is also the Dirac-δ type as given below: 4 X ~ ~ej ) π~α δ 3 (~rj ) HHFS = gp (Spj · S 3 j=A,B mM c !  3 ~pj S ~ej 4 X m S ~ 2 = gp · αmc πδ 3 (~rj ). 3 j=A,B M ~ ~ mc

(7.5)

(7.6)

HvdW in Eq. (7.1) denotes vdW hamiltonian of the system. Recalling the electrostatic interaction between two hydrogen atoms, as discussed in Section2, we have

ˆ vdW H

(A) (B) e2 X ri rj ≈ , βij 4π0 ij R3

(7.7)

where βij is a second rank tensor given by

βij = δij −

3Ri Rj . R2

(7.8)

Eq. (7.7) can equivalently be written as

ˆ vdW H

e2 ≈ 4π0

~ ~rB · R ~ ~rA · ~rB 3 ~rA · R − R3 R5

! .

(7.9)

158 ~ is along the quantization axis of the Let us assume that the atomic separation R system i.e., z-axis, we obtain

HvdW

  e2 (xA xB + yA yB + zA zB ) 3 (zA zB R2 ) ≈ − 4π0 R3 R5 (xA xB + yA yB − 2zA zB ) = α~c . R3

(7.10)

7.2. CONSERVED QUANTITY The total angular momentum of the system is the sum of the total angular momentum of the atoms A and B.

F~ = F~A + F~B .

(7.11)

The total angular momentum of each atom is defined by the sum

~ +S ~e + S ~p , F~ = L

(7.12)

~ is the orbital angular momentum, S ~ is the spin angular momentum of the where L ~p is the spin angular momentum of the proton. The z-component of electron, and S the total angular momentum is thus given by

Fz = Lz,A + Lz,B + Sez,A + Sez,B + Spz,A + Spz,B .

(7.13)

We are interested in the commutation relation [Fz , H]. Let us first compute the commutator [Lz,A + Lz,B , H]:

[Lz,A + Lz,B , H] = [Lz,A + Lz,B , HLS ] + [Lz,A + Lz,B , HHFS ] + [Lz,A + Lz,B , HvdW ], (7.14)

159 where 4 [Lz,A + Lz,B , HLS ] = α2 mc2 3



~ mc

3

 ln

# " X 1 Lz,A + Lz,B , δ 3 (~rj ) . (7.15) 2 α j=A,B

The spatial distribution of the electron of an electrically neutral hydrogen atom in its S-states is spherically symmetric. The position operator ~r of the electron in such a spherically symmetric distribution commutes with the orbital angular momentum operator of the same electron. This implies

[Lz,A + Lz,B , HLS ] = 0.

(7.16)

Furthermore, " # X 4 π~α 3 ~pj · S ~ej )δ (~rj ) . [Lz,A + Lz,B , HHFS ] = gp Lz,A + Lz,B , (S 3 mM c j=A,B

(7.17)

As explained earlier for commutation relation (7.15), the orbital angular momentum commutes with the position operator. Moreover, the orbital angular momentum and the spin commutes. Thus, we have

[Lz,A + Lz,B , HHFS ] = 0.

Let us now examine the commutator of Lz,A + Lz,B with HvdW :  h i  (xA xB + yA yB − 2zA zB ) Lz,A +Lz,B , HvdW = Lz,A + Lz,B , α~c R3 α~c = 3 [Lz,A + Lz,B , xA xB + yA yB − 2zA zB ] R  α~c  = 3 [Lz,A , xA xB + yA yB ] + [Lz,B , xA xB + yA yB ] R α~c  = 3 [xA Py,A − yA Px,A , xA xB + yA yB ] R

(7.18)

160  + [xB Py,B − yB Px,B , xA xB + yA yB ]  α~c  = 3 xA (−i~yB ) − yA (−i~xB ) − yA (−i~xB ) − xA (−i~yB ) = 0. R

(7.19)

To get the third line of Eq. (7.19), we have used the fact that [Lz , z] = 0. In the fourth line, we have expressed the z-component of the orbital angular momentum in terms of position components and the linear momenta as

Lz = xPy − yPx .

(7.20)

To get the fifth line of Eq. (7.19), we have made use of the following commutation relations:

[ri , ri ] = 0,

[Pi , Pi ] = 0,

[ri , Pj ] = i~δij ,

and

[A, B] = − [B, A] .

(7.21)

The spin angular momentum commutes with the spherically symmetric function of the position operator. Thus, we have

[Sez,A + Sez,B , HLS ] = 0,

(7.22a)

[Sez,A + Sez,B , HHFS ] = 0,

(7.22b)

[Sez,A + Sez,B , HvdW ] = 0,

(7.22c)

[Spz,A + Spz,B , HLS ] = 0,

(7.22d)

[Spz,A + Spz,B , HHFS ] = 0,

(7.22e)

[Spz,A + Spz,B , HvdW ] = 0.

(7.22f)

From Eqs. (7.16), (7.18), (7.19), and (7.22a)-(7.22f), we can conclude that the total angular momentum of the system containing two electrically neutral hydrogen atoms

161 commutes with the total Hamiltonian of the system i.e.

[Fz , H] = 0.

(7.23)

This clearly states that the total angular momentum Fz is a constant of motion [64].

7.3. HYPERFINE-RESOLVED BASIS STATES The Hyperfine splitting and the Lamb shift are of the same order to the vdW interaction for R > 100a0 , where R is the interatomic distance. However, the fine structure energy shift EF S is

EF S = E(2P3/2 ) − E(2S1/2 ) ≈ 10 × ELS .

(7.24)

In comparison to the 2P1/2 -state, the 2P3/2 -state is heavily displaced from the 2S1/2 state (see Figure 7.1). Thus, we can neglect the influence of the 2P3/2 -state. In other words, we concentrate only on the effects of the hyperfine splitting, the fine structure, and the vdW interaction on the 2S-2S system. If `, j, and F are the orbital angular momentum quantum number, the total electronic angular quantum number, and the total atomic quantum number, j is

1 2

and ` takes value ` = 0 for the 2S1/2 and ` = 1 for the 2P1/2 . However, F holds 1 1 1 1 − ≤ F ≤ + , 2 2 2 2

(7.25)

which indicates that F takes either 0 or 1. By the definition of the multiplicity,

gF = 2F + 1,

(7.26)

162

n=2

2P3/2

2P3/2

F=2 23.65 MHz F=1

Energy

F=9.911 GHz

2S1/2

177.6 MHz L=1.058 GHz 2S1/2 , 2P1/2 2P1/2

≈

F=1

Fz =+1 Fz =0 Fz = -1 Fz =0

F=0 F=1 59.86 MHz F=0

Fz +2 +1 0-1 -2 +1 0-1

Fz =+1 Fz =0 Fz = -1 Fz =0

n=1 43.52 GHz 1S1/2

1S1/2

F=1 1.420 GHz

8.172 GHz F=0

Bohr level

Dirac fine structure

Lamb shift

hyperfine

Fz =+1 Fz =0 Fz =-1 Fz =0

magnetic field

Figure 7.1: Fine and hyperfine levels of the hydrogen atom for n=1, 2. Here, L and F represent the Lamb shift and fine structure, F stands for the hyperfine quantum number and Fz indicates the z-component of the hyperfine quantum number, where z-axis is the axis of quantization. The numerical values presented in this figure are taken from Refs. [4; 5; 6; 7; 8; 9]. The spacing between the levels is not well scaled. In other words, some closed levels are also spaced widely for better visibility.

163 the (F = 0)-state is a singlet and the (F = 1)-state is a triplet. There are 8 states per atoms, viz. one state corresponding to the 2S1/2 -state with F = 0, three states corresponding to the 2S1/2 -state with F = 1, one state corresponding to the 2P1/2 state with F = 0, and three states corresponding to the 2P1/2 -state with F = 1. For the two-hydrogen atoms system, there are 8 × 8 = 64 states. Let Fz = Fz,A + Fz,B be the total hyperfine quantum number of the 64-dimensional Hilbert space. As Fz of either atom can have values 1, 0, or −1, the total hyperfine quantum number takes values

Fz = −2, −1, 0, +1, +2.

(7.27)

Let us denote the eigenstates of the unperturbed Hamiltonian

H0 = HHFS,A + HHFS,B + HLS,A + HLS,B ,

(7.28)

of the system as |`, F, Fz i. Let us first analyze the basis sets considering only the electronic contribution. The total angular quantum number j and the total magnetic projection quantum number µ are given by

j =`+

1 2

1 and µ = m ± , 2

(7.29)

where ` is 0 for S-state and 1 for P -state. The magnetic projection quantum number m ranges from −` to `. Let us denote the electronic basis state by |j, `, µi which can be expressed in terms of the orbital angular momentum and the spin angular momentum with the help of Clebsch-Gordan coefficients as

|j, `, µi =

` X X m=−` σ=±

1 1 µ 2 C`m 1 |`, mi| , σi. σ 2 2 1

2

(7.30)

164 As we ignore the influence of the 2P3/2 -state, we consider only the value j = 12 . We then have ` X X 1µ 1 1 2 | , `, µi = C`m 1 |`, mi| , σi. σ 2 2 2 1 m=−`

(7.31)

σ=± 2

For ` = 0, the total magnetic projection number µ can take either + 12 or − 21 . 1 1 1 1 1 1 1 1 2 2 | , 0, i = C00 i = |0, 0i| , i ≡ |0, 0ie |+ie . 1 1 |0, 0i| , 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 −2 2 | , 0, − i = C00 1 1 |0, 0i| , − i = |0, 0i| , − i ≡ |0, 0ie |−ie . −2 2 2 2 2 2 2 2

(7.32a) (7.32b)

For ` = 1, m can have any one value of 1, 0, or -1. However, the condition m ± 21 = µ is satisfied. 1 X 1 1 1 1 1 2 2 | , 1, i = C1m 1 |1, mi| , σi σ 2 2 2 2 m=−1 1 1 1 1 1 1 1 1 2 2 2 2 i + C11 = C10 1 1 |1, 0i| , 1 1 |1, 1i| , − i − 2 2 2 2 2 2 r 2 2 1 1 1 1 1 2 = − √ |1, 0i| , i + |1, 1i| , − i 2 2 3 2 2 3 r 1 2 ≡ − √ |1, 0ie |+ie + |1, 1ie |−ie . 3 3 1 X 1 1 1 1 − 12 2 | , 1, − i = C1m 1 |1, mi| , σi σ 2 2 2 2 m=−1 1 1 1 1 1 1 − 12 − 21 2 2 i + C10 = C1−1 1 1 |1, −1i| , 1 1 |1, 0i| , − i − 2 2 2 2 2 2 r 22 2 1 1 1 1 1 =− |1, −1i| , i + √ |1, 0i| , − i 3 2 2 2 2 3 r 1 2 ≡ √ |1, 0ie |−ie − |1, −1ie |+ie . 3 3

(7.33a)

(7.33b)

We add the proton spin to compute the hyperfine basis set of a single atom. As the spin of the proton of the hydrogen atom exerts a torque on the electron revolving

165 around it producing magnetic dipole field, the set of observables (J, Sp , mJ , mp ) can not be the CSCO anymore. Here, J and Sp are the total electronic angular momentum and the spin angular momentum of the proton whereas mJ and mp are the magnetic projections of J and Sp . On the other hand, the total angular momentum of the ~p and its z-component are conserved. In our case, the allowed system F~ = J~ + S values of F are 1 1 F = − Sp , ..., + Sp 2 2 = 0 and 1,

(7.34)

whereas Fz varies from −F , −F + 1, ...., F . Let us denote the state vectors by |`, F, Fz i. In our system, ` = 0 and ` = 1 refer to the 2S1/2 and 2P1/2 states respectively. F = 0 and F = 1 respectively indicate the hyperfine singlet and hyperfine triplet whereas Fz , the z-component of the total angular momentum of the system, is the magnetic projection of F . Then we have

|`, F, Fz i = =

j i X X

1 F Fz Cjµiβ |j, `, µie | , βip 2 µ=−j β=−i

µ=± 12

provided

1 2

1 1 C F1 µF1z β | , `, µie | , βip , 2 2 2 2 1

X X

β=± 2

+ µ = F is satisfied. For S-states |`, F, Fz i = |0, F, Fz i.

For ` = 0, F = 0 and Fz = 0,

|0, 0, 0i =

1 1 C 00 1 1 | , 0, µie | , βip µ β 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 1 1 1 1 = C 00 | , 0, ie | , − ip + C 00 | , 0, − ie | , ip 1 1 1 1 − 12 − 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 = − √ | , 0, ie | , − ip + √ | , 0, − ie | , ip 2 2 2 2 2 2 2 2 2 2

(7.35)

166   1 ≡ √ |+ie |−ip − |−ie |+ip |0, 0ie . 2

(7.36)

For ` = 0, F = 1 and Fz = 1,

|0, 1, 1i =

1 1 1 1 1 1 11 C 11 ie | , ip 1 1 | , 0, µie | , βip = C 1 1 1 1 | , 0, µ β 2 2 2 2 2 2 2 2 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 =| , 0, ie | , ip ≡ |+ie |+ip |0, 0ie . 2 2 2 2

(7.37)

For ` = 0, F = 1 and Fz = 0,

|0, 1, 0i =

1 1 C 10 1 1 | , 0, µie | , βip µ β 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 1 1 1 1 ie | , − ip + C 00 ie | , ip = C 00 1 1 1 1 | , 0, 1 | , 0, 1 1 1 − 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 = √ | , 0, ie | , − ip + √ | , 0, ie | , ip 2 2 2 2 2 2 2 2 2 2   1 ≡ √ |−ie |+ip + |+ie |−ip |0, 0ie . 2

(7.38)

For ` = 0, F = 1 and Fz = −1,

|0, 1, −1i =

1 1 1 1 1 1 1−1 C 1−1 1 1 | , 0, µie | , βip = C 1 1 1 1 | , 0, − ie | , − ip µ β − − 2 2 2 2 2 2 2 2 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 =| , 0, − ie | , − ip ≡ |−ie |−ip |0, 0ie . 2 2 2 2 For ` = 1, F = 0 and Fz = 0,

|1, 0, 0i =

1 1 C 00 1 1 | , 1, µie | , βip µ β 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 1 1 1 1 =C 00 ie | , − ip + C 00 ip 1 1 1 1 | , 1, 1 1 1 1 | , 1, − ie | , − − 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 = √ | , 1, ie | , − ip − √ | , 1, − ie | , ip 2 2 2 2 2 2 2 2 2 2

(7.39)

167 1 =√ 2

! 1 1 2 |1, 1ie |−ie | , − ip 3 2 2 ! r 1 2 1 1 √ |1, 0ie |−ie − |1, −1ie |+ie | , ip 3 2 2 3

1 − √ |1, 0ie |+ie + 3

1 −√ 2

r

1 1 ≡ √ |+ie |+ip |1, −1ie − √ |−ie |+ip |1, 0ie 3 6 1 1 + √ |−ie |−ip |1, 1ie − √ |+ie |−ip |1, 0ie . 3 6

(7.40)

There are four P -states in which |`, F, Fz i = |1, F, Fz i. For ` = 1, F = 1 and Fz = 1,

|1, 1, 1i =

1 1 C 11 1 1 | , 1, µie | , βip µ β 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 1 1 1 1 =C 11 ie | , ip = | , 1, ie | , ip 1 1 1 1 | , 1, 2 2 22 2 2 2 2 2 2 2! 2 r 1 2 1 1 = − √ |1, 0ie |+ie + |1, 1ie |−ie | , ip 3 2 2 3 r 1 2 ≡ − √ |+ie |+ip |1, 0ie + |−ie |+ip |1, 1ie . 3 3

(7.41)

For ` = 1, F = 1 and Fz = 0,

|1, 1, 0i =

1 1 C 10 1 1 | , 1, µie | , βip µ β 2 2 2 2 1

X X

µ=± 21 β=± 2

1 1 1 1 1 1 1 1 =C 10 ie | , − ip + C 10 ip 1 1 1 1 1 | , 1, 1 1 1 | , 1, − ie | , − − 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 = √ | , 1, ie | , − ip + √ | , 1, − ie | , ip 2 2 2 2 2 2 2 2 2 2 ! r 1 1 2 1 1 =√ − √ |1, 0ie |+ie + |1, 1ie |−ie | , − ip 3 2 2 2 3 ! r 1 1 2 1 1 √ |1, 0ie |−ie − +√ |1, −1ie |+ie | , ip 3 2 2 2 3 1 1 ≡ − √ |+ie |+ip |1, −1ie + √ |−ie |+ip |1, 0ie 3 6 1 1 + √ |−ie |−ip |1, 1ie − √ |+ie |−ip |1, 0ie . 3 6

(7.42)

168 Finally, for ` = 1, F = 1 and Fz = −1,

|1, 1, −1i =

1 1 C 1−1 1 1 | , 1, µie | , βip µ β 2 2 2 2 1

X X

µ=± 12 β=± 2

1 1 1 1 =C 1−1 1 1 1 1 | , 1, − ie | , − ip − − 2 2 2 2 2 2 2 2 1 1 1 1 =| , 1, − ie | , − ip 2 2 2 2 ! r 1 2 1 1 = √ |1, 0ie |−ie − |1, −1ie |+ie | , − ip 3 2 2 3 r 2 1 |+ie |−ip |1, −1ie . ≡ √ |−ie |−ip |1, 0ie − 3 3

(7.43)

These 8 states, namely 4 S-states and 4 P -states given by Eqs. (7.36) - (7.43), serve as the single-atom hyperfine basis states.

7.4. MATRIX ELEMENTS OF ELECTRONIC POSITION OPERATORS We use the definition of the spherical unit vectors as defined in Ref. [65]. 1 eˆ+ = − √ (ˆ ex + iˆ ey ) 2

(7.44a)

eˆ0 = eˆz

(7.44b)

1 eˆ− = √ (ˆ ex − iˆ ey ) 2

(7.44c)

Let us evaluate few r-matrix elements. 

   1  1 h0, 0, 0|~r|1, 1, 0i = √ e h+|p h−| − e h−|p h+| e h00| ~r − √ |+ie |+ip |1, −1ie 2 3  1 1 1 + √ |−ie |+ip |1, 0ie + √ |−ie |−ip |1, 1ie + √ |+ie |−ip |1, 0ie 6 3 6     1 1 1 1 √ e h00|~r|1, 0ie = √ − √ e h00|~r|1, 0ie − √ 2 6 2 6 √ 1 1 = − √ e h00|~r|1, 0ie = − √ (−3 a0 eˆz ) = 3a0 eˆz . (7.45) 3 3

169 

   1  1 h0, 0, 0|~r|1, 1, ±1i = √ e h+|p h−| − e h−|p h+| e h00| ~r ∓ √ |+ie |+ip |1, 0ie 2 3 r  2 |−ie |+ip |1 ± 1ie ± 3   √ 3a0 1 1 3a0 i √ eˆx ± √ eˆy = 3a0 eˆ± . = − √ e h00|~r|1 ± 1ie = − √ (7.46) 3 3 2 2



   1  1 h0, 1, 0|~r|1, 0, 0i = √ e h−|p h+| + e h+|p h−| e h00| ~r √ |+ie |+ip |1, −1ie 2 3  1 1 1 − √ |−ie |+ip |1, 0ie + √ |−ie |−ip |1, 1ie − √ |+ie |−ip |1, 0ie 6 3 6 1 1 = − √ e h00|~r|1, 0ie − √ e h00|~r|1, 0ie 12 12 √ 1 1 = − √ e h00|~r|1, 0ie = − √ (−3 a0 eˆz ) = 3a0 eˆz . (7.47) 3 3



  1 1 h0, 1, ±1|~r|1, 0, 0i = e h±|p h±|e h00| ~r √ |+ie |+ip |1, −1ie − √ |−ie |+ip |1, 0ie 3 6  1 1 + √ |−ie |−ip |1, 1ie − √ |+ie |−ip |1, 0ie 3 6 1 3a0 1 ex + iˆ ey ) = √ e h00|~r|1 ∓ 1ie = √ √ (∓ˆ 3 3 2   √ √ 1 = 3a0 ∓ √ (ˆ e ± )∗ . (7.48) ex ∓ iˆ ey ) = 3a0 (ˆ 2

r     1 2 h0, 1, ±1|~r|1, 1, ±1i = e h±|p h±|e h00| ~r ∓ √ |+ie |+ip |1, 0ie ± |−ie |+ip |1 ± 1ie 3 3 √ 1 1 = ∓ √ e h00|~r|1, 0ie = ∓ √ (−3a0 eˆz ) = ± 3a0 eˆz . (7.49) 3 3



  1 1 h0, 1, ±1|~r|1, 1, 0i = e h±|p h±|e h00| ~r − √ |+ie |+ip |1, −1ie + √ |−ie |+ip |1, 0ie 3 6  1 1 + √ |−ie |−ip |1, 1ie + √ |+ie |−ip |1, 0ie 3 6

170 √ 1 1 3a0 = ∓ √ e h00|~r|1 ∓ 1ie = ∓ √ √ (∓ˆ ex + iˆ ey ) = ± 3a0 eˆ∓ . 3 3 2

(7.50)

   1 1  h0, 1, 0|~r|1, 1, ±1i = √ e h−|p h+| + e h+|p h−| e h00| ~r ∓ √ |+ie |+ip |1, 0ie 2 3 r  2 |−ie |+ip |1 ± 1ie ± 3  1 3a0  1 ey = ± √ e h00|~r|1 ± 1ie = ± √ √ ± eˆx + iˆ 3 3 2  1  √ √ = ∓ 3a0 − (±ˆ ex + iˆ ey ) = ∓ 3a0 eˆ± . (7.51) 2 

All the other r-matrix elements are zero. For example, r  2 1 h1, 1, ±|~r|1, 1, ±1i = ∓ √ e h+|p h+|e h10| ± e h−|p h+|e h1 ± 1| 3 3 r   1 2 ~r ∓ √ |+ie |+ip |1, 0ie ± |−ie |+ip |1 ± 1ie 3 3 1 2 = e h10|~r|1, 0ie + e h1 ± 1|~r|1 ± 1ie = 0, 3 3   1 h0, 0, 0|~r|0, 0, 0i = √ (e h+|p h−| − e h−|p h+|) e h00| 2     1 ~r √ |+ie |−ip − |−ie |+ip |0, 0ie = e h00|~r|0, 0ie = 0, 2 

(7.52a)

(7.52b)

and so on.

7.5. SCALING PARAMETERS For the sake of simplicity, we define the following parameters α4 m2 2 gp c, H≡ 18  M  α5 1 L≡ ln mc2 , 6π α2 a2 V ≡ α~c 03 , R

(7.53) (7.54) (7.55)

171 which we use to scale the expectation values of the hyperfine Hamiltonian, the Lamb shift and the vdW interaction. Substituting the values of the fine-structure constant, g-factor of the proton, masses of the electron and the proton, and the speed of light, the hyperfine splitting constant H works out to

H = 3.924 × 10−26 J ≡ 5.921 × 107 Hz.

(7.56)

In terms of H, the Lamb shift L and the vdW interaction V are given as

L = 17.873 H,

(7.57a)

4.942 × 10−23 V= H. R3

(7.57b)

The expectation value of the Lamb shift Hamiltonian amounts to L and it is nonzero only if both atoms are in the S-states, i.e.,

h`, F, MF |HLS |`, F, MF i = L δ`0 .

(7.58)

The hyperfine triplets corresponding to the 2P1/2 are displaced from the corresponding hyperfine singlet by H, whereas the hyperfine triplet corresponding to the 2S1/2 is displaced by 3H from the corresponding hyperfine singlet. The triplet is lifted upward and the singlet is pushed downward [66]. Thus, we have 3 h0, 1, MF |HHFS |0, 1, MF i = H, 4 1 h1, 1, MF |HHFS |1, 1, MF i = H, 4 9 h0, 0, 0|HHFS |0, 0, 0i = − H, 4 3 h1, 0, 0|HHFS |1, 0, 0i = − H. 4

(7.59a) (7.59b) (7.59c) (7.59d)

172 7.6. GRAPH THEORY (ADJACENCY GRAPH) In the graph theory, an adjacency graph [67; 68] is a diagrammatic representation of a square matrix whose elements are boolean values. One vertex can be connected to the other vertex by one, or more than one edge. A vertex can be connected to itself as well. If each vertex is connected to every other vertex in some number of steps, then the graph is said to be connected. However, if two vertices are not connected at all, they do not talk with each other. The adjacency matrix corresponding to the undirected graph is symmetric in nature. Note that the eigenvalues of a symmetric matrix are real and it is always possible to get orthonormal eigenvectors [69]. The non-negative power Ak of an adjacency matrix tells us about the number of paths of length k of its elements. For example, (Ak )mn is the count of paths of k X length k from m to n. The sum Ai , which depicts the number of paths of length i=1

ranging from 1 to k between every pair of vertices, possesses impressive feature. If the final matrix obtained from the sum contains all the nonzero entries, this means the matrix is irreducible. In other words, if the sum contains any zero entries it indicates that the matrix can be reduced into irreducible matrices.The power A2 is of particular importance. It not only counts the number of paths of length 2 of its entries but also tells us about the connectedness of the corresponding adjacency graph. The adjacency graph G corresponding to an adjacency matrix A of order n is disconnected if and only if there exists a square matrix S = A2 of order n such that the matrix S can be written as  0  Bk×k :  S= : ··  ··  0 : C(n−k)×(n−k)

   .  

(7.60)

173 Detailed mathematical proof of the statement of disconnectivity is given in theorem 1.6 of Ref. [70]. The adjacency matrix A containing two disconnected components can be split-up as 



0  A(G1 ) :  A= : ··  ··  0 : A(G2 )

  ,  

(7.61)

where A(G1 ) and A(G2 ) stand for the adjacency matrices of the components of the adjacency graphs G1 and G2 . The components G1 and G2 do not share any edges between their vertices. Thus, there is no coupling between them. In later sections, we will notice that the adjacency graph is very useful to express a hyperfine subspace into two irreducible subspaces.

7.7. HAMILTONIAN MATRICES IN THE HYPERFINE SUBSPACES As we already mentioned, the 64-dimensional Hilbert space has five manifolds namely, Fz = +2, Fz = +1, Fz = 0, Fz = −1, and Fz = −2. The Fz = +2 and the Fz = −2 manifolds are 4-dimensional, the Fz = +1 and the Fz = −1 manifolds are 16-dimensional, and Fz = 0 manifold is 24-dimensional. We analyze all these manifolds separately. 7.7.1. Manifold Fz = +2. The four states in the Fz = +2 manifold, in the ascending order of quantum numbers, are

|φ1 i = |(0, 1, 1)A (0, 1, 1)B i,

|φ2 i = |(0, 1, 1)A (1, 1, 1)B i,

|φ3 i = |(1, 1, 1)A (0, 1, 1)B i,

|φ4 i = |(1, 1, 1)A (1, 1, 1)B i.

(7.62)

174 The first element of the matrix hφ1 |H|φ1 i is given by

hφ1 |H|φ1 i =h(0, 1, 1)A (0, 1, 1)B |H|(0, 1, 1)A (0, 1, 1)B i =e,A h+|

p,A h+| e,A h0, 0| e,B h+| p,B h+| e,B h0, 0|H|+ie,A

× |+ip,A |0, 0ie,A |+ie,B |+ip,B |0, 0ie,B ,

(7.63)

where H = HLS, A + HLS, B + HHFS, A + HHFS, B + HvdW . The Lamb shift due to each of the HLS, A and HLS, B is L and the hyperfine splitting due to each of the HHFS, A and HHFS, B is 43 H, whereas the vdW interaction does not contribute anything to the diagonal element. Thus, we have 3 hφ1 |H|φ1 i = H + 2L. 2

(7.64)

The matrix element hφ1 |H|φ2 i is given by

hφ1 |H|φ2 i =h(0, 1, 1)A (0, 1, 1)B |H|(0, 1, 1)A (1, 1, 1)B i =e,A h+| "

p,A h+| e,A h0, 0| e,B h+| p,B h+| e,B h0, 0|H|+ie,A |+ip,A

1 × − √ |+ie,B |+ip,B |1, 0ie,B + 3

The orthogonality relation

e,B h0, 0|1, 0ie,B

r

|0, 0ie,A #

2 |−ie,B |+ip,B |1, 1ie,B , 3

(7.65)

= 0 requires that the right-hand side of

Eq. (7.68) should vanish. hφ1 |H|φ2 i = 0 = (hφ2 |H|φ1 i)∗ = hφ2 |H|φ1 i

(7.66)

Swapping A and B in hφ1 |H|φ2 i, we get hφ1 |H|φ3 i. Thus, it is straightforward to note that

hφ1 |H|φ3 i = 0 = hφ3 |H|φ1 i.

(7.67)

175 The matrix element hφ1 |H|φ4 i is given by

hφ1 |H|φ4 i =h(0, 1, 1)A (0, 1, 1)B |H|(1, 1, 1)A (1, 1, 1)B i =e,A h+| p,A h+| e,A h0, 0| e,B h+| p,B h+| e,B h0, 0| r   1 2 H − √ |+ie,A |+ip,A |1, 0ie,A + |−ie,A |+ip,A |1, 1ie,A 3 3 r   1 2 − √ |+ie,B |+ip,B |1, 0ie,B + |−ie,B |+ip,B |1, 1ie,B 3 3 1 = e,A h0, 0| e,B h0, 0|HvdW |1, 0ie,A |1, 0ie,B 3 = − 2V = hφ4 |H|φ1 i.

(7.68)

In the similar manner, we determine all the element of the matrix H(Fz =+2) , which reads 

H(Fz =+2)

    =   

3 H 2

+ 2L 0 0

−2V

 −2V   H + L −2V 0   . −2V H + L 0    1 0 0 H 2 0

0

(7.69)

The adjacency matrix associated to the Hamiltonian matrix H(Fz =+2) is 

A(Fz =+2)

 1 0 0   0 1 1  =  0 1 1   1 0 0



1   0   , 0    1

(7.70)

which is obtained by the replacement of the nonzero entries of the matrix H(Fz =+2) by one. The adjacency graphs corresponding to adjacency matrix A(Fz =+2) is shown in Figure 7.2. With the help of the adjacency graph, we see that the Fz = +2 manifold

176 1

4

2

3

Figure 7.2: An adjacency graph of the matrix A(Fz =+2) . The first diagonal entry, i.e., first vertex is adjacent to the fourth diagonal entry, i.e., fourth vertex and vice versa. The second diagonal element, i.e., the second vertex is adjacent to the third diagonal element, i.e., third vertex and vice versa. However, the two pieces of the graph do not share any edges between the vertices.

can be decomposed into two subspaces. The subspace (I) is composed of the states (I)

(7.71)

(I)

(7.72)

|φ1 i = |φ1 i = |(0, 1, 1)A (0, 1, 1)B i, |φ2 i = |φ4 i = |(1, 1, 1)A (1, 1, 1)B i,

in which atoms are in S-S or P -P configuration while the subspace (II) is composed of the states (II)

(7.73)

(II)

(7.74)

|φ1 i = |φ2 i = |(0, 1, 1)A (1, 1, 1)B i, |φ2 i = |φ3 i = |(1, 1, 1)A (0, 1, 1)B i,

in which atoms are in S-P or P -S configuration. These two subspaces do not couple to each other. The Hamiltonian matrix corresponding to the subspace (I) reads 

  (I) H(Fz =+2) = 

3 H 2

+ 2L −2V  . 1 −2V H 2

(7.75)

In the subspace (I), the energy levels are non degenerate. The energy eigenvalues corresponding to the subspace (I) are given by

E (I) = H + L ±

1p 16V 2 + (H + 2L)2 , 2

(7.76)

177 Or, (I) E+ (I)

E−

3 V2 = H + 2L + 4 + O(V 4 ), 2 H + 2L 1 V2 = H−4 + O(V 4 ). 2 H + 2L

(7.77a) (7.77b)

This clearly shows that the eigenvalues in the subspace (I) do not experience the first (I)

order shift in the vdW interaction V, i.e., ∆E± ∼ R−6 . From Eqs. (7.77), one can write (I)

∆E± ∼ 4

V2 . H + 2L

(7.78)

We have H ≡ 0.055949L, and in the atomic units V = 3/R3 and L →

α3 6π

ln(α−2 ).

Thus, (I)

∆E± ∼

4×9 36 × 6π 8 × 107 ∼ ∼ . (0.055949L + 2L) R6 2.055949 α3 ln (α−2 ) R6 R6

(7.79)

Recognizing E = −C6 /R6 , we find that the vdW coefficient , C6 , for 2S(F = 0) → 2S(F = 1) or 2P (F = 0) → 2P (F = 1) hyperfine transition is in the order of 107 . (I)

(II)

The normalized eigenvectors associated to the eigenvalues E+ and E+ are (I) |φ+ i

=p

1



(I) a1 |φ1 i



,

(7.80a)

 (I) (I) (I) |φ− i = p 2 a |φ i − a |φ i . 2 1 1 2 a1 + a22

(7.80b)

a21 + a22  1

+

(I) a2 |φ2 i

where a1 and a2 are given by p

16V 2 + (H + 2L)2 4V "  1/2 # H + 2L 16V 2 1+ 1+ =− 4V (H + 2L)2

a1 = −

H + 2L +

178 H + 2L =− 2V

 1+

4V 2 (H + 2L)2



+ O(V 3 ),

a2 = 1.

(7.81a) (7.81b)

Note that for very large interatomic separation, 4V 2 /(H + 2L)2  1 and hence, |a1 | ≈ (H + 2L)/(2V)  a2 = 1. The Hamiltonian matrix associated to the subspace (II) is 



 H + L −2V  (II) H(Fz =+2) =  . −2V H + L

(7.82)

The energy levels are degenerate and coupled by the vdW interaction V. The eigenen(II)

ergies and eigenvectors of the Hamiltonian matrix H(Fz =+2) are (II)

E+ = H + L ± 2V,  1  (II) (II) (II) |φ± i = √ |φ1 i ± |φ2 i . 2

(7.83) (7.84)

The shift in the eigenenergies of the subspace (II) are linearly dependent with the vdW interaction energy V. More explicitly, (II)

∆E± = 4V

(7.85)

Thus, the hyperfine transition in the subspace (II) goes to R−3 . See Figure 7.3 for an evolution of energy levels as a function of interatomic distance in the Fz = +2 hyperfine manifold. For a sufficiently large interatomic distance, V → 0, and we have only three energy levels as expected from unperturbed energy values. However, as the interatomic distance decreases the vdW interaction comes into play and energy levels split and deviate from unperturbed values. The energy levels do not cross in the Fz = +2 hyperfine manifold.

179

Figure 7.3: Energy levels as a function of interatomic separation R in the Fz = +2 hyperfine manifold. The horizontal axis which represents the interatomic distance is expressed in the unit of Bohr’s radius, a0 , and the vertical axis, which is the energy divided by the plank constant, is in hertz. The energy levels in the subspace (I) deviate heavily from their unperturbed values 12 H and 32 H+L for R < 500a0 . The doubly degenerate energy level L+H splits up into two levels, which repel each other as the interatomic distance decreases.

7.7.2. Manifold Fz = +1. The Fz = +1 manifold has 16 states as listed below:

|ψ1 i = |(0, 0, 0)A (0, 1, 1)B i,

|ψ2 i = |(0, 0, 0)A (1, 1, 1)B i,

|ψ3 i = |(0, 1, 0)A (0, 1, 1)B i,

|ψ4 i = |(0, 1, 0)A (1, 1, 1)B i,

|ψ5 i = |(0, 1, 1)A (0, 0, 0)B i,

|ψ6 i = |(0, 1, 1)A (0, 1, 0)B i,

|ψ7 i = |(0, 1, 1)A (1, 0, 0)B i,

|ψ8 i = |(0, 1, 1)A (1, 1, 0)B i,

180 |ψ9 i = |(1, 0, 0)A (0, 1, 1)B i,

|ψ10 i = |(1, 0, 0)A (1, 1, 1)B i,

|ψ11 i = |(1, 1, 0)A (0, 1, 1)B i,

|ψ12 i = |(1, 1, 0)A (1, 1, 1)B i,

|ψ13 i = |(1, 1, 1)A (0, 0, 0)B i,

|ψ14 i = |(1, 1, 1)A (0, 1, 0)B i,

|ψ15 i = |(1, 1, 1)A (1, 0, 0)B i,

|ψ16 i = |(1, 1, 1)A (1, 1, 0)B i.

(7.86)

In Eq. (7.86), the 16 states are ordered in the ascending order of quantum numbers. We calculate all the 256 elements of the Hamiltonian matrix for Fz = +1. Then we replace all the nonzero off-diagonal element by 1 and all the diagonal elements by zero. This results an adjacency matrix A(Fz =+1) of order 16 as given below: 

A(Fz =+1)

                   =                   

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

1

0

1

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

                    .                   

(7.87)

181 See Figure 7.4 for adjacency graph for the matrix A(Fz =+1) which shows the linkage between neighboring vertices in the matrix (7.87). We notice that 

16 X i=1

Ai(Fz =+1)

                        =                       



a 0 b 0 b b 0 0 0 c 0 d 0 0 d d   0 a 0 b 0 0 b b c 0 d 0 d d 0 0     b 0 a 0 b b 0 0 0 d 0 c 0 0 d d    0 b 0 a 0 0 b b d 0 c 0 d d 0 0    b 0 b 0 a b 0 0 0 d 0 d 0 0 c d     b 0 b 0 b a 0 0 0 d 0 d 0 0 d c    0 b 0 b 0 0 a b d 0 d 0 c d 0 0    0 b 0 b 0 0 b a d 0 d 0 d c 0 0   , 0 c 0 d 0 0 d d a 0 b 0 b b 0 0     c 0 d 0 d d 0 0 0 a 0 b 0 0 b b    0 d 0 c 0 0 d d b 0 a 0 b b 0 0    d 0 c 0 d d 0 0 0 b 0 a 0 0 b b     0 d 0 d 0 0 c d b 0 b 0 a b 0 0    0 d 0 d 0 0 d c b 0 b 0 b a 0 0    d 0 d 0 c d 0 0 0 b 0 b 0 0 a b    d 0 d 0 d c 0 0 0 b 0 b 0 0 b a

(7.88)

where,

a = 12106896,

b = 12106888,

The presence of zeros in

P16

i=1

c = 4035624, and d = 4035632.

(7.89)

Ai(Fz =+1) indicates that A(Fz =+1) can be reduced into

at least two irreducible matrices. It can be clearly seen from the adjacency matrix (7.87) that 1 is adjacent to 12, 15, and 16. 16 is adjacent to 1, 3, and 5. 5 is adjacent to 10, 12, and 16. 12 is adjacent to 1, 5, and 6. 6 is adjacent to 10, 12, and 15. 10

182 13

8

2

11

15

4

9

14

6

7 (b)

(I) G(Fz =+1)

1

12

3

10

16

5 (b)

(II) G(Fz =+1)

Figure 7.4: An adjacency graph of the matrix A(Fz =+1) . The graph for A(Fz =+1) is (II) (I) disconnected having two components G(Fz =+1) and G(Fz =+1) which do not share any edges between the vertices.

is adjacent to 3, 5, and 6. 3 is adjacent to 10, 15, and 16. 15 is adjacent to 1, 3, and 6. However, these vertices are neither adjacent nor linked in any steps to the remaining other vertices. At the same time, 2 is adjacent to 11, 13, and 14. 14 is adjacent to 2, 4, and 7. 7 is adjacent to 9, 11, and 14. 11 is adjacent to 2, 7, and 8. 8 is adjacent to 9, 11, and 13. 13 is adjacent to 2, 4, and 8. 4 is adjacent to 9, 13, and 14. 9 is adjacent to 4, 7, and 8. The power A2(Fz =+1) of the adjacency matrix A(Fz =+1) contains two diagonal nonzero matrices of order 8 and two same sized off-diagonal zero matrices, which verifies that the adjacency graph corresponding to the matrix A(Fz =+1) has two disconnected components. The graph 7.4 clearly indicates that the 16-dimensional Fz = +1 manifold can be decomposed into two subspaces. These two subspaces do not talk with each other as they are uncoupled. Thus we can analyze each subspace independently. Firstly, we consider the subspace (I) of manifold Fz = +1. The subspace (I) is composed of

183 |ψ2 i, |ψ4 i, |ψ7 i, |ψ8 i, |ψ9 i, |ψ11 i, |ψ13 i, and |ψ14 i. We rename these states as below: (I)

|ψ2 i = |ψ4 i = |(0, 1, 0)A (1, 1, 1)B i,

(I)

(I)

|ψ4 i = |ψ8 i = |(0, 1, 1)A (1, 1, 0)B i,

(I)

|ψ6 i = |ψ11 i = |(1, 1, 0)A (0, 1, 1)B i,

|ψ1 i = |ψ2 i = |(0, 0, 0)A (1, 1, 1)B i,

(I)

|ψ3 i = |ψ7 i = |(0, 1, 1)A (1, 0, 0)B i,

(I)

|ψ5 i = |ψ9 i = |(1, 0, 0)A (0, 1, 1)B i, (I)

(I)

|ψ7 i = |ψ13 i = |(1, 1, 1)A (0, 0, 0)B i,

|ψ8 i = |ψ14 i = |(1, 1, 1)A (0, 1, 0)B i. (7.90)

The Hamiltonian matrix of the subspace (I) reads 

(I)

H(Fz =+1)

          =          

L − 2H

0

0

0

0

−2V

V

0

H+L

0

0

−2V

0

−V

0

0

L

0

V

−V

0

0

0

0

H+L

−V

V

−2V

0

−2V

V

−V

L

0

0

−2V

0

−V

V

0

H+L

0

V

−V

0

−2V

0

0

L − 2H

−V

V

−2V

0

0

0

0

−V



     −2V      0 .   0    0    0   H+L V

(7.91)

(I)

(I)

If A(Fz =+1) is the adjacency matrix corresponding to H(Fz =+1) , we have           8 X (I)
i  A(Fz =+1) =    i        

11135

10880

10880

10880

10710

10965

10965

10880

11135

10880

10880

10965

10710

10965

10880

10880

11135

10880

10965

10965

10710

10880

10880

10880

11135

10965

10965

10965

10710

10965

10965

10965

11135

10880

10880

10965

10710

10965

10965

10880

11135

10880

10965

10965

10710

10965

10880

10880

11135

10965

10965

10965

10710

10880

10880

10880

(I)


i

As all of the elements of the

P8 i

A(Fz =+1)

states are connected with each other.

10965



  10965    10965     10710  .  10880    10880     10880   11135

(7.92)

are nonzero, we confirm that all the

184 The energy level L − 2H and L are doubly degenerate and coupled with the nonzero off-diagonal entries V. However, the energy level L + H is four-fold (I)

(A)

(I)

(A)

degenerate. Consider the subspace spanned by |ψ1 i ≡ |ψ1 i and |ψ7 i ≡ |ψ2 i. (A)

The Hamiltonian matrix H(Fz =+1) reads 



V  L − 2H  (A) H(Fz =+1) =  . V L − 2H

(7.93)

The eigenvalues and the corresponding eigenvectors of the Hamiltonian matrix (7.93) are (A)

E± = L − 2H ± V,  1  (A) (A) (A) |ψ1 i ± |ψ2 i . |χ± i = 2

(7.94a) (7.94b) (I)

(B)

(I)

The other doubly degenerate energy level L is spanned by |ψ3 i ≡ |ψ1 i and |ψ5 i ≡ (B)

(B)

|ψ2 i. The Hamiltonian matrix H(Fz =+1) is 



 L V  (B) H(Fz =+1) =  . V L

(7.95)

The eigenvalues and the corresponding eigenvectors of the Hamiltonian matrix (7.95) are (B)

E± = L ± V,  1  (B) (B) (B) |χ± i = √ |ψ1 i ± |ψ2 i . 2

(7.96a) (7.96b)

185 The four-fold degenerate Hamiltonian matrix 

(C)

H(Fz =+1)

0 0 V  H+L   0 H+L V 0  =  0 V H+L 0   V 0 0 H+L

        

(7.97)

is spanned by the following vectors (I)

(C)

|ψ2 i ≡ |ψ1 i,

(I)

(C)

|ψ4 i ≡ |ψ2 i,

(I)

(C)

|ψ6 i ≡ |ψ3 i,

(I)

(C)

|ψ8 i ≡ |ψ4 i.

(7.98)

(C)

The Hamiltonian matrix H(Fz =+1) can again be decomposed into two identical 2 × 2 matrices. 





V  L+H  (C),1 H(Fz =+1) =  , V L+H



V  L+H  (C),2 and H(Fz =+1) =   . (7.99) V L+H

(C),1

(C)

(C)

(C),2

The Hamiltonian matrix H(Fz =+1) is associated with |ψ1 i and |ψ4 i while H(Fz =+1) (C)

(C)

is associated with |ψ2 i and |ψ3 i. The eigenvalues of both the matrix are given by (C)

E± = L + H ± V,

(7.100)

whereas the eigenvectors are given as  1  (C) (C) (B) |χ±,1 i = √ |ψ1 i ± |ψ4 i , 2

 1  (C) (C) (B) |χ±,2 i = √ |ψ2 i ± |ψ3 i . 2

(7.101)

Figure 7.5 is a Born-Oppenheimer potential curve for subspace(I) of Fz = +1 hyperfine manifold. For large interatomic distance, V → 0, and as the interatomic distance decreases, energy levels split, repel with each other, and experience V → R−3 shift.

186

Figure 7.5: Evolution of the energy levels as a function of interatomic separation R in the subspace (I) of the Fz = +1 hyperfine manifold. For infinitely long interatomic distance, we observe three distinct energy levels same as in the unperturbed case. However, for small interatomic separation, the energy levels split and deviate from the unperturbed energies and become separate and readable.

Let us now focus on the subspace (II) of manifold Fz = +1. The subspace (II) is spanned by |ψ1 i, |ψ3 i, |ψ5 i, |ψ6 i, |ψ10 i, |ψ12 i, |ψ15 i, and |ψ16 i. We rename these states as below: (II)

|ψ2 i = |ψ3 i = |(0, 1, 0)A (0, 1, 1)B i,

(II)

|ψ4 i = |ψ6 i = |(0, 1, 1)A (0, 1, 0)B i,

|ψ1 i = |ψ1 i = |(0, 0, 0)A (0, 1, 1)B i, |ψ3 i = |ψ5 i = |(0, 1, 1)A (0, 0, 0)B i, (II)

|ψ5 i = |ψ10 i = |(1, 0, 0)A (1, 1, 1)B i,

(II)

(II)

(II)

|ψ6 i = |ψ12 i = |(1, 1, 0)A (1, 1, 1)B i,

187 (II)

(II)

|ψ7 i = |ψ15 i = |(1, 1, 1)A (1, 0, 0)B i,

|ψ8 i = |ψ16 i = |(1, 1, 1)A (1, 1, 0)B i. (7.102)

The atoms in the subspace (II) are in S-S or P -P configurations. The Hamiltonian matrix of the subspace (II) reads 

(II) H(Fz =+1)

          =          

2L − 32 H

0

0 3H 2

0

−2V

0

0

0

−2V

0

0

2L − 32 H

0

V

−V

0

0

0

2L + 23 H

−V

V

0

−2V

V

−V

− 21 H

0

−2V

0

−V

V

0

1 H 2

V

−V

0

−2V

0

0

−V

V

−2V

0

0

0

0

2L +

0

V

−V



  −V V    0 −2V     −2V 0  .  0 0    0 0     1 0  −2H  1 0 H 2 (7.103)

In this subspace, no two degenerate levels are coupled by V in first order. Thus, all the energy levels experience R−6 vdW shift as shown in Figure 7.6. Thus the states (II)

listed in Eq. (7.102) serve as eigenstates of the Hamiltonian matrix, H(Fz =+1) , of the (II)

system. The hyperfine transition goes second order in V. If A(Fz =+1) is the adjacency
i P (II) (II) matrix of H(Fz =+1) , then the sum 8i A(Fz =+1) is identical to Eq. (7.92).

7.7.3. Manifold Fz = 0. The Fz = 0 hyperfine manifold is composed of

|Ψ1 i = |(0, 0, 0)A (0, 0, 0)B i,

|Ψ2 i = |(0, 0, 0)A (0, 1, 0)B i,

|Ψ3 i = |(0, 0, 0)A (1, 0, 0)B i,

|Ψ4 i = |(0, 0, 0)A (1, 1, 0)B i,

|Ψ5 i = |(0, 1, −1)A (0, 1, 1)B i, |Ψ7 i = |(0, 1, 0)A (0, 0, 0)B i,

|Ψ6 i = |(0, 1, −1)A (1, 1, 1)B i, |Ψ8 i = |(0, 1, 0)A (0, 1, 0)B i,

188

Figure 7.6: Evolution of the energy levels as a function of interatomic separation R in the subspace (II) of the Fz = +1 hyperfine manifold. For infinitely long interatomic distance, we observe four distinct energy levels same as in the unperturbed case. However, for small interatomic separation, the energy levels split and deviate from the unperturbed energies and become distinct and readable.

|Ψ9 i = |(0, 1, 0)A (1, 0, 0)B i, |Ψ11 i = |(0, 1, 1)A (0, 1, −1)B i,

|Ψ10 i = |(0, 1, 0)A (1, 1, 0)B i, |Ψ12 i = |(0, 1, 1)A (1, 1, −1)B i,

|Ψ13 i = |(1, 0, 0)A (0, 0, 0)B i,

|Ψ14 i = |(1, 0, 0)A (0, 1, 0)B i,

|Ψ15 i = |(1, 0, 0)A (1, 0, 0)B i,

|Ψ16 i = |(1, 0, 0)A (1, 1, 0)B i,

|Ψ17 i = |(1, 1, −1)A (0, 1, 1)B i, |Ψ19 i = |(1, 1, 0)A (0, 0, 0)B i,

|Ψ18 i = |(1, 1, −1)A (1, 1, 1)B i, |Ψ20 i = |(1, 1, 0)A (0, 1, 0)B i,

189 |Ψ21 i = |(1, 1, 0)A (1, 0, 0)B i,

|Ψ22 i = |(1, 1, 0)A (1, 1, 0)B i,

|Ψ23 i = |(1, 1, 1)A (0, 1, −1)B i,

|Ψ24 i = |(1, 1, 1)A (1, 1, −1)B i.

(7.104)

The Hamiltonian matrix H(Fz =0) is a square matrix of order 24. We first evaluate H(Fz =0) , and then replace each of the off-diagonal nonzero entries by 1 and each of the diagonal elements by 0. Thus constructed square matrix, whose entries are of boolean values, is an adjacency matrix A(Fz =0) which reads

A(Fz =0) =                                                

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

1



0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

1

1

0

0

1

0

(7.105)

0

0

0

0

0

1

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

                       .                       

1

1

0

0

0

0

1

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

P24

Ai(Fz =0) ,

Figure 7.7 is an adjacency graph corresponding to A(Fz =0) . The sum

i=1

which counts the number of neighbors of length(d) given by 1 ≤ d ≤ 24 that by every

190 pair of nodes shares, satisfies 24 X

Ai(Fz =0) =

i=1

                                 

P

Q

0

0

S

Q

P

0

0

0

0

Q

S

S

0

0

Q

Q

0

0

P

Q

0

0

S

S

0

Q

Q

0

S

0

0

P

0

S

0

0

Q

0

Z

0

S

S

0

0

0

0

V

V

0

0

S

0

Q

Q

0

S

Q

0

S

V

V

0

A

0

0

0

0

W

0

0

V

V

V

0

0

0

V

X

V

0

W

0

W

0

0

0

W

0

X

V

W

W

0

B

0

0

0

W

0

0

X

V

X

0

V

0

W

0

W

0

0

0

W

0

W

W

0

C

0

0

S

S

0

Z

0

0

S

S

0

A

W

W

0

0

B

0

W

W

0

0

C

0

Q

Q

0

0

S

0

P

Q

0

0

S

0

0

0

V

X

0

W

0

0

V

V

0

W

Q

Q

0

0

S

0

Q

P

0

0

S

0

0

0

X

V

0

W

0

0

V

V

0

W

0

0

Q

Q

0

S

0

0

P

Q

0

S

V

X

0

0

W

0

V

V

0

0

W

0

0

0

Q

Q

0

S

0

0

Q

P

0

S

X

V

0

0

W

0

V

V

0

0

W

0

S

S

0

0

A

0

S

S

0

0

Z

0

0

0

W

W

0

C

0

0

W

W

0

B

0

0

S

S

0

A

0

0

S

S

0

Z

W

W

0

0

C

0

W

W

0

0

B

0

0

0

V

V

0

W

0

0

V

X

0

W

P

Q

0

0

S

0

Q

Q

0

0

S

0

0

0

V

V

0

W

0

0

X

V

0

W

Q

P

0

0

S

0

Q

Q

0

0

S

0

V

V

0

0

W

0

V

X

0

0

W

0

0

0

P

Q

0

S

0

0

Q

Q

0

S

V

V

0

0

W

0

X

V

0

0

W

0

0

0

Q

P

0

S

0

0

Q

Q

0

S

0

0

W

W

0

B

0

0

W

W

0

C

S

S

0

0

Z

0

S

S

0

0

A

0

W

W

0

0

B

0

W

W

0

0

C

0

0

0

S

S

0

Z

0

0

S

S

0

A 0

0

0

V

X

0

W

0

0

V

V

0

W

Q

Q

0

0

S

0

P

Q

0

0

S

0

0

X

V

0

W

0

0

V

V

0

W

Q

Q

0

0

S

0

Q

P

0

0

S

0

V

X

0

0

W

0

V

V

0

0

W

0

0

0

Q

Q

0

S

0

0

P

Q

0

S

X

V

0

0

W

0

V

V

0

0

W

0

0

0

Q

Q

0

S

0

0

Q

P

0

S

0

0

W

W

0

C

0

0

W

W

0

B

S

S

0

0

A

0

S

S

0

0

Z

0

W

W

0

0

C

0

W

W

0

0

B

0

0

0

S

S

0

A

0

0

S

S

0

Z

(7.106) where

P =13185279766584,

Q = 13185279766572,

R = 17374576685400,

S =18646800486300,

T = 13185279766572,

U = 18646800486300,

V =3444045886572,

W = 4870616940000,

X = 3444045886584,

Y =4870616940000,

Z = 26370559533156,

A = 26370559533144,

B =6888091773156,

and C = 6888091773144.

Not all the elements of

P24

i=1

(7.107)

Ai(Fz =0) are nonzero. Thus, the matrix H(Fz =0) can be

reduced into irreducible sub-matrices. The adjacency matrix squared A2(Fz =0) takes

                 ,                

191 22

11

21

5

15

1

8

20

16

24

2

18

7

12

6

13

3

10

(I)

19

14

23

4

17

9

(II)

(a) G(Fz =0)

(b) G(Fz =0)

Figure 7.7: An adjacency graph of the matrix A(Fz =0) . The graph for (I) (II) A(Fz =0) has two disconnected components G(Fz =0) and G(Fz =0) which do not share any edges between the vertices.

the form  A2(Fz =0)

 B12×12 : 012×12  = ·· : ··   012×12 : C12×12

   ,  

(7.108)

where B12×12 and C12×12 are nonzero matrices of order 12 while 012×12 represents a null matrix of order 12. Eq. (7.108) confirms that the adjacency graph corresponding to the adjacency matrix A(Fz =0) has two disconnected components. Each component (I)

(II)

G(Fz =0) and G(Fz =0) of the adjacency graph has 12 vertices. Eq. (7.108) and Figure 7.7 imply that we can partition the Hamiltonian matrix H(Fz =0) of the Fz = 0 hyperfine

192 manifold as 



H(Fz =0)

(I)

:

0

··

:

··

0

:

H(Fz =0)

 H(Fz =0) =  

 . 

(II)

(7.109)

Thus, our 24-dimensional hyperfine manifold Fz = 0 reduces into two irreducible 12dimensional sub-manifolds. The subspace (I) of the Fz = 0 manifold is composed of |Ψ1 i, |Ψ2 i, |Ψ5 i, |Ψ7 i, |Ψ8 i, |Ψ11 i, |Ψ15 i, |Ψ16 i, |Ψ18 i, |Ψ21 i, |Ψ22 i, and |Ψ24 i. Thus, (I)

(I)

|Ψ1 i = |Ψ1 i = |(0, 0, 0)A (0, 0, 0)B i, |Ψ2 i = |Ψ2 i = |(0, 0, 0)A (0, 1, 0)B i, (I)

(I)

|Ψ3 i = |Ψ5 i = |(0, 1, −1)A (0, 1, 1)B i, |Ψ4 i = |Ψ7 i = |(0, 1, 0)A (0, 0, 0)B i, (I)

(I)

|Ψ5 i = |Ψ8 i = |(0, 1, 0)A (0, 1, 0)B i, |Ψ6 i = |Ψ11 i = |(0, 1, 1)A (0, 1, −1)B i, (I)

(I)

|Ψ7 i = |Ψ15 i = |(1, 0, 0)A (1, 0, 0)B i, |Ψ8 i = |Ψ16 i = |(1, 0, 0)A (1, 1, 0)B i, (I)

(I)

|Ψ9 i = |Ψ18 i = |(1, 1, −1)A (1, 1, 1)B i, |Ψ10 i = |Ψ21 i = |(1, 1, 0)A (1, 0, 0)B i, (I)

(I)

|Ψ11 i = |Ψ22 i = |(1, 1, 0)A (1, 1, 0)B i, |Ψ12 i = |Ψ24 i = |(1, 1, 1)A (1, 1, −1)B i, (7.110) (I)

are the corresponding basis vectors. The Hamiltonian matrix H(Fz =0) reads (I)

H(Fz =0) =  2L −

                    

9H 2

0

0

2L −

0

0

0 0 0 0 0 −V 0 −2V −V

3H 2

0 0 0 0 0 V −2V 0 −V

0

0

0

0

0

0

−V

0

−2V

−V

0

0

0

0

0

0

V

−2V

0

−V

0

0

−V

V

2V

−V

V

0

0

0

−2V

−V

0

0

V

−2V

0

V

0

0

V

−V

−V

0

V

V

2V

− 23 H

0

0

0

0

0

0

−H 2

0

0

0

0

0

H 2

0

0

0

0

−H 2

0

0

0

H 2

0

0

H 2

2L +

3H 2

0 0 0 −V V 2V −V V 0

0 2L −

3H 2

0 0 0 −2V −V 0 0 V

0 2L +

3H 2

0 −2V 0 V 0 0 V

0 2L +

3H 2

−V −V 0 V V 2V

0 0 0 0

0 0 0

0 0

0

           .          

(7.111)

193

No two degenerate levels of the matrix (7.111) are coupled. Thus, the states listed in (7.110) serve as the eigenvectors of the matrix (7.111). Figure 7.8 is a BornOppenheimer potential curve for Fz = 0 in subspace (I). For a very large value of R, V → 0, however, the interaction energy experiences a R−6 type energy shift as R decreases. Surprisingly, we notice several level crossings and the level crossings are unavoidable. According to the no-crossing rule, this is an unusual outcome. (I)

On the other hand, if A(Fz =0) is the adjacency matrix corresponding to the P  (I) i (I) Hamiltonian matrix H(Fz =0) , the sum 12 is given by i=1 A(Fz =0)                 12  i  X  (I) A(Fz =0) =    i=1               

 α β γ β β γ δ

δ

 δ

ζ

β α γ β β γ δ

δ

 ζ

δ

γ γ η γ γ θ





ι





β β γ α β γ δ

ζ

 δ

δ

β β γ β α γ ζ

δ

 δ

δ

γ γ θ γ γ η

 κ 





δ

δ

 δ

ζ

 α β γ β β

δ

δ

 ζ

δ

 β α γ β β





ι

 κ γ γ η γ γ

δ

ζ

 δ

δ

 β β γ α β

ζ

δ

 δ

δ

 β β γ β α



 κ 



ι γ γ θ γ γ



        κ             ι  ,  γ    γ     θ    γ    γ    η

(7.112)

where

α = 12697599, β = 12693504, γ = 17881088, δ = 12618606,  = 17918537, ζ = 12622701, η = 25391103, θ = 25387008, ι = 25241307, κ = 25237212. (7.113)

194

Figure 7.8: Evolution of the energy levels as a function of interatomic separation R in the subspace (I) of Fz = 0 hyperfine manifold. The energy levels are asymptotic for large interatomic separation. Although at the large separation, there are six unperturbed energy levels, the degeneracy is removed in small separation and hence, the energy levels spread widely. The small figure inserted on the right top of the main figure is the magnified version of a small portion as indicated in the figure. The figure shows several level crossings.

195 The absence of the zero in the sum

P12  i=1

(I) A(Fz =0)

i

(I)

indicates that the matrix H(Fz =0)

can not be reduced anymore. Now we focus on the subspace (II) of manifold Fz = 0. The subspace (II) of the Fz = 0 manifold is composed of |Ψ3 i, |Ψ4 i, |Ψ6 i, |Ψ9 i, |Ψ10 i, |Ψ12 i, |Ψ13 i, |Ψ14 i, |Ψ17 i, |Ψ19 i, |Ψ20 i, and |Ψ23 i. Let us rename these states as (II)

(II)

|Ψ1 i = |Ψ3 i = |(0, 0, 0)A (1, 0, 0)B i,

|Ψ2 i = |Ψ4 i = |(0, 0, 0)A (1, 1, 0)B i,

(II)

|Ψ4 i = |Ψ9 i = |(0, 1, 0)A (1, 0, 0)B i,

(II)

|Ψ6 i = |Ψ12 i = |(0, 1, 1)A (1, 1, −1)B i,

(II)

|Ψ8 i = |Ψ14 i = |(1, 0, 0)A (0, 1, 0)B i,

|Ψ3 i = |Ψ6 i = |(0, 1, −1)A (1, 1, 1)B i, |Ψ5 i = |Ψ10 i = |(0, 1, 0)A (1, 1, 0)B i, |Ψ7 i = |Ψ13 i = |(1, 0, 0)A (0, 0, 0)B i,

(II)

(II)

(II)

(II)

|Ψ10 i = |Ψ19 i = |(1, 1, 0)A (0, 0, 0)B i,

(II)

|Ψ12 i = |Ψ23 i = |(1, 1, 1)A (0, 1, −1)B i.

|Ψ9 i = |Ψ17 i = |(1, 1, −1)A (0, 1, 1)B i, |Ψ11 i = |Ψ20 i = |(1, 1, 0)A (0, 1, 0)B i,

(II)

(II)

(7.114) (II)

The Hamiltonian matrix H(Fz =0) of the subspace (II) reads (II)

H(Fz =0) =                      

L − 3H

0

0

0

0

0

0

0

−V

0

−2V

−V

0

L − 2H

0

0

0

0

0

0

V

−2V

0

−V

0

0

L+H

0

0

0

−V

V

2V

−V

V

0

0

0

0

L

0

0

0

−2V

−V

0

0

V

0

0

0

0

L+H

0

−2V

0

V

0

0

V

0

0

0

0

0

L+H

−V

−V

0

V

V

2V

0

0

−V

0

−2V

−V

L − 3H

0

0

0

0

0

0

0

V

−2V

0

−V

0

L

0

0

0

0

−V

V

2V

−V

V

0

0

0

L+H

0

0

0

0

−2V

−V

0

0

V

0

0

0

L − 2H

0

0

−2V

0

V

0

0

V

0

0

0

0

L+H

0

−V

−V

0

V

V

2V

0

0

0

0

0

L+H

           .          

(7.115)

196 (II)

It is interesting to note that if A(Fz =0) is the adjacency matrix corresponding to P12  (II) i P12  (I) i (II) H(Fz =0) , then the sum i=1 A(Fz =0) is identical to i=1 A(Fz =0) . Notice that, (II)

there are four degenerate subspaces in the H(Fz =0) . The two-fold degenerate level L−3H has vanishing off-diagonal elements. Thus, the degeneracy remains unresolved (II)

(II)

in the first order correction. The states |Ψ1 i and |Ψ7 i serve as eigenvectors. The energy level L − 2H is two-fold degenerate. The Hamiltonian matrix of this degenerate subspace is 



 L − 2H −2V  (A) H(Fz =0) =  , −2V L − 2H

(7.116)

which is spanned by (A)

(II)

|Ψ1 i = |Ψ2 i

and

(A)

(II)

|Ψ2 i = |Ψ10 i.

(7.117)

The eigenvalues and the eigenvectors are (A)

E± = L − 2H ± 2V,  1  (A) (A) (A) |χ± i = √ |Ψ1 i ± |Ψ2 i . 2

(7.118a) (7.118b)

The third degenerate subspace with doubly degenerate energy L is spanned by (B)

(II)

|Ψ1 i = |Ψ4 i

and

(B)

(II)

|Ψ2 i = |Ψ8 i.

(7.119)

The Hamiltonian matrix reads as   (B) H(Fz =0) = 

 L

−2V  . −2V L

(7.120)

197 The eigenvalues and the eigenvectors of the system are (B)

E± = L ± 2V,  1  (B) (B) (B) |χ± i = √ |Ψ1 i ± |Ψ2 i . 2

(7.121a) (7.121b)

The fourth degenerate subspace is spanned by the states (C)

(II)

|Ψ2 i = |Ψ5 i,

(I)

|Ψ4 i = |Ψ9 i,

(II)

|Ψ6 i = |Ψ12 i,

|Ψ1 i = |Ψ3 i, (C)

|Ψ3 i = |Ψ6 i, (C)

|Ψ5 i = |Ψ11 i,

(C)

(II)

(C)

(II)

(C)

(II)

(7.122)

with the 6-fold degenerate Hamiltonian matrix 

(C)

H(Fz =0)



0 0 2V V 0  H+L   0 H+L 0 V 0 V     0 0 H+L 0 V 2V =   2V V 0 H+L 0 0    V 0 V 0 H+L 0   0 V 2V 0 0 H+L

       .       

(7.123)

The eigenvalues of the Hamiltonian matrix (7.123) are (C)

E±,1 = L + H ± 2V, √  (C) E±,2 = L + H ± 3 + 1 V, √  (C) E±,3 = L + H ± 3 − 1 V.

(7.124a) (7.124b) (7.124c)

198 The degeneracy is completely removed and the energy shifts are first order in V. The corresponding normalized eigenvectors are  1 (C) (C) (C) (C) = ∓|Ψ1 i ± |Ψ3 i − |Ψ4 i + |Ψ6 i , (7.125a) 2  √  1 (C) (C) (C) (C) (C) |χ±,2 i = p ± |Ψ1 i ± 3 − 1 |Ψ2 i ± |Ψ3 i + |Ψ4 i √ 2 3− 3   √ (C) (C) 3 − 1 |Ψ5 i + Ψ6 i , (7.125b) +  √  1 (C) (C) (C) (C) (C) |χ±,3 i = p ∓ |Ψ1 i ± 3 + 1 |Ψ2 i ∓ |Ψ3 i + |Ψ4 i √ 2 3+ 3  √  (C) (C) − 3 + 1 |Ψ5 i + Ψ6 i . (7.125c)

(C) |χ±,1 i

As interatomic distance decreases, the unperturbed L−3H energy level experience an energy shift which is second order in V i.e. ∼ R−6 , whereas rest of other unperturbed energy levels experience R−3 type energy shift (see Figure 7.9). 7.7.4. Manifold Fz = −1. The Fz = −1 hyperfine manifold has 16 states. We write the 16 states in this manifolds in the ascending order of quantum numbers as given below:

|ψ10 i = |(0, 0, 0)A (0, 1, −1)B i,

|ψ20 i = |(0, 0, 0)A (1, 1, −1)B i,

|ψ30 i = |(0, 1, −1)A (0, 0, 0)B i,

|ψ40 i = |(0, 1, −1)A (0, 1, 0)B i,

|ψ50 i = |(0, 1, −1)A (1, 0, 0)B i,

|ψ60 i = |(0, 1, −1)A (1, 1, 0)B i,

|ψ70 i = |(0, 1, 0)A (0, 1, −1)B i,

|ψ80 i = |(0, 1, 0)A (1, 1, −1)B i,

|ψ90 i = |(1, 0, 0)A (0, 1, −1)B i,

0 |ψ10 i = |(1, 0, 0)A (1, 1, −1)B i,

0 |ψ11 i = |(1, 1, −1)A (0, 0, 0)B i,

0 |ψ12 i = |(1, 1, −1)A (0, 1, 0)B i,

0 |ψ13 i = |(1, 1, −1)A (1, 0, 0)B i,

0 |ψ14 i = |(1, 1, −1)A (1, 1, 0)B i,

0 |ψ15 i = |(1, 1, 0)A (0, 1, −1)B i,

0 |ψ16 i = |(1, 1, 0)A (1, 1, −1)B i.

(7.126)

199

Figure 7.9: Evolution of the energy levels as a function of interatomic separation R in the subspace (II) of Fz = 0 hyperfine manifold. The vertical axis is the energy divided by the plank constant, and the horizontal axis is the interatomic distance in the unit of Bohr’s radius a0 . The energy levels are asymptotic for large interatomic separation. Although at the large separation, there are six unperturbed energy levels, the degeneracy is removed in small separation and hence, the energy levels spread widely. We observe two level crossings for small atomic separation. The arrow, ‘ ↑ 0 , shows the location of crossings.

The Hamiltonian matrix for Fz = −1 hyperfine manifold is a square symmetric matrix of order 16. We replace all the nonzero off-diagonal element of the Hamiltonian matrix by 1 and all the diagonal elements by zero to construct corresponding undirected

200 adjacency matrix A(Fz =−1) which reads 

A(Fz =−1)

                        =                       



0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1   0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0     0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1    0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1    0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0     0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0    0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0    0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0   . 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0     0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0    0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0    0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0     1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0    1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0    0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0    1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

Figure 7.10 is an adjacency graph corresponding to A(Fz =−1) . The sum

(7.127)

P16

i=1

Ai(Fz =−1)

counts the number of neighbors of length(d) given by 1 ≤ d ≤ 16 which is shared by every pair of nodes. Interestingly, we find 16 X i=1

Ai(Fz =−1)

=

16 X

Ai(Fz =+1) .

(7.128)

i=1

Thus, similar to the H(Fz =+1) matrix, the H(Fz =−1) matrix can also be reduced into irreducible sub-matrices. The square of the adjacency matrix A(Fz =−1) can be expressed

201 12

8

2

11

14

5

9

15

7

6

1

13

3

10

(I)

16

4 (II)

(b) G(Fz =−1)

(b) G(Fz =−1)

Figure 7.10: An adjacency graph of the matrix A(Fz =−1) . The graph for (II) (I) A(Fz =−1) is disconnected having two components G(Fz =−1) and G(Fz =−1) which do not share any edges between the vertices.

as  A2(Fz =−1)

 B8×8 : 08×8  = : ··  ··  08×8 : C8×8

   ,  

(7.129)

where B8×8 and C8×8 are nonzero matrices of order 8 while 08×8 represents a null matrix of order 8. Eq. (7.129) confirms that the adjacency graph corresponding to the adjacency matrix A(Fz =−1) has two disconnected components. Each component (I)

(II)

G(Fz =−1) and G(Fz =−1) of the adjacency graph has 8 vertices (see Figure 7.10). The graph clearly indicates that the 16-dimensional Fz = −1 manifold can be decomposed into two subspaces each of dimension 8. These two subspaces are uncoupled to each other. The first subspace, subspace (I) of the manifold Fz = −1, is composed of |ψ20 i,

202 0 0 0 |ψ50 i, |ψ60 i, |ψ80 i, |ψ90 i, |ψ11 i, |ψ12 i, and |ψ15 i. We rename these states as below:

0(I)

0(I)

|ψ2 i = |ψ50 i = |(0, 1, −1)A (1, 0, 0)B i,

0(I)

|ψ4 i = |ψ80 i = |(0, 1, 0)A (1, 1, −1)B i,

0(I)

0 |ψ6 i = |ψ11 i = |(1, 1, −1)A (0, 0, 0)B i,

|ψ1 i = |ψ20 i = |(0, 0, 0)A (1, 1, −1)B i,

0(I)

|ψ3 i = |ψ60 i = |(0, 1, −1)A (1, 1, 0)B i,

0(I)

|ψ5 i = |ψ90 i = |(1, 0, 0)A (0, 1, −1)B i, 0(I)

0 |ψ7 i = |ψ12 i = |(1, 1, −1)A (0, 1, 0)B i,

0(I)

0 i = |(1, 1, 0)A (0, 1, −1)B i. |ψ8 i = |ψ15

(7.130) The Hamiltonian matrix of the subspace (I) reads 

(I)

H(Fz =−1)

          =          

L − 2H

0

0

0

V

0

V



2V

          .          

0

L

0

0

V

0

2V

V

0

0

L+H

0

V

2V

0

V

0

0

0

L + H 2V

V

V

0

0

V

V

2V

L

0

0

0

V

0

2V

V

0

L − 2H

0

0

V

2V

0

V

0

0

L+H

0

2V

V

V

0

0

0

0

L+H (7.131)

Notice that, there are three degenerate subspaces. The energy levels L−2H and L are doubly degenerate whereas the energy level L + H is four-fold degenerate. Consider 0(I)

0(A)

0(I)

0(A)

the subspace spanned by |ψ1 i ≡ |ψ1 i and |ψ6 i ≡ |ψ2 i. The Hamiltonian (A)

matrix H(Fz =−1) reads 



V  L − 2H  (A) H(Fz =−1) =  . V L − 2H

(7.132)

203 The eigenvalues and the corresponding eigenvectors of the Hamiltonian matrix (7.132) are (A)

E± = L − 2H ± V,  1  0(A) (A) 0(A) |χ± i = √ |ψ1 i ± |ψ2 i . 2 0(I)

(7.133a) (7.133b) 0(B)

0(I)

The doubly degenerate energy level L is spanned by |ψ2 i ≡ |ψ1 i and |ψ5 i ≡ 0(B)

(B)

|ψ2 i. The Hamiltonian matrix H(Fz =−1) is 



 L V  (B) H(Fz =−1) =  . V L

(7.134)

The eigenvalues and the corresponding eigenvectors of the Hamiltonian matrix (7.134) are (B)

E± = L ± V,  1  0(B) 0(B) (B) |χ± i = √ |ψ1 i ± |ψ2 i . 2

(7.135a) (7.135b)

The four-fold degenerate Hamiltonian matrix 

(C)

H(Fz =−1)

0 0 V  L+H   0 L+H V 0  =  0 V L+H 0   V 0 0 L+H

        

(7.136)

is spanned by the following vectors 0(I)

0(C)

|ψ3 i ≡ |ψ1 i,

0(I)

0(C)

|ψ4 i ≡ |ψ2 i,

0(I)

0(C)

|ψ7 i ≡ |ψ3 i,

0(I)

0(C)

|ψ8 i ≡ |ψ4 i. (7.137)

204 (C)

The Hamiltonian matrix H(Fz =−1) can again be decomposed into two identical 2 × 2 sub-matrices. 



V  L+H  (C),1 H(Fz =−1) =  , V L+H





V  L+H  (C),2 and H(Fz =−1) =  . V L+H (7.138) 0(C)

(C),1

0(C)

(C),2

The Hamiltonian matrix H(Fz =−1) is associated with |ψ1 i and |ψ4 i while H(Fz =−1) 0(C)

0(C)

is associated with |ψ2 i and |ψ3 i. The eigenvalues for both the matrix are given by (C)

E± = L + H ± V,

(7.139)

whereas the eigenvectors are given as  1  0(C) 0(B) √ |ψ1 i ± |ψ4 i , = 2  1  0(C) 0(B) (C) √ |ψ2 i ± |ψ3 i . |χ±,2 i = 2

(C) |χ±,1 i

(7.140a) (7.140b)

See Figure 7.11 for an evolution of energy levels as a function of interatomic separation, R, in the subspace (I) of the Fz = −1 hyperfine manifold. As the interatomic distance increases, each of the unperturbed energy levels experience a R−3 type energy shift. In contrast to the Fz = 0 hyperfine manifold, there is no level-crossing in the subspace (I) of the Fz = −1 hyperfine manifold. Notice that, energy curves for subspace (I) of Fz = ±1 are alike. The subspace (II) of the Fz = −1 manifold is spanned by |ψ10 i, |ψ30 i, |ψ40 i, |ψ70 i, 0 0 0 0 |ψ10 i, |ψ13 i, |ψ14 i, and |ψ16 i. Let us rename these state vectors as below:

0(II)

|ψ1 i = |ψ10 i = |(0, 0, 0)A (0, 1, −1)B i,

0(II)

|ψ2 i = |ψ30 i = |(0, 1, −1)A (0, 0, 0)B i,

205

Figure 7.11: Energy levels as a function of interatomic separation R in the subspace (I) of the Fz = −1 hyperfine manifold. For infinitely long interatomic separation, there are three distinct energy levels, as expected from (I) the unperturbed energy values of the Hamiltonian matrix, H(Fz =−1) , given by Eq. (7.131). However, for small interatomic separation, the energy levels split.

0(II)

|ψ3 i = |ψ40 i = |(0, 1, −1)A (0, 1, 0)B i,

0(II)

|ψ4 i = |ψ70 i = |(0, 1, 0)A (0, 1, −1)B i,

0(II)

0 |ψ6 i = |ψ13 i = |(1, 1, −1)A (1, 0, 0)B i,

0(II)

0 |ψ8 i = |ψ16 i = |(1, 1, 0)A (1, 1, −1)B i.

0 |ψ5 i = |ψ10 i = |(1, 0, 0)A (1, 1, −1)B i, 0 |ψ7 i = |ψ14 i = |(1, 1, −1)A (1, 1, 0)B i,

0(II)

0(II)

(7.141)

206 The Hamiltonian matrix of the subspace (II) reads (II)

H(Fz =−1) =  2L − 32 H 0    0 2L − 23 H    0 0     0 0    0 V    V 0    V 2V   2V V

0

3 H 2

0

0

V

V

2V

0

0

V

0

2V

V

+ 2L

0

V

2V

0

V

+ 2L

2V

V

V

0

0

3 H 2

V

2V

− 21 H

0

0

0

2V

V

0

− 21 H

0

0

0

V

0

0

1 H 2

0

V

0

0

0

0

1 H 2

            . (7.142)          

In this subspace, no two degenerate levels are coupled to each other. Thus, the diagonal elements serve as the eigenvalues, and the state vectors serve as the eigenvectors. An evolution of energy levels as a function of interatomic distance is presented in Figure 7.12. As interatomic distance decreases, energy levels experience the second order shift in V and evolve as R−6 type shift. Analysis shows very interesting feature in the comparison of the Fz = +1 and Fz = −1 manifolds. The components of the adjacency graph corresponding to the matrix A(Fz =+1) and A(Fz =−1) look identical though ordering of the vertices is not identical. Furthermore, the Hamiltonian matrix H(Fz =+1) is not exactly same to that of H(Fz =−1) . However, they do have the same eigenvalues. 7.7.5. Manifold Fz = −2. Similar to the Fz = +2 hyperfine manifold, the Fz = −2 manifold is also a 4-dimensional subspace. It is composed of |φ01 i = |(0, 1, −1)A (0, 1, −1)B i,

|φ02 i = |(0, 1, −1)A (1, 1, −1)B i,

|φ03 i = |(1, 1, −1)A (0, 1, −1)B i,

|φ04 i = |(1, 1, −1)A (1, 1, −1)B i.

(7.143)

207

Figure 7.12: Energy levels as a function of interatomic separation R in the subspace (II) of the Fz = −1 hyperfine manifold. For infinitely long interatomic separation, there are four distinct energy levels, as expected from the (II) unperturbed energy values of the Hamiltonian matrix, H(Fz =−1) , given by Eq. (7.142). However, for small interatomic separation, the energy levels split and deviate from the unperturbed values.

The Hamiltonian matrix of the Fz = −2 reads 

H(Fz =−2)

    =   

3 H 2

+ 2L 0 0

−2V



−2V   H + L −2V 0   . −2V H + L 0    1 0 0 H 2 0

0

(7.144)

208 This is same to that of the Fz = +2 manifold. The Hamiltonian matrix H(Fz =−2) can (II)

(I)

be decoupled into two 2 × 2 matrices H(Fz =−2) and H(Fz =−2) which read   (I) H(Fz =−2) = 





3H 2

+ 2L −2V   H −2V 2



 H + L −2V  (II) and H(Fz =−2) =   . (7.145) −2V H + L (I)

The subspace (I) with the Hamiltonian matrix H(Fz =−2) is spanned by 0(I)

0(I)

|φ1 i = |φ01 i = |(0, 1, −1)A (0, 1, −1)B i, and |φ2 i = |φ04 i = |(1, 1, −1)A (1, 1, −1)B i. (7.146) (I)

The eigenvalues of the Hamiltonian matrix H(Fz =−2) are (I)

E± = H + L ±

1p 16V 2 + (H + 2L)2 , 2

(7.147)

Or, V2 3 (II) + O(V 4 ), E+ = H + 2L + 4 2 H + 2L 2 1 V (I) E− = H − 4 + O(V 4 ), 2 H + 2L

(7.148a) (7.148b)

with the corresponding eigenvectors 0(I) |φ+ i = p 0(I) |φ− i

1



α12 + α22  1

=p 2 α1 + α22

 0(I) 0(I) α1 |φ1 i + α2 |φ2 i ,

0(I) α2 |φ1 i

−

0(I) α1 |φ2 i



,

(7.149a) (7.149b)

where α1 and α2 are given by p 16V 2 + (H + 2L)2 + H + 2L ≡ a1 , α1 = − 4V

(7.150a)

209 α 2 = 1 ≡ a2 .

(7.150b)

The subspace(II) is composed of 0(II)

0(II)

|φ1 i = |φ02 i = |(0, 1, −1)A (1, 1, −1)B i, and |φ2 i = |φ03 i = |(1, 1, −1)A (0, 1, −1)B i. (7.151) (II)

with the hamiltonian matrix H(Fz =−2) given in Eq. (7.145). The eigenenergies and (II)

eigenvectors of the Hamiltonian matrix H(Fz =−2) are given by (I)

E+ = H + L ± 2V,  1  0(II) 0(II) 0(II) |φ± i = |φ1 i ± |φ2 i . 2

(7.152) (7.153)

See Figure 7.13 for evolution of energy levels as a function of interatomic separation R in the Fz = −2 hyperfine manifold. Note that, the eigenvalues of the Fz = −2 manifold are identical to that of Fz = +2 and eigenvectors of one manifold can be acquired from the other one just by swapping |φj i ↔ |φ0j i.

7.8. REPUDIATION OF NON-CROSSING RULE The non-crossing theorem for a polyatomic system [71] says that for a system with N atoms, there will be 3N − 6 coupling parameters, as a result, level-crossing would occur, however, the number of level-crossing does not exceed 3N − 6, where N ≥ 2. For example, for a system containing three atoms, there are three coupling parameters. Thus the potential curves can have maximum three level-crossings. Similarly, a four-atom system can have maximum six level-crossings. On the other hand, a system containing two atoms has just one coupling parameter. In the long-range interaction, this coupling parameter is the interatomic distance R. Thus, the two

210

Figure 7.13: Energy levels as a function of interatomic separation R in the Fz = −2 hyperfine manifold. For large interatomic separation, there are three distinct energy levels. However, for small interatomic separation, the degenerate energy level L + H splits into two, and the level repulsion occurs.

atom system is supposed not to have any level-crossing, which requires no level crossings also in our system of two neutral hydrogen atoms both of them being in the first excited states. For Fz = ±2 hyperfine manifolds of the 2S-2S system, each of the irreducible subspaces is of dimension two. As expected from the non-crossing rule, we also do not see the level crossings in either of the four subspaces. In the Fz = ±1 hyperfine manifolds, each of irreducible subspaces is of dimension 8. There is no level crossing within the irreducible subspaces although some of the energy curves from different irreducible subspaces cross. Peculiar things happen in the Fz = 0 hyperfine subspace. In the subspace in which the atoms are in S-P or P -S configurations, we

211 witness two level crossings. On the other subspace in which the atoms are either in S-S or P -P configurations, several level crossings occur. In our work, we have employed an extended-precision arithmetic near the crossing point and confirmed that the crossing points are not due to the numerical insufficiency [72]. This finding confirms that the level crossings do present and are unavoidable, which indicates that the non-crossing theorem discussed in the literature so far does not hold true in higher dimensional quantum mechanical systems. Taking the example of water dimer, authors in Ref. [73] also have shown possibility of the curve crossings between two Born-Oppenheimer potential energy surfaces. Interestingly, they also found several curve crossings of potential energy surfaces. Their results also favor our findings. The following rewording seems appropriate: “A system with two energy levels follows non-crossing theorem. However, the higher-dimensional irreducible matrices do not always follow the non-crossing theorem”.

7.9. HYPERFINE SHIFT IN SPECIFIC SPECTATOR STATES In this section, we investigate the energy differences of 2S singlet and triplet hyperfine sub levels. The spectator can be in any arbitrary atomic state. We present detailed calculation of the Hamiltonian matrices, the normalized eigenvectors and the corresponding eigenvalues in all three possible hyperfine manifolds, viz. Fz = +1, Fz = 0 and Fz = −1. 7.9.1. Manifold Fz = +1. The atom A, in the following states (II)

(I)

|ψ1 i = |(0, 0, 0)A (0, 1, 1)B i and |ψ1 i = |(0, 0, 0)A (1, 1, 1)B i

(7.154)

is in the hyperfine singlet whereas the atom B is in the hyperfine triplet in the states (II)

(I)

|ψ2 i = |(0, 1, 0)A (0, 1, 1)B i and |ψ2 i = |(0, 1, 0)A (1, 1, 1)B i.

(7.155)

212 (II)

(II)

The spectator atom, i.e., the atom B is in 2S1/2 state in the transition |ψ1 i → |ψ2 i (I)

(I)

while the spectator atom B is in the 2P1/2 state in the transition |ψ1 i → |ψ2 i. Note (I)

(I)

(I)

that the state |ψ1 i is same to that of |ψ7 i = |(1, 1, 1)A (0, 0, 0)B i and the state |ψ2 i (I)

is also same to that of |ψ8 i = |(1, 1, 1)A (0, 1, 0)B i under the interchange of the (I)

(I)

subscripts A and B. Thus, the state |ψ1 i is energetically degenerate to |ψ7 i and (I)

(I)

the state |ψ2 i is energetically degenerate to |ψ8 i which are coupled with each other through the off-diagonal elements V. We have (I)

(I)

(7.156a)

(I)

(I)

(7.156b)

hψ1 |HvdW |ψ7 i = V, hψ2 |HvdW |ψ8 i = V.

Eqs. (7.156a) and (7.156b) tell us that, in the Fz = +1 manifold, if the spectator atom is at 2P1/2 -state, the hyperfine transition is linear to V. On the other hand, (II)

(II)

|ψ1 i and |ψ2 i are not coupled to any other energetically degenerate level. This implies that there is no first order vdW shift proportional to V. The absence of the (II)

(II)

first order shift does not guarantee that |ψ1 i and |ψ2 i are completely decoupled. Let us define the effective Hamiltonian Heff as

Heff =

() lim Heff →0

 = lim H1 · →0

1 E0,ψ − H0 + 

 · H1 ,

(7.157)

where H1 is the off-diagonal part of the Hamiltonian matrix of respective hyperfine manifold and E0,ψ is the energy corresponding to the reference state |ψi. We take the limit  → 0 at the end of the calculation. (II)

Interchanging the subscripts A and B in the state |ψ1 i, we get the state (II)

(II)

|ψ3 i = |(0, 1, 1)A (0, 0, 0)B i. This implies that the state |ψ1 i with energy L − 32 H is

213 (II)

energetically degenerate with respect to |ψ3 i. The Hamiltonian matrix H1,3 reads   H1,3 = lim  →0

(II) () (II) hψ1 |Heff |ψ1 i (II)

()

(II) () (II) hψ1 |Heff |ψ3 i

(II)

(II)

()

(II)

hψ3 |Heff |ψ1 i hψ3 |Heff |ψ3 i

  .

(7.158)

Let us now evaluate the elements of the matrix H1,3 .

(II) () (II) (H1,3 )11 = lim hψ1 |Heff |ψ1 i →0

=

(II) lim hψ1 |H1 →0

·

E0,ψ(II) 1

1 − H0 + 

! (II)

· H1 |ψ1 i. (7.159)

Let us introduce a completeness relation: X

|βihβ| = 1.

(7.160)

β

This is the so-called spectral decomposition of unity [74]. Using relation (7.160) in Eq. (7.159), we get

(H1,3 )11 = lim →0

X X (II) hψ1 |H1 |mihm| m



n

1

(II)

= lim hψ6 | →0

E0,ψ(II) 1

+

(II) hψ7 |

+

(II) hψ8 |

E0,ψ(II)

E0,ψ(II) 1

E0,ψ(II) 1

1 (II) |nihn|H1 |ψ1 i − H0 + 

1 (II) (II) (II) (II) (II) |ψ6 ihψ1 |H1 |ψ6 ihψ6 |H1 |ψ1 i − H0 + 

1 (II) (II) (II) (II) (II) |ψ7 ihψ1 |H1 |ψ7 ihψ7 |H1 |ψ1 i − H0 +   1 (II) (II) (II) (II) (II) |ψ ihψ1 |H1 |ψ8 ihψ8 |H1 |ψ1 i − H0 +  8

4V 2 V2 V2 = + + 2L − 32 H − 21 H 2L − 23 H + 12 H 2L − 32 H − 21 H V2 5V 2 = + . 2(L − H) 2L − H

(7.161)

214

(H1,3 )12 = lim →0

(II) () (II) hψ1 |Heff |ψ3 i

= lim →0

(II) hψ1 |H1

·

E0,ψ(II) 1

= lim →0

XX m

(II) hψ1 |H1 |mihm|

n

 (II) = lim hψ1 | →0

1

E0,ψ(II) 1

1 (II) (II) (II) (II) (II) |ψ ihψ1 |H1 |ψ1 ihψ1 |H1 |ψ3 i − H0 +  1

E0,ψ(II) E0,ψ(II)

1 (II) (II) (II) (II) (II) |ψ ihψ1 |H1 |ψ8 ihψ8 |H1 |ψ3 i − H0 +  8

1

+

1 (II) |nihn|H1 |ψ3 i − H0 + 

1 (II) (II) (II) (II) (II) |ψ ihψ1 |H1 |ψ7 ihψ7 |H1 |ψ3 i − H0 +  7

(II)

(II) hψ8 |

(II)

· H1 |ψ3 i

E0,ψ(II) 1

+ hψ7 |

!

1 (II) (II) (II) (II) (II) |ψ ihψ1 |H1 |ψ6 ihψ6 |H1 |ψ3 i − H0 +  6

(II)

+ hψ6 |

E0,ψ(II)

1 − H0 + 

1



(−2V)(−V) (−V)(−2V) =0 + +0+ 3 1 2L − 2 H − 2 H 2L − 32 H − 21 H 2V 2 . = L−H

(7.162) (II)

(II)

To obtain the second last line of Eq. (7.162), we substituted the values hψ1 |H1 |ψ3 i = (II)

(II)

(II)

0, hψ7 |H1 |ψ3 i = 0 and then we took  = 0. The Hamiltonian matrix H(Fz =+1) is symmetric. Thus, we have

(H1,3 )12 = (H1,3 )21 .

(7.163)

Similarly, (II)

()

(II)

(II)

(H1,3 )22 = lim hψ3 |Heff |ψ3 i = lim hψ3 |H1 · →0

→0

E0,ψ(II) 3

= lim →0

XX m

n

 (II) = lim hψ3 | →0

(II)

hψ3 |H1 |mihm|

3

E0,ψ(II) 3

+

(II) hψ5 |

E0,ψ(II)

E0,ψ(II) 3

1 − H0 + 

! (II)

· H1 |ψ3 i

1 (II) |nihn|H1 |ψ3 i − H0 + 

1 (II) (II) (II) (II) (II) |ψ3 ihψ3 |H1 |ψ3 ihψ3 |H1 |ψ3 i − H0 + 

1 (II) (II) (II) (II) (II) |ψ5 ihψ3 |H1 |ψ5 ihψ5 |H1 |ψ3 i − H0 + 

215

E0,ψ(II)

1 (II) (II) (II) (II) (II) |ψ6 ihψ3 |H1 |ψ6 ihψ6 |H1 |ψ3 i − H0 + 

E0,ψ(II)

1 (II) (II) (II) (II) (II) |ψ8 ihψ3 |H1 |ψ8 ihψ8 |H1 |ψ3 i − H0 + 

(II)

+ hψ6 |

3

+

(II) hψ8 |

3

V2 V2 4V 2 + + = 2L − 32 H + 12 H 2L − 23 H − 12 H 2L − 32 H − 21 H V2 5V 2 + . = 2(L − H) 2L − H



(7.164)

The matrix H1,3 given in Eq. (7.158) reads   H1,3 = 

5V 2 2(L−H)

+

V2 2L−H



2V 2 L−H

2V 2

5V 2

L−H

2(L−H)

+

 .

V2

(7.165)

2L−H

The matrix (7.165) has the following eigenvalues and eigenvectors: V2 2V 2 5V 2 + ± , 2(L − H) 2L − H L − H  1  (II) (II)± (II) |ψ1,3 i = √ |ψ1 i ± |ψ3 i . 2

± E1,3 =

(7.166a) (7.166b) (II)

(II)

Let us now discuss the reference state |ψ2 i. The state |ψ2 i is degenerate (II)

with the state |ψ4 i. The unperturbed energy corresponding to these states is 2L + 3 H. 2

The Hamiltonian matrix H2,4 is given by   H2,4 = lim  →0

(II) () (II) hψ2 |Heff |ψ2 i (II)

()

(II) () (II) hψ2 |Heff |ψ4 i

(II)

(II)

()

(II)

hψ4 |Heff |ψ2 i hψ4 |Heff |ψ4 i

  .

(7.167)

The first diagonal element (H2,4 )11 is given by (II)

()

(II)

(II)

(H2,4 )11 = lim hψ2 |Heff |ψ2 i = lim hψ2 |H1 · →0

→0

E0,ψ(II) 2

= lim →0

XX m

n

(II)

hψ2 |H1 |mihm|

E0,ψ(II) 2

1 − H0 + 

!

1 (II) |nihn|H1 |ψ2 i − H0 + 

(II)

· H1 |ψ2 i

216 = lim →0

X

(II)

hψk |

k=5,7,8 2

E0,ψ(II) 2

1 (II) (II) (II) (II) (II) |ψk ihψ2 |H1 |ψk ihψk |H1 |ψ2 i − H0 + 

5V V2 = + . 2(L + H) 2L + H

(7.168)

The next diagonal element (H2,4 )22 is given by

(H2,4 )22 = lim →0

(II) () (II) hψ4 |Heff |ψ4 i

= lim →0

(II) hψ4 |H1

·

E0,ψ(II) 4

X

(II) = lim hψk | →0 E0,ψ(II) k=5,6,7 4 2 2

=

1 − H0 + 

! (II)

· H1 |ψ4 i

1 (II) (II) (II) (II) (II) |ψk ihψ4 |H1 |ψk ihψk |H1 |ψ4 i − H0 + 

5V V + . 2(L + H) 2L + H

(7.169)

The off-diagonal elements are given by (II)

()

(II)

(II)

(H2,4 )12 = lim hψ2 |Heff |ψ4 i = lim hψ2 |H1 · →0

→0

E0,ψ(II) 2

= lim →0

=

X

(II)

hψk |

k=5,7 2

E0,ψ(II) 2

1 − H0 + 

! (II)

· H1 |ψ4 i

1 (II) (II) (II) (II) (II) |ψ ihψ2 |H1 |ψk ihψk |H1 |ψ4 i − H0 +  k

2V = (H2,4 )21 . L+H

(7.170)

The Hamiltonian matrix H2,4 thus reads   H2,4 = 

5V 2 2(L+H)

+

V2 2L+H



2V 2 L+H

2V 2

5V 2

L+H

2(L+H)

+

V2

 

(7.171)

2L+H

which has the eigenvalues

± E2,4

5V 2 V2 2V 2 = + ± 2(L + H) 2L + H L + H

(7.172)

217 with eigenvectors (II)± |ψ2,4 i

 1  (II) (II) √ = |ψ2 i ± |ψ4 i . 2

(7.173)

The first order vdW shift V is proportional to R−3 , where R is the interatomic distance. This clearly indicates that the transition energies are R-dependent. In the (II)

(II)

|ψ1 i → |ψ2 i transition, the energy difference between the symmetric superposi    (II) (II) (II) (II) tions √12 |ψ1 i + |ψ3 i and √12 |ψ2 i + |ψ4 i and the energy difference between     (II) (II) (II) (II) the antisymmetric superposition √12 |ψ1 i − |ψ3 i and √12 |ψ2 i − |ψ4 i are (I)

(I)

given in the Table 7.1. In the |ψ1 i → |ψ2 i transition, the energy difference be(+)

Table 7.1: The energy differences between the symmetric superposition ∆EII (−) and the antisymmetric superposition ∆EII in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2S1/2 state. R ∞ 750 a0 500 a0 250 a0

(+)

∆EII 0 -0.007 01 0.383 31 37.042 26

0 -0.00133 0.043 94 22.835 56

 (I) (I) |ψ2 i + |ψ8 i and   (I) (I) the energy difference between the antisymmetric superposition √12 |ψ1 i − |ψ7 i   (I) (I) and √12 |ψ2 i − |ψ8 i are given in the Table 7.2.

tween the symmetric superpositions

√1 2



(−)

∆EII

 (I) (I) |ψ1 i + |ψ7 i and

√1 2



7.9.2. Manifold Fz = 0. The 24-dimensional Fz = 0 hyperfine manifold has 4 states having atom A in the 2S singlet level as given below: (I)

|Ψ1 i = |(0, 0, 0)A (0, 0, 0)B i, (II)

|Ψ1 i = |(0, 0, 0)A (1, 0, 0)B i,

(I)

|Ψ2 i = |(0, 0, 0)A (0, 1, 0)B i, (II)

|Ψ2 i = |(0, 0, 0)A (1, 1, 0)B i.

(7.174)

218 (+)

Table 7.2: The energy differences between the symmetric superposition ∆EI (−) and the antisymmetric superposition ∆EI in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2P1/2 state. (+)

R ∞ 750 a0 500 a0 250 a0

∆EI 0 2.637 62 13.267 84 125.041 24

(−)

∆EI

0 2.817 36 13.322 29 125.042 34

The atom A is in the hyperfine triplet in the following 4 states: (I)

|Ψ4 i = |(0, 1, 0)A (0, 0, 0)B i, (II)

|Ψ4 i = |(0, 1, 0)A (1, 0, 0)B i,

(I)

|Ψ5 i = |(0, 1, 0)A (0, 1, 0)B i, (II)

|Ψ5 i = |(0, 1, 0)A (1, 1, 0)B i.

(7.175)

The spectator atom B is in the state |2S1/2 i, which is preserved in the transitions (I)

(I)

(I)

(I)

|Ψ1 i → |Ψ4 i and |Ψ2 i → |Ψ5 i whereas the state |2P1/2 i of the spectator atom B (II)

(II)

(II)

(II)

(II)

is preserved in the transitions |Ψ1 i → |Ψ4 i and |Ψ2 i → |Ψ5 i. The state |Ψ4 i (II)

is energetically degenerate with |Ψ8 i and coupled each other by the first order vdW interaction −2V, i.e., (II)

(II)

hΨ4 |HvdW |Ψ8 i = −2V. (II)

(7.176) (II)

(II)

Same thing happens for |Ψ2 i which is degenerate to |Ψ10 i and |Ψ5 i which is (II)

(II)

degenerate to |Ψ9 i and |Ψ12 i, i.e., (II)

(II)

(II)

(II)

hΨ2 |HvdW |Ψ10 i = −2V, (II)

(7.177) (II)

hΨ5 |HvdW |Ψ9 i = hΨ5 |HvdW |Ψ12 i = −2V.

(7.178)

219 (I)

(I)

(I)

Interesting thing happens in the transition |Ψ1 i → |Ψ4 i as the state |Ψ1 i is nondegenerate. Here is the detailed calculation of the energy ∆EΨ(I) . 1

∆EΨ(I) = lim →0

1

(I) () (I) hΨ1 |Heff |Ψ1 i

= lim →0

(I) hΨ1 |H1

·

E0,Ψ(I) 1

= lim →0

= lim →0

XX m

(I) hΨ1 |H1 |mihm|

n

E0,Ψ(I) 1

X

(I) hΨk |

k=9,11,12 2

E0,Ψ(I) 1

1 − H0 + 

! (I)

· H1 |Ψ1 i

1 (I) |nihn|H1 |Ψ1 i − H0 + 

1 (I) (I) (I) (I) (I) |Ψk ihΨ1 |H1 |Ψk ihΨk |H1 |Ψ1 i − H0 + 

(−V) (−2V)2 (−V)2 = + + 2L − 29 H − 12 H 2L − 92 H − 12 H 2L − 92 H − 21 H 6V 2 . = 2L − 5H (I)

(7.179)

(I)

The state |Ψ4 i is energetically degenerate to the |Ψ2 i. However, the degenerate states are not coupled directly. So, the first order vdW shift is absent. We want to determine the matrix H2,4 given by   H2,4 = lim  →0

() (I) (I) hΨ2 |Heff |Ψ2 i (I)

()

() (I) (I) hΨ2 |Heff |Ψ4 i

(I)

(I)

()

(I)

hΨ4 |Heff |Ψ2 i hΨ4 |Heff |Ψ4 i

  .

(7.180)

Here, we have used a new symbol H to denote the Hamiltonian matrix H2,4 instead of H2,4 just to distinguish the matrix (7.180) from (7.167). The new symbol does not carry new physical meaning. Following the same procedure which we applied to calculate H2,4 in the subspace (II) of the Fz = +1 manifold, we can easily the calculate the Hamiltonian matrix H2,4 in the subspace (I) of the Fz = 0 manifold which yields   H2,4 = 

V2 L−H

+

V2 −L+H

4V 2 2L−H

V2

V2

−L+H

L−H

+

4V 2 2L−H

  .

(7.181)

220 The eigenvalues and the eigenvectors corresponding to the Hamiltonian matrix H2,4 are V2 4V 2 V2 = + ± , L − H 2L − H −L + H  1  (I) (I) √ i = |Ψ± |Ψ i ± |Ψ i . 2 4 2,4 2

E± 2,4

(7.182a) (7.182b)

(I)

Let us now analyze the state |Ψ5 i corresponding to the unperturbed energy 2L+ 32 H, (I)

(I)

which is degenerate to the states |Ψ3 i and |Ψ6 i. The Hamiltonian matrix H3,5,6 is given by  H3,5,6

() (I) (I) hΨ3 |Heff |Ψ5 i

() (I) (I) hΨ3 |Heff |Ψ3 i

() (I) (I) hΨ3 |Heff |Ψ6 i

  (I) () (I) (I) () (I) (I) () (I) = lim  hΨ5 |Heff |Ψ3 i hΨ5 |Heff |Ψ5 i hΨ5 |Heff |Ψ6 i →0   (I) () (I) (I) () (I) (I) () (I) hΨ6 |Heff |Ψ3 i hΨ6 |Heff |Ψ5 i hΨ6 |Heff |Ψ6 i

   .  

(7.183)

The elements of the matrix H3,5,6 can be calculated in a similar way to H1,3 in the subspace (II) of the Fz = +1 manifold. We have,

(H3,5,6 )11 = lim →0

(I) hΨ3 |H1

·

E0,Ψ(I) 3

X

= lim →0

=

1 − H0 + 

(I)

hΨk |

k=7,8,9,10,11 2

E0,Ψ(I) 3

! (I)

· H1 |Ψ3 i

1 (I) (I) (I) (I) (I) |Ψ ihΨ3 |H1 |Ψk ihΨk |H1 |Ψ3 i − H0 +  k

2

V V 5V 2 + + . 2L + 3H L + H 2L + H

(I)

(H3,5,6 )12 = lim hΨ3 |H1 · →0

E0,Ψ(I) 3

= lim →0

=

X

(I)

hΨk |

k=7,9 2

E0,Ψ(I) 3

1 − H0 + 

(7.184)

! (I)

· H1 |Ψ5 i

1 (I) (I) (I) (I) (I) |Ψ ihΨ3 |H1 |Ψk ihΨk |H1 |Ψ5 i − H0 +  k

2

2V 2V + = (H3,5,6 )21 . 2L + 3H 2L + H

(7.185)

221

(H3,5,6 )13 = lim →0

(I) hΨ3 |H1

·

E0,Ψ(I) 3

1 − H0 + 

X

(I) = lim hΨk | →0 E0,Ψ(I) k=7,8,10,11 3 2 2

=

! (I)

· H1 |Ψ6 i

1 (I) (I) (I) (I) (I) |Ψk ihΨ3 |H1 |Ψk ihΨk |H1 |Ψ6 i − H0 + 

V V V2 V2 − − + 2(L + H) 2(L + H) 2L + H 2L + H

=0 = (H3,5,6 )31 .

(H3,5,6 )22 = lim →0

(I) hΨ5 |H1

(7.186)

·

E0,Ψ(I) E0,Ψ(I)

1 (I) (I) (I) (I) (I) |Ψk ihΨ5 |H1 |Ψk ihΨk |H1 |Ψ5 i − H0 + 

5

= lim →0

=

X

(I)

hΨk |

k=7,9,12 2

5

→0

2

(I) hΨ5 |H1

·

(7.187)

E0,Ψ(I) 5

X

(I) = lim hΨk | →0 E0,Ψ(I) k=7,9,12 5 2 2

1 − H0 + 

! (I)

· H1 |Ψ6 i

1 (I) (I) (I) (I) (I) |Ψk ihΨ5 |H1 |Ψk ihΨk |H1 |Ψ6 i − H0 + 

2V 2V + = (H3,5,6 )32 . 2L + 3H 2L + H

(I)

(H3,5,6 )33 = lim hΨ6 |H1 · →0

E0,Ψ(I) 6

= lim →0

=

(I)

· H1 |Ψ5 i

4V 2V + . 2L + 3H 2L + H

(H3,5,6 )23 = lim

=

!

1 − H0 + 

X

(I)

hΨk |

k=7,8,10,11,12 2 2

1 − H0 + 

E0,Ψ(I) 6

(7.188)

! (I)

· H1 |Ψ6 i

1 (I) (I) (I) (I) (I) |Ψ ihΨ6 |H1 |Ψk ihΨk |H1 |Ψ6 i − H0 +  k

V 5V 2 V + + . 2L + 3H L + H 2L + H

(7.189)

222 The matrix H3,5,6 takes the following form  H3,5,6

  =  

V2 2L+3H

+

V2 L+H

+

+

2V 2

4V 2

2L+H

2L+3H

2V 2 2L+3H

5V 2 2L+H

2V 2 2L+3H

2V 2

0

2L+3H



+

2V 2 2L+H

+

2V 2

2V 2

2L+H

2L+3H

+

0

2V 2

V2

2L+H

2L+3H

+

+

2V 2 2L+H

V2 L+H

+

5V 2 2L+H

  .   (7.190)

If we apply the additional approximation H  L in Eq. (7.190), the matrix H3,5,6 reduces to the following simpler form:  H3,5,6

  ≈  

4V 2 L

2V 2 L

2V 2 L

3V 2 L

0

2V 2 L

 0   4V 2  . L  

(7.191)

4V 2 L

The eigenvalues of the the matrix (7.191) are (1) E3,5,6 (2)

=

7+

√
2 33 V , 2L

4V 2 , L √
7 − 33 V 2 = , 2L

E3,5,6 = (3)

E3,5,6

(7.192a) (7.192b) (7.192c)

with the corresponding eigenvectors   √ 1 (I) (I) (I) q 4|Ψ3 i + ( 33 − 1)|Ψ5 i + 4|Ψ6 i , = √ 2(33 − 33)  1  (I) (2) (I) |Ψ3,5,6 i = − √ |Ψ3 i − |Ψ6 i , 2   √ 1 (3) (I) (I) (I) |Ψ3,5,6 i = q i − ( i + 4|Ψ i . 4|Ψ 33 + 1)|Ψ 3 5 6 √ 2(33 + 33) (1) |Ψ3,5,6 i

(7.193a)

(7.193b) (7.193c)

223 (I)

(I)

In the |Ψ1 i → |Ψ4 i transition, the energy difference between the symmetric super    (I) (I) (I) (I) 1 1 √ √ positions 2 |Ψ1 i + |Ψ2 i and 2 |Ψ1 i + |Ψ4 i and the energy difference be    (I) (I) (I) (I) 1 1 √ √ tween the antisymmetric superpositions 2 |Ψ1 i − |Ψ2 i and 2 |Ψ1 i − |Ψ4 i (II)

(II)

are given in the Table 7.3). In the |Ψ1 i → |Ψ4 i transition, the energy difference (+)

Table 7.3: The energy differences between the symmetric superposition ∆EI (−) and the antisymmetric superposition ∆EI in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2S1/2 state. R ∞ 750 a0 500 a0 250 a0

(+)

(−)

∆EI 0 0.054 21 -0.249 93 21.732 18

between the symmetric superpositions

√1 2

∆EI

0 0.018 16 -0.077 32 -2.882 35

  (II) (II) |Ψ1 i + |Ψ7 i and

√1 2

and the energy difference between the antisymmetric superposition   (I) (I) 1 √ and 2 |Ψ4 i − |Ψ8 i are given in the Table 7.4.



√1 2

(II) |Ψ4 i



+

(I) |Ψ1 i



(I) |Ψ8 i

−

(I) |Ψ7 i



(+)

Table 7.4: The energy differences between the symmetric superposition ∆EII (−) and the antisymmetric superposition ∆EII in the unit of the hyperfine splitting constant H. In this transition, the spectator atom is in the 2P1/2 state. R ∞ 750 a0 500 a0 250 a0

(+)

∆EII 0 -1.586 70 -2.571 28 -2.938 37

(−)

∆EII

0 2.186 39 12.317 98 38.704 42

7.9.3. Manifold Fz = −1. The difference in the Hamiltonian matrix between the Fz = +1 and Fz = −1 manifolds tells us that we need a detailed analysis of the Fz = −1 manifold as well. We have the following two states, in which the atom A is in the hyperfine singlet 0(I)

0(II)

|ψ1 i = |(0, 0, 0)A (1, 1, −1)B i and |ψ1 i = |(0, 0, 0)A (0, 1, −1)B i

(7.194)

224 whereas the atom A is in the hyperfine triplet in the states 0(I)

0(II)

|ψ4 i = |(0, 1, 0)A (1, 1, −1)B i and |ψ4 i = |(0, 1, 0)A (0, 1, −1)B i.

(7.195)

0(II)

0(II)

0(I)

0(I)

The spectator atom, i.e., the atom B is in 2S1/2 state in the transition |ψ1 i → |ψ4 i while the spectator atom B is in the 2P1/2 state in the transition |ψ1 i → |ψ4 i. 0(I)

0(I)

0(I)

Interchanging the subscripts A and B of the states |ψ1 i and |ψ4 i we get |ψ6 i = 0(I)

0(I)

|(1, 1, −1)A (0, 1, 0)B i and |ψ7 i = |(1, 1, −1)A (0, 1, 0)B i. Thus, the state |ψ1 i is 0(I)

0(I)

energetically degenerate to |ψ6 i and the state |ψ4 i is energetically degenerate to 0(I)

|ψ7 i. These states are coupled with each other through the off-diagonal elements V indicating that the interaction energy is proportional to R−3 . We have 0(I)

0(I)

(7.196a)

0(I)

0(I)

(7.196b)

hψ1 |HvdW |ψ6 i = V, hψ4 |HvdW |ψ7 i = V. 0(II)

0(II)

The state |ψ1 i and |ψ4 i are not coupled to any other energetically degenerate levels which implies that the first order vdW shift is absent. Hence we expect the leading order shift to be of second order in V. Exchanging the subscripts A and B, 0(II)

0(II)

0 in the state |ψ1 i, we get, |ψ2 i. We now calculate the Hamiltonian H1,2 as we did

for H1,3 in Fz = +1 manifold.   0 H1,2 = lim  →0

0(II) () 0(II) hψ1 |Heff |ψ1 i 0(II)

0(II)

()

0(II) () 0(II) hψ1 |Heff |ψ2 i 0(II)

()

0(II)

hψ2 |Heff |ψ1 i hψ2 |Heff |ψ2 i

  .

(7.197)

where ()

Heff = H1 ·

E0,ψ0(II) 1

1 − H0 + 

! · H1 .

(7.198)

225 0 The elements of the matrix H1,2 can be calculated as we did for H1,3 , which yields

  0 H1,2 =

5V 2 2(L−H)

+

V2 2L−H

5V 2 2(L−H)

2V 2 L−H



2V 2 L−H

+

 .

V2 2L−H

(7.199)

This is same to that of matrix H1,3 . The unperturbed energy corresponding to the 0(II)

0(II)

states |ψ1 i and |ψ2 i is 2L− 32 H. The matrix (7.199) has the following eigenvalues and eigenvectors: V2 2V 2 5V 2 + ± , 2(L − H) 2L − H L − H  1  0(II) 0(II)± 0(II) |ψ1,2 i = √ |ψ1 i ± |ψ2 i . 2

0± E1,2 =

0(II)

(7.200a) (7.200b)

0(II)

Let us now turn to the reference state |ψ4 i. The state |ψ4 i is degenerate with 0(II)

0 the state |ψ3 i. The Hamiltonian matrix H4,3 is found to be identical to H2,4 which

reads   0 H4,3 =

5V 2 2(L+H)

+

V2 2L+H



2V 2 L+H

2V 2

5V 2

L+H

2(L+H)

+

V2

(7.201)

 

2L+H

with eigenvalues

0± E4,3 =

5V 2 V2 2V 2 + ± 2(L + H) 2L + H L + H

(7.202)

and eigenvectors  1  0(II) 0(II)± 0(II) |ψ4,3 i = √ |ψ4 i ± |ψ3 i . 2 In the and

0(II) |ψ1 i

√1 2



→

0(II) |ψ4 i

(7.203)

√1 2



0(II) |ψ1 i



0(II) |ψ2 i

transition, the symmetric superpositions are +      0(II) 0(II) (II) (II) 0(II) 0(II) 1 1 √ √ |ψ4 i + |ψ3 i whereas 2 |ψ1 i − |ψ2 i and 2 |ψ4 i − |ψ3 i are

226 the antisymmetric superpositions. The energy differences between the symmetric lev0(+)

els ∆EII

0(−)

and the antisymmetric levels ∆EII

can be read from the Table 7.2 with

the substitutions (II)

0(II)

0(II)

(II)

(II)

0(II)

(II)

0(II)

|ψ1 i → |ψ1 i, |ψ3 i → |ψ2 i, |ψ2 i → |ψ4 i, |ψ4 i → |ψ3 i. 0(I)

(7.204)

0(I)

Similarly, in the case of |ψ1 i → |ψ4 i transition, the energy differences between     0(I) 0(I) 0(I) 0(II) the symmetric superposition √12 |ψ1 i + |ψ6 i and √12 |ψ4 i + |ψ7 i as well as     0(I) 0(I) 0(I) 0(II) the antisymmetric superposition √12 |ψ1 i − |ψ6 i and √12 |ψ4 i − |ψ7 i can be read from Table 7.1 with the substitutions (I)

0(I)

(I)

0(I)

(I)

0(I)

(II)

0(II)

|ψ1 i → |ψ1 i, |ψ7 i → |ψ6 i, |ψ2 i → |ψ4 i, |ψ8 i → |ψ7 i.

(7.205)

227 8. LONG-RANGE INTERACTION IN nS-1S SYSTEMS

8.1. DIRECT INTERACTION ENERGY IN THE vdW RANGE In this Section , we concentrate on the interaction between two hydrogen atom in which one of them is in the ground state, and the other one is in the higher excited state of the atom. In such a system, an extra contribution to the interaction energy naturally arises as the Wick-rotated contour enclosed poles. The source of the poles is the low-lying virtual states of the reference atom available by a dipole transition. We here focus on the nS-1S system with n = 3, 4, 5. We refer to Ref. [75] for a detailed analysis of nS-1S systems for 3 ≤ n ≤ 12. The 1S-state is a nondegenerate state while the nS-state has nP -states as its quasi-degenerate neighbors. The state corresponding to the |nP1/2 i is shifted from the nS-state by the Lamb shift Ln and the state corresponding to the |nP3/2 i is shifted from the nS-state by the fine structure Fn i.e.

E(nS1/2 ) − E(nP1/2 ) ≡ Ln ,

(8.1a)

E(nP3/2 ) − E(nS1/2 ) ≡ Fn .

(8.1b)

The Lamb shift and the fine structure of hydrogen for n = 3, 4, and 5 can be found in Refs. [76; 77; 78]. In the units of Hartree energy, Eh , Ln and Fn , for 3 ≤ n ≤ 5, are given as

L3 = 4.78 × 10−8 Eh ,

F3 = 4.46 × 10−7 Eh ,

(8.2a)

L4 = 2.02 × 10−8 Eh ,

F4 = 1.88 × 10−7 Eh ,

(8.2b)

L5 = 9.82 × 10−9 Eh ,

F5 = 9.45 × 10−8 Eh .

(8.2c)

228 A few more values of Ln and Fn can be found in Refs. [6; 51]. Notice that, Fn ≈ 10 Ln . Furthermore, both the Lamb shift, Ln and the fine structure splittings, Fn decreases approximately as 1/n3 as the principal quantum number n increases. As oscillator strength of |nP1/2 i and |nP3/2 i states with respect to the nS 1 3

state are distributed in a ratio

÷ 23 , the matrix element P nS (ω) is given by

3 3 e2 X X |hn, 0, 0|xi |n, `, mi|2 2e2 X X |hn, 0, 0|xi |n, `, mi|2 P nS (ω) = + 9 i=1 µ −Ln + ~ω − i 9 i=1 µ Fn + ~ω − i   3 1 2 e2 X X i 2 |hn, 0, 0|x |n, `, mi| + . (8.3) = 9 i=1 µ −Ln + ~ω − i Fn + ~ω − i

The polarizability to the nS state αnS (ω) is the sum of the matrix elements PnS (ω) and PnS (−ω), thus the Wick-rotated form of the degenerate polarizability αnS (iω) can be written as

αnS (iω) = P nS (iω) + P nS (−iω)   3 e2 X X −2Ln 4Fn i 2 = |hnS|x |nP (m = µ)i| + . 9 i=1 µ (−Ln − i)2 + (~ω)2 (Fn − i)2 + (~ω)2 (8.4) We substitute Eq. (8.4) in the following expression  direct

W nS;1S (R) = −

3~  π(4π0 )2 R6

Z∞

 dω αnS (iω)α1S (iω) ,

(8.5)

0

to determine the degenerate contribution to the interaction energy in vdW range, direct

W nS;1S (R). Namely, direct

W nS;1S (R) = −

3 X X e2 ~ |hnS|xi |nP (m = µ)i|2 2 6 3 π(4π0 ) R i=1 µ

229 Z∞ ×



 −2Ln 4Fn dω + α1S (iω) (−Ln − i)2 + (~ω)2 (Fn − i)2 + (~ω)2

0

  3 ~ e2 X X π 2π i 2 |hnS|x |nP (m = µ)i| + =− α1S (0) 3 i=1 µ π(4π0 )2 R6 ~ ~  2 3 X X e2 ~ e2 9 i 2 =− |hnS|x |nP (m = µ)i| (4π0 )2 R6 2 αmc α2 mc2 i=1 µ  2  4 3 e2 9 XX ~ i 2 2 1 =− |hnS|x |nP (m = µ)i| mc 6 2 i=1 µ 4π0 ~c R αmc 3

9 XX a4 =− |hnS|xi |nP (m = µ)i|2 Eh 06 . 2 i=1 µ R

(8.6)

Here, we have used α = e2 /(4π0 ~c) and a0 = ~/(αmc). The nondegenerate confnS;1S (R) arising due to the virtual kP states, where tribution to vdW interaction W k ≥ n, can be calculated numerically using 3~ fdirect (R) = − W nS;1S π(4π0 )2 R6

Z∞ dω α enS (iω)α1S (iω).

(8.7)

0

Then, we get the Wick-rotated contribution as the sum direct direct direct fnS;1S WnS;1S (R) = W nS;1S (R) + W (R).

(8.8)

direct In the short range limit, the direct pole term, PnS;1S (R), is given by

direct PnS;1S (R)

  X EmP − EnS X 2e2 α1S ω = |hnS|xi |mP i|2 . =− 2 6 (4π0 ) R m ~ i

(8.9)

The pole term also follows the R−6 power law in the vdW range. 8.1.1. 3S-1S System. For the 3S-1S system, we have 3 X X i=1

µ

|h3S|xi |3P (m = µ)i|2 = 162a20 .

(8.10)

230 direct

Thus, the degenerate contribution W 3S;1S (R) reads direct W 3S;1S (R)

 a 6 a40 9 0 2 = − × 162 a0 Eh 6 = −729 Eh . 2 R R

(8.11)

fdirect (R) is calculated numerically using The nondegenerate contribution W 3S;1S 3~ fdirect (R) = − W 3S;1S π(4π0 )2 R6

Z∞ dω α e3S (iω)α1S (iω).

(8.12)

0

As usual, α e3S (iω) is the sum α e3S (iω) = Pe3S (iω)+ Pe3S (−iω), where the matrix element −1/2

Q3S (±iω) in terms of frequency, can be acquired replacing t = (1 ± i18~ω/(α2 mc2 )) in the matrix element derived in section (3.4.3). The numerical calculation yields

fdirect (R) = −180.320 073 947 Eh W 3S;1S

 a 6 o

R

.

(8.13)

direct fdirect (R) is the total contribution due to the WickThe sum of the W 3S;1S (R) and W 3S;1S direct rotated term, W3S;1S (R), which reads

direct

direct fdirect (R) = −729 Eh W3S;1S (R) = W 3S;1S (R) + W 3S;1S  a 6 0 = −909.320 073 947 Eh . R

 a 6 0

R

− 180.320 073 947 Eh

 a 6 0

R (8.14)

direct The pole term, P3S;1S (R), arises due to the presence of the virtual 2P state, which

reads X e2 h3S|xi |2P ih2P |xi |3Sih1S|xj |kihk|xj |1Si 2 e2 3(4π0 )2 R6 ±,k E1S ± (E2P − E3S )   2e2 E2P − E3S X =− α1S ω = h3S|xi |2P ih2P |xi |3Si (4π0 )2 R6 ~ i 2 2 2 X 2e 5α mc =− α1S (ω = − ) |h3S|xi |2P i|2 2 6 (4π0 ) R 72~ i

direct P3S;1S (R) = −

231   15 2e2 6 6 2 × 38 ~2 √ √ =− Q (t = ) + Q (t = ) 1S 1S (4π0 )2 R6 512 α2 m2 c2 41 31 2 4.632338310 ~2 215 × 38 a20 = − 6 ~2 c2 α2 × × R α4 m3 c4 512  a 6 0 = −8.158497517 Eh . (8.15) R In the second line of Eq. (8.15), we have used ω = −α2 mc2 /(8~) + α2 mc2 /(18~) = p −5α2 mc2 /(72~) for ω in the expression t = 1/ 1 ± E(n=1) /(~ω) to calculate the corresponding values of the P-matrix element. On the fifth line of Eq. (8.15), we have used Eh = α2 mc2 and a0 = ~/(αmc) to express our result in terms of the Hartree energy and the Bohr radius. direct (R), is the sum The total vdW interaction to direct term, E3S;1S

direct direct direct E3S;1S (R) = W3S;1S (R) + P3S;1S (R)  a 6  a 6 0 0 = −909.320 073 947 Eh − 8.158497517 Eh R R  a 6 0 = −917.478 571 464 Eh . R

(8.16)

8.1.2. 4S-1S System. For the 4S-1S system, we have 3 X X i=1

|h4S|xi |4P (m = µ)i|2 = 540a20 .

(8.17)

µ

direct

Thus, the degenerate contribution W 4S;1S (R) reads 3

direct W 4S;1S (R)

a4 9 XX |h4S|xi |4P (m = µ)i|2 Eh 06 =− 2 i=1 µ R  a 6 9 a40 0 2 = − × 540 a0 Eh 6 = −2430 Eh . 2 R R

(8.18)

232 fdirect (R) is calculated numerically using The nondegenerate contribution W 4S;1S 3~ fdirect (R) = − W 4S;1S π(4π0 )2 R6

Z∞ dω α e4S (iω)α1S (iω),

(8.19)

0

which yields

direct f4S;1S W (R) == −415.860 208 974 Eh

 a 6 0

R

.

(8.20)

direct On the other hand, the pole term, P4S;1S (R), is given by

direct P4S;1S (R)

 X  X EkP − E4S 2e2 α1S ω = × |h4S|xi |kP i|2 =− 2 6 (4π0 ) R 2≤k 2.

259 9. CONCLUSION

To study the long-range interaction between two neutral hydrogen atoms, we have used the time ordered perturbation theory. We observed that the odd order perturbations vanish and the second order terms correspond to the self-energy contribution and talk only about the Lamb shift of the individual atoms. Thus, what we care about here is the fourth order perturbation term, which finally gives the CP interaction. The functional form of the interatomic interaction depends on the range of the interatomic distance. If both the interacting atoms are in the ground state, the interaction follows the usual C6 (1S; 1S)/R6 functional form for a0 ≤ R ≤ a0 /α. The distance R, which ranges from the Bohr radius a0 to the wavelength of the typical optical transition a0 /α is the so-called vdW range. For the 1S-1S system, we find C6 (1S; 1S) = 6.499 026 705Eh a60 , which agrees with the previously reported result [53; 80; 81; 82]. The interatomic interaction has the well known R−7 functional form if the distance is larger than the wavelength of optical transition i.e. a0 /α ≤ R. Thus, when both atom are in ground state fourth-order time-ordered perturbation theory is applied and retardation regime is achieved for a0 /α  R The situation is different if the atom interacting with the ground state atom is in an excited state. For excited reference states, we match the scattering amplitude and the effective Hamiltonian of the system. If the atom interacting with the ground state atom is in the first excited state, quasi-degenerate levels are present. In this case, we have to differentiate three ranges for the interatomic distance: vdW range (a0 ≤ R ≤ a0 /α), CP range (a0 /α ≤ R ≤ ~c/L), and Lamb shift range (R ≥ ~c/L). In the vdW range, the interatomic interaction between the atoms A and B can be formulated in the functional form −C6 (2S; 1S)/R6 . A complication

260 arises as |1SiA |2SiB and |2SiA |1SiB are energetically degenerate. Thus, we have the mixing term contribution as well. We have thus expressed the vdW interaction as the sum, C6 (2S; 1S) = D6 (2S; 1S) ± M6 (2S; 1S), where D6 (2S; 1S) represents the direct term and M6 (2S; 1S) depicts the mixing term contributions. For the 2S-1S system, there is a clear discrepancy in the literature among various results. In Ref [80], Tang and Chan reported that the direct coefficient D6 (2S; 1S) is 56.7999Eh a60 and they did not calculate mixing terms contribution. In Ref [53], Chibisov presented the calculation of both the direct and the mixing term. Chibisov claimed that D6 (2S; 1S) = 55.5 (0.5) Eh a60 and M6 (2S; 1S) = 27.9819 (2) Eh a60 .

In Ref.

[83], Deal and Young reported that D6 (2S; 1S) = 176.7523Eh a60 and M6 (2S; 1S) = 27.9832Eh a60 . Our finding, ignoring the relativistic correction, shows C6 (2S; 1S) = (176.752 266 285 ± 27.983 245 543) Eh a60 . We confirm all the significant figures of the result reported in Ref [83], and we add few more significant figures. We noticed that in both publications [80] and [53], authors did not include the contribution of the quasidegenerate 2P levels of the excited atom. In the CP range, the interaction is still of the R−6 functional form. We find C6 (2S; 1S) = (243/2 ∓ 46.614 032 414) Eh a60 , which is smaller than that of C6 (2S; 1S) coefficient in the vdW range. For the very large interatomic distance, the interaction energy is the sum of the CP type −C7 (2S; 1S)/R−7 term and the pole term which has an oscillatory distance dependance whose amplitude falls off as R−2 . The pole term arises as the Wick-rotated contour of the complex ω-plane picks up a pole at ω = L2 + i, where L2 = E(2S1/2 ) − E(2P1/2 ) is the Lamb shift. We have examined the Dirac-δ perturbation to the interaction energy of the 1S-1S and 2S-1S system. The Dirac-δ perturbation is a local potential which is nonzero only at the origin. It is the first time that the δ-perturbation to the interaction energy for S-states have been studied. The Dirac-δ modification of the interaction energy is of great interest as the fine structure, the hyperfine correction, and the

261 leading radiative correction to the vdW interaction for S-states [84] are of Dirac-δ type. The δ-perturbation has vanishing contribution to the Hamiltonian of the system as the probability density of the P -states vanish at the origin. However, it modifies both the energy and the wave function of the system. The Dirac-δ correction to the interaction energy is in the order of α2 times the plain interaction. If both atoms are in the ground state, there is no degenerate state to consider. The δ-perturbed interaction energy for the 1S-1S system ignoring the relativistic correction is found to
6 be δE(1S; 1S) = −34.685 544 399 aR0 Eh . In the CP region, where the contribution is chiefly due to the degenerate one, both the energy part and the wave function part of the 1S-1S follow 1/R7 behavior which is negligible. On the other hand, the energy part and the wave function part of perturbed interaction energy in the Lamb shift range are found to be (E)

δE1S;1S (R) = −

387 α  a0 7 729 α  a0 7 (ψ) Eh and δE1S;1S (R) = − Eh . 8 π R 16 π R

(9.1)

For 2S-1S system, we observed that the δ-perturbed interaction energy, in the vdW range, is

δE(2S; 1S) = −

δD6 (2S; 1S) ± δM6 (2S; 1S) R6

=(367.914605710 ∓ 58.095351093)α2 Eh

 a 6 0

R

,

(9.2)

which is clearly in the 1/R6 functional form. A very peculiar behavior is observed as the δ perturbed interaction energy follows the 1/R5 power law in the CP range. The energy type contribution, δE (E) (2S; 1S), and the wave function type correction, δE (ψ) (2S; 1S), both of them are solely the contributions given by the quasi-degenerate

262 states, which are (E)

(E)

δD (2S; 1S) ± δM5 (2S; 1S) δE (2S; 1S) = − 5 R5  891  α 3  a 5 0 =− ± 10.682382428 Eh , 32 π R (ψ) (ψ) δD (2S; 1S) ± δM6 (2S; 1S) δE (ψ) (2S; 1S) = − 6 R6    a 6 81 0 ∓ 58.439051900 α2 Eh . =− 4 R (E)

(9.3)

(9.4)

It is observed that, in the van der Waals range, the dominant contribution comes from the wave function type correction, however, in the CP range, the energy type correction is the dominant one. In the Lamb shift range, the interaction energy is the sum of the CP term which follows a R−7 power law and the long range cosine term with amplitude falling off as R−2 and it is in the order of 10−36 Hz. From the experimental point of view, this is too small quantity to consider. In this work, we have also analyzed the hyperfine resolved 2S-2S system composed of two electrically neutral hydrogen atoms. The analysis of the 2S-2S system involves fascinating interplay of degenerate and nondegenerate perturbations theory with a full account of hyperfine splitting. Our approach to investigating the 2S2S system allows us to do the hyperfine calculation and to estimate the hyperfine pressure shift for the 2S hyperfine interval measurement. Each hydrogen atom has four hyperfine states for S-states, namely the hyperfine singlet for F = 0 and the hyperfine triplet for F = 1 and similarly the four hyperfine states corresponding to P -states. Thus, the basis set of the two hydrogen atom system has 64 states. The 64-dimensional Hilbert space corresponding to the hyperfine resolved 2S-2S system decomposes into five manifolds. We noticed that the adjacency graph serves as a great tool to study a higher dimensional matrix. Interestingly, each manifold further decomposes into two sub-manifolds of the same

263 dimension. In each of these sub-manifolds, the Hamiltonian matrix can be solved analytically. In these sub-manifolds, there are several degenerate subspaces which are first order in vdW shift, i.e., proportional to R−3 . However, the hyperfine transition where both atoms are in S-states undergoes the second order in vdW shift, i.e., proportional to R−6 . We also study the evolution of the energy levels, which are so-called the Born-Oppenheimer potential curves, in all the hyperfine subspaces. We observed a strange but a highly impressive feature of level crossings in the hyperfine resolved 2S-2S system. In Fz = +2 and Fz = −2 manifolds, no level crossing occurs. However, in Fz = +1 and Fz = −1 manifolds, the level-crossings occur between the levels of different irreducible sub-manifolds, while in the Fz = 0 manifolds, the level-crossings present not only between the levels of different manifolds but also the levels of the same irreducible manifolds may cross. We reveal that the crossings are unavoidable, which repudiates the non-crossing theorem discussed in the literature so far. Thus, we can conclude that the system with two energy levels follows noncrossing theorem; however, the higher-dimensional irreducible matrices do not always follow the non-crossing rule. We are not much aware of the applicability of such phenomenon in spectroscopy; however, this certainly gives an insight understanding the Born-Oppenheimer potential curves. We also studied higher excited S-states of the hydrogen atom interacting with the ground state atom. For excited reference states, interaction energy is calculated matching the scattering amplitude and the effective hamiltonian of the system. For mathematical simplicity, we have also employed the method of Wick-rotation, which is one of a standard calculation tricks of rotating the integration contour. The Wickrotated integration contour enclosed poles at ω = −EmP,nS /~ + i, where mP with 2 ≤ m ≤ n are the low lying virtual P -states of the atom which is at nS-state. The pole term contribution to the interaction energy thus arises naturally. In the vdW range, both the Wick-rotated and the pole terms of both the direct and the mixing

264 type contributions to the interaction energy are of R−6 type, although the dominant contribution comes from the Wick-rotated term. We notice that the higher principal (direct)

quantum number of the atom, larger the direct contribution EnS;1S (R) and smaller (mixing)

the mixing contribution EnS;1S (R) to the interaction energy. In the CP range, the Wick-rotated type input is the sum of the R−6 and R−7 terms, and the pole type contribution has cosine terms obeying the power law R−2 , R−4 , and R−6 and sine terms obeying power law R−3 and R−5 . An nS-1S system for a particular value of n has n − 2 poles arising from the low-lying virtual mP -states, where 2P -states have the dominant contribution. In the Lamb shift range, the Wick-rotated term follows the R−7 . In the case of pole term, the dominant contribution comes from the cosine term whose magnitude is of R−2 type. In this range, the Wick-rotated term and the pole term are of the same order of magnitude. The interaction energy of the nS-1S systems in the vdW range is negative, which indicates that there exists the electrostatic force of attraction between the atoms which establishes a weak chemical bond between them in the vdW range. However, in the CP and Lamb shift ranges, the electrostatic force is not necessarily attractive. Indeed, its attractive and repulsive nature oscillates. For excited reference states, in vdW range both the Wick-rotated and the pole term follow the same R−6 functional form, in Casimir-Polder range, there is a competition between Wick-rotated (R−6 ) and the oscillatory pole term, and in the Lamb shift range, an oscillatory pole term whose magnitude falls off as R−2 dominates the Wick-rotated term. In short, if an atom interacting with the other atom in the ground state is in an excited state, the system never reach to the retardation regime.

APPENDIX A

DISCRETE GROUND STATE POLARIZABILITY

266 A.1. DISCRETE RADIAL GREEN FUNCTION With the help of the completeness relation in discrete representation, one can write h~r1 |~r2 i as below:

h~r1 |1|~r2 i =

X

=

X

h~r1 |n`mihn`m|~r2 i =

n`m

X

∗ ψn`m (r1 , θ1 , ϕ1 )ψn`m (r2 , θ2 , ϕ2 )

n`m ∗ Rn` (r1 )Rn` (r2 )Y`m (θ1 , ϕ1 )Y`m (θ2 , ϕ2 )

n`m

 `  `  X (n − ` − 1)!  2 3 2r1 2r2 r1 + r2 exp − = 2n(n + `)! na na0 na0 na0 0 n`m     2r1 2r2 L2`+1 L2`+1 . (A.1) n−`−1 n−`−1 na0 na0 Here, h~r|n`mi = ψn`m (r, θ, ϕ) is the complete eigenfunction for Schr¨odinger-Coulomb Hamiltonian. We have used an ansatz which states that the total eigenfunctions can be expressed as the product of a radial part and an angular part as

ψn`m (r, θ, ϕ) =Rn` (r)Y`m (θ, ϕ),

(A.2)

where the radial wave function Rn` (r) is given by [85] 

(n − ` − 1)! Rn` (r) = (n + `)!

1/2

2`+1 1 n2 a03/2



r na0

`



r exp − na0



L2`+1 n−`−1



2r na0

 ,

(A.3)

and the angular part Y`m (θ, ϕ) is the usual spherical harmonics given by 

(2` + 1)(` − m)! Y`m (θ, ϕ) = 4π(` + m)! Here,

L2`+1 n−`−1



2r na0



1/2

P`m (cos(θ)) eimϕ .

(A.4)

and P`m (cos(θ)) are respectively the associated Laguerre and the

associated Legendre polynomials. In what follows, we generalize this concept to derive

267 discrete radial Green function. The total discrete Green function is given by X 1 1 |~r2 i = hn`m|~r2 i h~r1 |n`mi H −E H −E n`m X ψn`m (r1 , θ1 , ϕ1 )ψ ∗ (r2 , θ2 , ϕ2 )

Gdis (~r1 , ~r2 , E) =h~r1 | =

n`m

En − E

n`m

 3  `  `  1 (n − ` − 1)! 2 2r1 2r2 r1 + r2 = exp − E na na0 na0 na0 n − E 2n(n + `)! 0 n`m     2r1 2r2 2`+1 ∗ L2`+1 Ln−`−1 Y`m (θ1 , ϕ1 )Y`m (θ2 , ϕ2 ). (A.5) n−`−1 na0 na0 X

Here, En is the energy eigenvalues corresponding to the eigenvalue equation   ~2 ~ 2 α~c HRn` (r) = − ∇ − Rn` (r) = En Rn` (r). 2me r

(A.6)

Let us rewrite Rn` (r) as

Rn` (r) = Cn`

    2r r 2`+1 Ln−`−1 , r exp − na0 na0 `

(A.7)

where 

Cn`

(n − ` − 1)! = (n + `)!

1/2

2`+1 1 n2 a3/2 0



1 na0

` ,

(A.8)

is a constant independent of r. In the following derivation, we will be using L in place   2r of L2`+1 just to save some space. Now, we have n−`−1 na0     r ~2 ~ 2 α~c ` HRn` (r) = − ∇ − Cn` r exp − L 2me r na0   2     ~2 ∂ 2 ∂ `(` + 1) α~c r ` = − + − − Cn` r exp − L 2me ∂r2 r ∂r r2 r na0         ~2 ∂ r r` r r ∂ `−1 ` =− Cn` `r exp − L− exp − L + r exp − L 2me ∂r na0 na0 na0 na0 ∂r

268         ~2 2 r 1 ` r r ∂ `−1 ` − Cn` `r exp − L− r exp − L + r exp − L 2me r na0 na0 na0 na0 ∂r     `(` + 1) 2me αc r ~2 ` Cn` − r exp − L − + 2me r2 ~r na0  ~2 1 ∂ `(` − 1) ` ∂ ` 1 ` =− L − Cn` L + L − L + L L − 2 2me r2 na0 r r ∂r na0 r na0 ∂r (na0 )      1 ∂ ∂2 ~2 2 ` 1 ∂ r ` ∂ ` L− L + 2 L r exp − − Cn` L− L+ L + r ∂r na0 ∂r ∂r na0 2me r r na0 ∂r       2 ~ `(` + 1) 2me αc r r − Cn` − r` exp − L × r` exp − + 2 na0 2me r ~r na0   2    ~2 r ∂ ∂ 2r `−1 =− Cn` r exp − r 2 L + 2` + 2 − L + (n − ` − 1)L 2me na0 ∂r na0 ∂r   2` 2 1 2me αc 2n 2` 2 ~2 Cn` − − + + − + + − 2me na0 r na0 r (na0 )2 ~r na0 r na0 r na0 r   r ` × r exp − L. (A.9) na0 Using the fact that L ≡

2`+1 Ln−`−1



2r na0



satisfies the associated Laguerre differential

equation:   2r ∂ ∂2 L + (n − ` − 1)L = 0, r 2 L + 2` + 2 − ∂r na0 ∂r

(A.10)

Eq. (A.9) reduces to     1 2 r ~2 2me αc ` + − Cn` r exp − L HRn` (r) = − 2me (na0 )2 ~r a0 r na0     ~2 ~αc ~2 r ` = − − + Cn` r exp − L 2me n2 a20 r m e a0 r na0   α2 me c2 r α2 me c2 ` =− C r exp − L = − Rn` (r). n` 2n2 na0 2n2

(A.11)

Here, we have used a0 = ~/(αme c). From Eq. (A.11), the eigenvalues En can be written as

En = −

α 2 m e c2 . 2n2

(A.12)

269 If we define a new quantum number k such that k = n−`−1, the associated Laguerre     2`+1 2`+1 2r 2r polynomials Ln−`−1 na0 becomes Lk and the energy eigenvalues, in (k+`+1)a0 this condition, can be written as α2 me c2 Ek` = − . 2 (k + ` + 1)2

(A.13)

The energy difference En − E in Eq. (A.5) is thus given by   α2 me c2 α2 me c2 En − E = − − − 2n2 2ν 2    2  2 2 α me c 1 1 ~2 n − ν2 = − = . 2 ν 2 n2 2me a20 n2 ν 2

(A.14)

Substituting the value of the energy difference En − E from Eq. (A.14) to Eq. (A.5), we get  3   ` 2 r1 + r2 2r1 2me X a20 n2 ν 2 (n − ` − 1)! exp − G (~r1 , ~r2 , ν) = 2 ~ n`m n2 − ν 2 2n(n + `)! na0 na0 na0 `      2r2 2r1 2r2 2`+1 2`+1 ∗ Ln−`−1 Ln−`−1 Y`m (θ1 , ϕ1 )Y`m (θ2 , ϕ2 ) na0 na0 na0  2`+1   4me X ν 2 (n − ` − 1)! 2 r1 + r2 = 2 exp − (r1 r2 )` 2 2 ~ n`m n − ν n(n + `)! na0 na0     2r1 2r2 ∗ L2`+1 L2`+1 Y`m (θ1 , ϕ1 )Y`m (θ2 , ϕ2 ). (A.15) n−`−1 n−`−1 na0 na0 dis

The total discrete Green function is defined as

Gdis (~r1 , ~r2 , ν) =

X n`m

∗ (θ2 , ϕ2 ). g`dis (r1 , r2 , ν)Y`m (θ1 , ϕ1 )Y`m

(A.16)

270 Comparing Eq. (A.16) with Eq. (A.15), we get the discrete radial Green function g`dis (r1 , r2 , ν) as g`dis (r1 , r2 , ν)

 2`+1   ∞ 4me X ν 2 2 (n − ` − 1)! r1 + r2 = 2 (r1 r2 )` exp − 2 2 ~ n=0 n − ν n(n + `)! na0 na0     2r1 2r2 2`+1 L2`+1 Ln−`−1 . (A.17) n−`−1 na0 na0

dis The (`=1)-component of the discrete radial Green function g`=1 (r1 , r2 , ν) reads

dis g`=1 (r1 , r2 , ν)

 3   ∞ 4me X ν2 2 r1 + r2 = 2 exp − (r1 r2 ) ~ n=2 (n2 − ν 2 ) n2 (n2 − 1) na0 na0     2r1 2r2 3 3 Ln−2 Ln−2 . (A.18) na0 na0

The sum over n starts from 2 not from zero as L3−2 (x) = 0 = L3−1 (x). A.2. DISCRETE GROUND STATE POLARIZABILITY The ground state static polarizability due to discrete energy levels is given by   2 1 e dis dis r2 1S 1S r1 α1S (ω = 0) =2P1S (ω = 0) = 2 3 H −E Z ∞ 2 Z ∞ 2e dis = r2 dr1 r22 dr2 R10 (r1 ) r1 g`=1 (r1 , r2 , ν) R10 (r2 ) r2 3 0 1 0   ∞ Z Z ∞ 32me e2 ∞ 4 r1 + r2 X 1 4 r1 dr1 r2 dr2 exp − = 3 2 2 2 3~ a0 0 a0 n (n − 1)2 0 n=2    3     2 2r1 2r2 r1 + r2 3 3 exp − Ln−2 Ln−2 . (A.19) na0 na0 na0 na0 dis Here, we have used the value of g`=1 (r1 , r2 , ν) from Eq. (A.18) with ν = 1 and

substituted the radial part of ground state wave function of hydrogen which reads 2 e−r/a0 R10 (r) = p 3 . a0

(A.20)

271 Let us use dimensionless variables ρ defined as ρi = 2ri /(na0 ). Then Eq. (A.19) becomes

dis (ω α1S

Z ∞ ∞  na 7 Z ∞ 1 32me e2 X 0 4 = 0) = ρ1 dρ1 ρ42 dρ2 2 3 2 2 2 3 ~ a0 n=2 n (n − 1) 2 0 0     ρ1 + ρ2 ρ1 n + ρ2 n exp − L3n−2 (ρ1 ) L3n−2 (ρ2 ) exp − 2 2   Z ∞ ∞ 2 X 4 5 a0 m e e (1 + n)ρ1 n 4 ρ1 exp − L3n−2 (ρ1 ) dρ1 = 12~2 n=2 (n2 − 1)2 0 2   Z ∞ (1 + n)ρ2 4 ρ2 exp − (A.21) L3n−2 (ρ2 ) dρ2 . 2 0

Interestingly, the ρ1 -integral is identical to the ρ2 -integral. Hence, one can write Eq. (A.21) as

dis α1S (ω

Z ∞ 2 ∞ e2 a20 X n5 4 −(1+n)u/2 3 = 0) = u e Ln−2 (u) du . 12 Eh n=2 (n2 − 1)2 0

(A.22)

Here, we have also used α = ~/(a0 me c), and Eh = α2 me c2 , where α and Eh are respectively the fine-structure constant and the Hartree energy. We can evaluate the u-integral in Eq. (A.22) using the standard integral identity [43] ∞

Z

dρ esρ ργ Lµn (ρ) =

0

1
Γ(γ + 1)Γ(n + µ + 1) (−s)−(γ+1) 2 F1 − n, γ + 1; µ + 1; − , n!Γ(µ + 1) s (A.23)

which yields Z

∞ 4 −(1+n)u/2

u e 0

L3n−2

Γ(5)Γ(n + 2) (u) du = (n − 2)!Γ(4)



2 1+n

5

 2 F1 2 − n, 5; 4;

2 1+n

 .

(A.24)

272 Substituting the value of the integral in Eq. (A.22) and simplifying the expression using standard integral identity [43]

2 F1

(−k, a + 1; a; z) = (1 − z)k

(z − 1)a + kz , a(z − 1)

(A.25)

we get

dis (ω α1S

2n  ∞ e2 a20 X 1024 n9 n−1 = 0) = , Eh n=2 3 (n − 1)6 (n + 1)8 n + 1

(A.26)

which yields

dis α1S (ω = 0) =0.362 240 952

e2 a20 . Eh

(A.27)

Recalling the total ground state static polarizability of a hydrogen atom

α1S (ω = 0) =

9 e2 a20 , 2 Eh

(A.28)

we come to the conclusion that, the major contribution in the ground state static polarizability comes from the continuum wave functions.

APPENDIX B

MAGIC WAVELENGTHS OF nS-1S SYSTEMS

274 B.1. ORIENTATION The AC Stark shift, which is the shifting of spectral lines in the presence of an oscillating electric field, can be used to trap neutral atoms. The AC Stark can also be useful in optical lattice clock experiments [86; 87; 88]. We here concentrate only in the trapping of the neutral hydrogen atoms. The AC Stark shift for the ground state is different to that of the excited states of the transition. The AC Stark shift vanishes if the trapping laser of particular wavelength called magic wavelength is turned on [86; 89; 90]. The calculation involves the determination of the point, where the AC Stark shift for the ground state is equal to that of the excited state, and the shifts due to the laser cancel. In other words, the atom does not feel the presence of light if the laser wavelength matches its magic wavelength value. The AC Stark shift for the state |φi of an atom depends on the intensity of the laser field and the optical frequency of the photon which is given by

∆EAC = −

IL α(φ, ωL ), 2 0 c

(B.1)

where IL stands for the intensity of the laser field. The IL is proportional to the square of the amplitude of the electric field EL , mathematically, IL = 21 0 cEL2 . α(φ, ωL ) in Eq. (B.1) represents the dipole polarizability of the state |φi [91; 92; 93]. The magic wavelengths corresponding to the 2S-1S transition of a hydrogen atom is computed in reference [94]. However, the author took only the discrete states of the hydrogen atom into account. The fact is the contribution of the continuum states can not be ignored. We already saw in Appendix A, for the ground state, the dominant contribution comes from the continuum states, not from the discrete one.

275 The following definition of the dipole polarizability  X  e2 X 1 α(φ, ωL ) = φ ~r ~r φ = Pφ (±ωL ), 3 ± HA − E ± ~ωL ±

(B.2)

includes the contributions of both discrete and continuous parts of the spectrum. The HA = p~ 2 /(2me ) − (α~c)/r, E, and Pφ in Eq. (B.2) are the atomic Hamiltonian of the system, the energy eigenvalues, and the P matrix element for the atomic reference state φ. The P1S , P2S , P3S , P4S , and P5S matrix elements are given by Eqs. (3.44), (3.56), (3.66), (3.73), and (3.74) respectively. The P6S matrix element reads "  ~2 e2 432 t2 P (6S,t) = 4 3 4 39439108405t24 − 3444722282t23 α m c 25(t − 1)14 (t + 1)12 − 113551229560t22 + 9795349850t21 + 135698822058t20 − 11514250414t19 − 87425932088t18 + 7253828382t17 + 33018970995t16 − 2654366212t15 − 7439943344t14 + 569035620t13 + 971507820t12 − 65940540t11 − 67724400t10 + 1350940t9 + 2692555t8 + 1404750t7 − 509400t6 − 501150t5 + 200650t4  442368 t9 (−1 + 36t2 ) 3 2 + 99850t − 45400t − 9050t + 4525 − 25(−1 + t2 )14  
2 (t − 1)2 8 6 4 2 × 2023t − 2932t + 1410t − 260t + 15 2 F1 1, −6t; 1 − 6t; (t + 1)2 #   −1/2 72~ω 68040 t2 − ; where t = 1 + . (B.3) 1 − t2 α2 mc2 The contribution of the degenerate P -states has been excluded from P (6S, t) subh i ~2 e2 68040 t2 tracting α4 m3 c4 (1−t2 ) . Work to the magic wavelengths for the 2S-1S and 3S-1S transitions in hydrogen atoms including the relativistic correction is presented in Ref. [95]. However, the relativistic correction, which is in the order of α2 ∼ 10−4 depends on the laser-field configuration. It is different for the different experimental setup. On the other hand,

276 the dominant correction to the magic wavelengths in the non-relativistic one-particle approximation comes from the reduced mass correction [96]. More explicitly, the reduced mass correction to the wavelength is of order me /mp ∼ 10−3 . The P matrix elements are proportional to the square of Bohr radius, a0 = ~/(αmc), and inversely proportional to the Hartree energy, Eh = α2 mc2 . Thus, the reduced mass correction on the dipole polarizability and hence the AC Stark shift has overall factor of (me /mr )3 , where the reduced mass mr of the system is given by

mr =

me mp , me + mp

(B.4)

where me and mp are the masses of an electron and a proton respectively. In this work, we also calculate the reduced mass correction of the magic wavelength, AC Stark shift, and the slope of the AC Stark shift at the magic wavelengths.

B.2. MAGIC WAVELENGTHS AND AC STARK SHIFT Let us recall the AC Stark shift corresponding to the nS-state, which reads

∆EAC (nS) = −

IL α(nS, ωL ). 2 0 c

(B.5)

Then the difference in AC Stark shift between an excited state and the ground state, i.e.,

∆EAC (nS) − ∆EAC (1S) = −

IL [α(nS, ωL ) − α(1S, ωL )] , 2 0 c

(B.6)

IL f1SnS (ωL ), 2 0 c

(B.7)

can be written as

∆EAC (nS) − ∆EAC (1S) = −

277 Table B.1: Influence of the reduced-mass correction (RMC) on the magic wavelengths λM , and the AC Stark shifts ∆EM for nS-1S transitions, where n = 2, 3, 4, 5, 6. Quantity

λM (in nm) ∆EM (in

IL kW/cm2 Hz)

condition 2S-1S

3S-1S

transitions 4S-1S

5S-1S

6S-1S

without RMC

514.366

1371.11

2811.24

4935.99

7588.47

with RMC without RMC

514.646 -221.222

1371.86 -212.290

2812.77 -211.249

4938.68 -211.026

7592.60 -210.964

with RMC

-221.584

-212.637

-211.595

-211.371

-211.309

where

f1SnS (ωL ) = α(nS, ωL ) − α(1S, ωL ).

(B.8)

The magic angular frequency satisfies the condition f1SnS (ωL = ωM ) = 0, which is the point of interaction of the polarizability of the ground state and that of the excited state of interest. Alternatively, this is the point where the difference of AC Stark shifts corresponding to the ground state, and the excited state vanishes nullifying the systematic uncertainties (see Figures (B.1) - (B.5)) The magic wavelength, λM , for the hydrogen nS-1S transition is given as

~ ωM = EM =

hc hc =⇒ λM = . λM ~ ωM

(B.9)

The magic wavelengths, λM , and the AC Stark shifts, ∆EAC , for the 2S-1S, 3S-1S, 4S-1S, 5S-1S, and 6S-1S transitions are listed in Table (B.1). The magic wavelength λM = 514.646 nm for the 2S-1S transition lies in between the 2S-3P transition (656.387 nm) and 2S-4P transition (486.213 nm) of a hydrogen atom. The magic wavelength λM = 1371.86 nm for the 3S-1S transition lies in between the 3S4P transition (1875.39 nm) and 3S-5P transition (1282.01 nm) of a hydrogen atom. The magic wavelength λM = 2812.77 nm for the 4S-1S transition lies in between

278

ΔΕAC in kHz/(kW/cm2 )

ΔΕAC in kHz/(kW/cm2)

300 200 100 0 -100 -200 -300

0.5

1.0

1.5 2.0 ℏωL (in eV)

2.5

-2.200 -2.205 -2.210 -2.215 -2.220 -2.225 2.41040 2.41041 2.41042 2.41043 2.41044 2.41045 ℏωL (in eV)

3.0

(a)

(b)

Figure B.1: AC Stark shift coefficients for the 1S- and 2S-states for intensity, IL = 10 kW/cm2 as a function of laser photon energy Eγ = ~ωL . In Figure (a), the dashed line at ∆EAC ≈ 0 represents the AC Stark shift coefficient of the 1S-state while the solid curved lines represent the AC Stark shift coefficient of the 2S-state. Figure (b) shows the AC stark shifts near the magic wavelength, λM , for 2S-1S transition. The AC Stark shifts of the 1S-state (dashed line) and the 2S-state (solid line) intersect at (2.41043 eV, -2.21222 kHz)).

-2.110 ΔΕAC in kHz/(kW/cm2 )

ΔΕAC in kHz/(kW/cm2)

1500 1000 500 0 -500 -1000 -1500

0.2

0.4

0.6

0.8 1.0 ℏωL (in eV)

(a)

1.2

1.4

-2.115 -2.120 -2.125 -2.130 -2.135 -2.140 0.904262 0.904263 0.904264 0.904265 0.904266 ℏωL (in eV)

(b)

Figure B.2: AC Stark shift coefficients for the 1S- and 3S-states by a laser light of intensity, IL = 10 kW/cm2 as a function of laser photon energy Eγ = ~ωL . In Figure (a), the dashed line at ∆EAC ≈ 0 represents the AC Stark shift coefficient of the 1S-state while the solid curved lines represent the AC Stark shift coefficient of the 3S-state. Figure (b) shows the AC Stark shifts near the magic wavelength, λM , for 3S-1S transition. The AC Stark shifts of the 1S-state (dashed line) and the 3S-state (solid line) intersect at (0.904264 eV, -2.12290 kHz/(kW/cm2 )).

279

ΔΕAC in kHz/(kW/cm2)

ΔΕAC in kHz/(kW/cm2)

10 000 5000 0 -5000

-2.06 -2.08 -2.10 -2.12 -2.14 -2.16 -2.18

-10 000 0.2

0.3

0.4 0.5 ℏωL (in eV)

0.6

0.7

0.441030

(a)

0.441030

0.441031 0.441031 ℏωL (in eV)

0.441032

(b)

Figure B.3: AC Stark shift coefficients for the 1S- and 4S-states for intensity, IL = 10 kW/cm2 as a function of laser photon energy Eγ = ~ωL . In Figure (a), the dashed line at ∆EAC ≈ 0 represents the AC Stark shift coefficient of the 1S-state while the solid curved lines represent the AC Stark shift coefficient of the 4S-state. Figure (b) shows the AC Stark shifts near the magic wavelength λM , for 4S-1S transition. The AC Stark shifts of the 1S-state (dashed line) and the 4S-state (solid line) intersect at (0.441031 eV, -2.11249 kHz/(kW/cm2 )).

40 ΔΕAC in kHz/(kW/cm2)

ΔΕAC in kHz/(kW/cm2)

40 000 20 000 0 -20 000 -40 000

0.15

0.20

0.25 0.30 0.35 ℏωL (in eV)

(a)

0.40

0.45

20 0 -20

0.25110

0.25115 0.25120 ℏωL (in eV)

0.25125

(b)

Figure B.4: AC Stark shift coefficients for the 1S- and 5S-states for intensity, IL = 10 kW/cm2 as a function of laser photon energy Eγ = ~ωL . In Figure (a), the dashed line at ∆EAC ≈ 0 represents the AC Stark shift coefficient of the 1S-state while the solid curved lines represent the AC Stark shift coefficient of the 5S-state. Figure (b) shows the AC Stark shifts near the magic wavelength λM , for 5S-1S transition. The AC Stark shifts of the 1S-state (dashed line) and the 55-state (solid line) intersect at (0.251184 eV, -2.11026 kHz/(kW/cm2 )).

280

ΔΕAC in kHz/(kW/cm2)

ΔΕAC in kHz/(kW/cm2)

20 000 10 000 0 -10 000

500

0

-500

-20 000 0.15

0.20

0.25 ℏωL (in eV)

0.30

0.16336

0.16338 0.16340 ℏωL (in eV)

(a)

0.16342

(b)

Figure B.5: AC Stark shift coefficients for the 1S- and 6S-states for intensity, IL = 10 kW/cm2 as a function of laser photon energy Eγ = ~ωL . In Figure (a), the dashed line at ∆EAC ≈ 0 represents the AC Stark shift coefficient of the 1Sstate while the solid curved lines represent the AC Stark shift coefficient of the 6S-state. Figure (b) shows the AC Stark shifts near the magic wavelength λM , for 6S-1S transition. The AC Stark shifts intersect at (0.163385 eV, -2.10964 kHz/(kW/cm2 )).

the 4S-5P transition (4051.77 nm) and 4S-6P transition (2625.55 nm) of a hydrogen atom. Similarly, the magic wavelength magic wavelength for 5S-1S, λM = 4938.68 nm lies between the 5S-6P transition (7458.94 nm) and 5S-7P transition (4653.21 nm) of a hydrogen atom. Likewise, the magic wavelength magic wavelength for 6S-1S, λM = 7592.60 nm lies between the 6S-7P transition (12370.4 nm) and 6S-8P transition (7501.57 nm) of a hydrogen atom. It is evident from Figures (B.1), (B.2), and (B.3) that, in addition to the magic wavelength tabulated above in Table ??, there are few other magic wavelength as well for each transition. For example, for the 2S1S transition, other magic wavelengths with reduced mass correction are 443.212 IL nm, 414.484 nm, 399.451 nm and so on with AC Stark shifts −225.203 (kW/cm 2 ) Hz, IL IL −227.404 (kW/cm 2 ) Hz, and −228.776 (kW/cm2 ) Hz respectively.

As shown in Figures (B.1) - (B.5), the AC Stark shift for 1S-state is almost constant. The AC Stark shift for 1S-state, ∆EAC (1S), is almost a horizontal line at

281 zero AC Stark shift. Numerically,

∆EAC (1S) = −

IL α(1S, ωL ). 2 0 c

(B.10)

A lawful approximation to the dynamic polarizability of the ground state hydrogen is that it is roughly equal to its static polarizability, i.e., α(1S, ωL ) ≈ α(1S, ωL = 0) = 9e2 a20 /(2Eh ). Thus, Eq. (B.10) yields

∆EAC (1S) ≈ −210.921

IL Hz. kW/cm2

(B.11)

One of the most important features we observe in the AC Stark shift for the 4S, 5S, and 6S reference states is the double pole structures in their energy versus AC Stark shift plots. For the 4S-state, the AC Stark shift has a double pole at 0.661388 eV. Similarly, for the 5S- and 6S- states, poles appear respectively at 0.306128 eV and 0.166292 eV. Let us now discuss the origin of such double pole structures. As given by Eq. (24) of Ref. [91], the AC Stark shift of the unperturbed state |φ, nL i reads   hφ|z|mihm|z|φi e2 ~ωL X hφ|z|mihm|z|φi nL + (nL + 1) , (B.12) ∆EAC (φ) = − 20 V m Em − Eφ − ~ωL Em − Eφ + ~ωL which reduces to Eq. (B.5) in the classical limit, nL → ∞, V → ∞, and nL /V = constant. Here, ωL , V, Em , and Eφ are the laser field frequency, normalization volume, energy corresponding to a virtual intermediate state |mi, and energy corresponding to the reference state |φi respectively. If the laser frequency is same to the energy difference between the energy of the reference state and one of the virtual level, we observe the pole structures as seen in the Figures. (B.3), (B.4), and (B.5) in the Stark shifts. More interestingly, the double pole structure in the AC Stark shift of 4S-state

282 can be eliminated by subtracting the following term IL e2 X X h4S|xj |3P (m = µ)ih3P (m = µ)|xj |4Si − . 2 0 c 3 µ j (E3P − E4S ) + ~ω

(B.13)

Similarly, the double pole structure in the Ac Stark shift of 5S- and 6S- states gets eliminated if we subtract

−

IL e2 X X h5S|xj |4P (m = µ)ih4P (m = µ)|xj |5Si , 2 0 c 3 µ j (E4P − E5S ) + ~ω

(B.14)

−

IL e2 X X h6S|xj |5P (m = µ)ih5P (m = µ)|xj |6Si , 2 0 c 3 µ j (E5P − E6S ) + ~ω

(B.15)

and

respectively from the total AC Stark shifts of the respective states. This double pole structure suggests that there exist a resonant emission into the laser field. The emitted photon has energy,

∆E = ~ω = EnS − E(n−1)P ,

n ≥ 4.

(B.16)

Our investigation shows that Eq. (B.16) exactly predict the position of the double poles in the AC Stark shifts of 4S-, 5S-, and 6S- states.

B.3. SLOPE OF THE AC STARK SHIFTS The slope η of the AC Stark shift at the magic wavelength is given by ∂ η= (∆EAC (nS, ωL ) − ∆EAC (1S, ωL )) , ∂ωL ωL =ωM

(B.17)

which measures how fast the difference of AC Stark shifts between the nS-state and

283 Table B.2: Influence of the reduced-mass correction (RMC) on the slope of Hz Stark shifts at the magic wavelengths in unit of GHz(kW/cm 2 ) for 2S-1S, 3S-1S, 4S-1S, 5S-1S and 6S-1S transitions. condition 2S-1S without RMC with RMC

-0.215044 -0.215395

transitions 3S-1S 4S-1S - 3.20155 - 3.20679

- 28.4212 - 28.4677

5S-1S

6S-1S

- 201.627 - 201.737

-8036.57 -8049.72

the 1S-state changes with the laser frequency. The slope of the AC Stark shifts at magic wavelengths are presented in Table (B.2). The magic wavelengths listed in Table (B.1) are the longest magic wavelengths for the corresponding transitions, and the slope of these transitions in Table (B.2) are the minimum slopes. The value of η with the reduced mass correction is 1.001637 times that of the η without the reduced mass correction. This factor comes from (me /mr )3 . In the laser trapping process, a large slope of the AC Stark shift should be avoided. With no surprise, the slope of the AC Stark shift in nS-1S transition is larger for the higher value of n. So far the feasibility of optical trapping [97] is concerned, difficulty increases as the value of n and hence the value of η increases.

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291 VITA

Chandra Mani Adhikari was born in a small village of Bharte in Lamjung district of Nepal. After completing his high school in Bharte, he moved to Pokhara city, where he completed his Bachelor’s degree in Physics with chemistry and mathematics minors from Prithivi Narayan Campus, Pokhara of Tribhuvan University, Nepal in 2008. He received the merit-based scholarship while studying his Bachelor’s degree. He then moved to the capital city of Nepal, Kathmandu, and enrolled in the Central Department of Physics on the Kirtipur Campus of Tribhuvan University to pursue his Master’s degree in physics. After completion of his coursework, he joined Prof. Mookerjee’s group in S. N. Bose National Center for Basic Sciences, Kolkata, India to complete his thesis work under the joint supervision of Prof. Abhijit Mookerjee and Prof. Narayan P. Adhikari. He received his Master’s degree in Physics from Tribhuvan University in December 2012. Chandra then came to Rolla, USA, in August 2013, and enrolled to Missouri University of Science and Technology to continue his graduate studies in physics. He received his MS degree in Physics in May 2015 and his PhD degree in Physics in December 2017 from Missouri University of Science and Technology. While he was in Rolla, he greatly enjoyed his research works under the supervision of Prof. Ulrich D. Jentschura. Chandra won the third prize of the “Graduate Seminar Series” in 2015 and third place of the Schearer Prize in 2016. Chandra also worked as graduate teaching assistant at Physics Department of Missouri University of Science and Technology. He won “An Outstanding Graduate Teaching Assistant Award of the Year” in 2015 and 2016 for two consecutive years.