Variational-Integral Perturbation Corrections for ...

ISSN: 0256 - 307 X

中国物理快报

Chinese Physics Letters

Volume 30 Number 1 January 2013

A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y Institute of Physics PUBLISHING

CHIN. PHYS. LETT. Vol. 30, No. 1 (2013) 013101

Variational-Integral Perturbation Corrections for Hydrogen Atoms in Magnetic Fields * ZHAO Yun-Hui(赵云辉)1** , PAN Yi-Qing(潘谊清)1 , LI Wen-Juan(李文娟)1 , DENG Xia(邓霞)1 , HAI Wen-Hua(海文华)2 1

Department of Primary Education, Changsha Normal College, Changsha 410100 2 Department of Physics, Hunan Normal University, Changsha 410082

(Received 10 December 2012) A variational-integral perturbation method (VIPM) has been established by combining the variational perturbation with the integral perturbation. We present the results obtained by the application of VIPM to the ground and excited states associated with the Zeeman effect on a hydrogen atom. We construct correction wave functions and calculate energy corrections for the ground state and some high excited states. These values are compared with results of Smith and Jason et al.; our results are more accurate than those from Smith and Jason, especially in the 4𝑓 excited state where we find a very interesting result that when magnetic fields increase, the energy level trap appears.

PACS: 31.15.−p, 32.60.+i, 03.65.Ge, 02.30.Rz

DOI: 10.1088/0256-307X/30/1/013101

The Zeeman effect plays a fundamental role in many aspects including astrophysics, solid state and plasma physics.[1−4] Although a lot of research has been carried out in this field,[5−15] the problems are still waiting to be solved. The nonseparability of the Schrödinger equation for an electron in a combined Coulomb and magnetic field makes the theoretical description of the problem quite difficult. The hydrogen atom in a magnetic field has been tackled by many researchers with the aid of various approaches ranging from the perturbation theory for the weak-field’s regime[16] to the adiabatic approximation for the opposite limit of very strong magnetic fields.[17] Considerable techniques for large order calculation have recently been reported,[18−23] the firstorder perturbation theory,[24,25] combining variational calculation with Laguerre Polynomials in cylindrical coordinates[26] and the Zeeman Hamiltonian plus series method.[27] Jason[28] used a simple basis for hydrogen to obtain the evolution of the energy spectrum of atoms in strong and uniform magnetic fields. Kravchenko and Liberman[29] investigated the asymptotic behavior of the solution and transformed the boundary conditions to a form which makes it possible to reduce the problem to the infinite set of algebraic equations and calculated energy levels and wave functions for the ground states and for several excited states of the hydrogen atom in a uniform magnetic field. Bayrak et al.[30] obtained the effect of the velocity-dependent potentials on the bound state energy eigenvalues. Li et al.[31] studied coherence-like states of two Coulomb-correlated ions confined in a

Paul trap. Duan et al.[32] obtained corrections to the nonrelativistic ground energy of a helium atom. In the present study, applying the improved Rayleigh–Schrödinger perturbation theory based on an integral equation to helium-like ions in ground states and treating electrons as perturbations,[33,34] we employ the improved variational-perturbation method (VIPM) based on the integral equation to solve the heavy quarkonium in the 2𝑆 state.[35] We apply the VIPM to the hydrogen atom in a strong and uniform magnetic field,[36] construct corrected wave functions and calculate energy corrections for ground state and some excited states such as 4𝑑2, 4𝑑1, 4𝑑0, 4𝑑.1, 4𝑑.2, 4𝑓 3, 4𝑓 2, · · ·. These values are compared with those of the elaborate calculations of Smith et al.[10] and the simple basis variational calculations of Jason.[28] Our data clearly demonstrate that the results of the VIPM are more accurate than those of others as mentioned above, especially the VIPM can keep convergence of the wave function.[33−37] We do not take into account relativistic effects, since for fields below 2.3505 × 109 T they are negligible. The Hamiltonian for a uniform magnetic field in a spinless hydrogen atom can be written in the form by a vector potential 𝐴, 𝐻=

1 [︁ 𝑒 ]︁ 𝑒2 𝑃+ 𝐴 − . 2𝑚𝑒 𝑐 𝑟

(1)

Choosing the usual gauge 𝐴 = 𝐵 ×𝑟/2 spherical coordinates, the atomic units, such as ~ = 𝑒 = 𝑚𝑒 = 𝑐 = 1,

* Supported by the National Natural Science Foundation of China under Grant No 11175064, and the Foundation of the Science and Technology of Hunan Province under Grant No 2011CK3013. ** Corresponding author. Email: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd

013101-1


are used throughout, and the Zeeman Hamiltonian is 1 1 1 𝛾2 𝐻 = − ∆ − + 𝛾𝐿𝑧 + 𝑟2 sin2 𝜃, 2 𝑟 2 8

(2)

where 𝐿𝑧 is the angular momentum in the 𝑧 direction and takes along the magnetic field direction; 𝛾 is a dimensionless coupling parameter describing the strength of the magnetic fields, and 𝛾=1 corresponds to 2.3505 × 109 G. We can choose a trial wave function in the form 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙) = 𝑅𝑛𝑙 (𝜆, 𝑟)𝑌𝑙𝑚 (𝜆, 𝜃, 𝜙),

(3)

where 𝜆 is a variational parameter and 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙) are the usual normalized eigensolutions of the generating Hamiltonian 1 𝜆 𝐻𝜆 = − ∇21 − , 2 𝑟

(0)

Combining Eq. (7) with Eq. (9) and 𝐸𝑛𝑙𝑚 , the total energies of the variational method are 𝐸𝑛𝑙𝑚 = −

𝜆2 (𝜆 − 1)𝜆 𝑚𝛾 𝛾 2 𝑝(𝑛, 𝑙, 𝑚) + + . + 2𝑛2 𝑛2 2 8𝜆2 (10) (0)

𝐻 ′ (𝜆, 𝑟, 𝜃) =

(4)

where 𝐻𝜆 is the generating Hamiltonian, and the (0) (0) zeroth-order energy 𝐸𝑛𝑙𝑚 is 𝐻𝜆 |𝜑(𝜆)⟩ = 𝐸𝑛𝑙𝑚 |𝜑(𝜆)⟩, (0) 𝐸𝑛𝑙𝑚 = −(𝜆2 )/(2𝑛2 ). The original Hamiltonian is

(1)

The total energies are the sum of 𝐸𝑛𝑙𝑚 and 𝐸𝑛𝑙𝑚 . We only consider the magnetic quantum number 𝑚 = 0. Combining the variational-perturbation with the integral perturbation, we establish the VIPM. Applying the VIPM to a hydrogen atom in a strong and uniform magnetic field, we investigate the corrected wave functions and energy corrections. If 𝑚 = 0, the perturbed Hamiltonian is 𝜆 − 1 𝛾2 2 2 + 𝑟 sin 𝜃. 𝑟 8

It is independent of the variable 𝜙. If 𝑚 is nonzero, the perturbed Hamiltonian is 𝐻 ′ (𝜆, 𝑟, 𝜃) =

𝛾2 𝜆 − 1 𝑚𝛾 + + 𝑟2 sin2 𝜃. 𝑟 2 8

𝐻 = 𝐻𝜆 + 𝐻 ′ 𝜆 𝜆 − 1 𝛾𝐿𝑧 𝛾2 1 + + 𝑟2 sin2 𝜃. = − ∇21 − + 2 𝑟 𝑟 2 8

𝜆−1 1 𝛾2 𝐻 = + 𝛾𝐿𝑧 + 𝑟2 sin2 𝜃. 𝑟 2 8

𝜑𝑛𝑙0 (𝜆, 𝑟, 𝜃, 𝜙) = 𝑟−1 𝜒𝑛𝑙0 (𝜆, 𝑟, 𝜃, 𝜙)

(5)

= 𝑟−1 𝜒𝑛𝑙 (𝜆, 𝑟, 𝜃)𝜑(0) 𝑚 (𝜙), (13)

𝜆2 (𝜆 − 1)𝜆 𝑚𝛾 𝛾 2 𝑝(𝑛, 𝑙, 𝑚) + + + 2 2 2𝑛 𝑛 2 8𝜆2

(𝜆 − 1)𝜆 𝑚𝛾 𝛾 2 𝑝(𝑛, 𝑙, 𝑚) + + . 𝑛2 2 8𝜆2

𝜀𝑛𝑙0 (𝜃, 𝜙) = 𝜀𝑛𝑙0 (𝜃)𝜑(0) 𝑚 (𝜙).

(15)

Applying the Rayleigh–Schrödinger expansions 𝐸𝑛𝑙𝑚 = 𝐸𝑛𝑙𝑚 +

(7)

(8)

∞ ∑︁

(𝑖)

𝐸𝑛𝑙𝑚 ,

𝑖=1

𝜒𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃) =

∞ ∑︁

(𝑖)

𝜒𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃),

(16)

𝑖=1

we can construct 𝑖th-order corrected wave function and obtain 𝑖th-order energy correction[36,37] ∫︁ 𝜃 (𝑖) (0) (0) (𝑖) ˜ 𝜗𝑛𝑙𝑚 (𝜆, 𝜃, 𝜙) = 𝜗𝑙𝑚 𝜗𝑙𝑚 (𝜃)𝜀𝑛𝑙𝑚 (𝜃, 𝜙) sin 𝜃𝑑𝜃

where 𝜆 is the variational parameter, 𝑛 the main quantum number, 𝑙 and 𝑚 the angular and magnetic quantum numbers. We can obtain the nonzero and minimum real number values for variational parameter 𝜆 for solving Eq. (8). The first-order corrected energies are ∫︁ (1) (0) (0) 𝐸𝑛𝑙𝑚 = 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙)* 𝐻 ′ 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙)𝑑𝜏 =

(14)

(0)

with 𝑝(𝑛, 𝑙, 𝑚) = 𝑛2 [5𝑛2 − 3𝑙(𝑙 + 1) + 1][𝑙(𝑙 + 1) + 𝑚2 − 1]/[4𝑙(𝑙 + 1) − 3]. Under the condition of minimal en¯ 𝐻 ergy 𝜕𝜕𝜆 = 0, we establish the equation[36] 𝛾 2 𝑛2 𝜆4 − 𝜆3 − [5𝑛2 − 3𝑙(𝑙 + 1) + 1] 8 [𝑙(𝑙 + 1) + 𝑚2 − 1] · = 0, 4𝑙(𝑙 + 1) − 3

𝜗𝑛𝑙0 (𝜃)𝜑(0) 𝑚 (𝜙),

𝜗𝑛𝑙0 (𝜃, 𝜙) =

(6)

For 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙), the average values of energy are ∫︁ ¯ 𝐻 = 𝜑*𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙)𝐻𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙)𝑑𝜏 =−

(12)

Therefore, we assume the wave functions in the form

The perturbation Hamiltonian is ′

(11)

(9) 013101-2

𝐶𝑖

∫︁ 𝜃 (0) (0) (𝑖) − 𝜗𝑙𝑚 𝜗˜𝑙𝑚 𝜀𝑛𝑙𝑚 (𝜃, 𝜙) sin 𝜃𝑑𝜃 𝐷𝑖 ∫︁ ∞ (0) (𝑖) = 𝑅𝑛𝑙 (𝜆, 𝑟)𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙)𝑑𝑟, 0 (17) ∫︁ ∞ [︁ ∑︁ 𝑖 (𝑖) (𝑗) (𝑖−𝑗) 𝜀𝑛𝑙𝑚 (𝜆, 𝜃, 𝜙) = 2 𝐸𝑛𝑙𝑚 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙) 0

𝑗=1

]︁ (𝑖−1) − 𝐻 (𝜆, 𝑟, 𝜃)𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃, 𝜙) ′

(0)

· 𝑅𝑛𝑙 (𝜆, 𝑟)𝑟2 𝑑𝑟,

(18)

CHIN. PHYS. LETT. Vol. 30, No. 1 (2013) 013101 𝜋

∫︁

(𝑖)

𝜀𝑛𝑙𝑚 = − ∫︁

2𝜋

−

(0)

0

(0)

(𝑖)

𝜀𝑙𝑚 (𝜃)𝜀𝑛𝑙𝑚 (𝜆, 𝜙) sin(𝑚𝜙)𝑑𝜙 = 0,

0

{︁ 𝛾 2 cos2 𝜃 𝛾2 (1) 𝜑100 = 𝑟2 𝑒−𝜆𝑟 𝑟2 √ 1/2 + √ 1/2 8 𝜋𝜆 16 𝜋𝜆 [︁ · cos(2𝜃) + 4 cos(𝜃)[− ln cos(𝜃/2) ]︁}︁ + ln sin(𝜃/2)] sin2 𝜃 . (29)

(𝑖)

𝜗𝑙𝑚 (𝜃, 𝜙)𝜀𝑛𝑙𝑚 (𝜆, 𝜃, 𝜙) sin 𝜃𝑑𝜃 = 0, (19) (20)

for 𝑖 = 1, 2, · · · , ∞, (𝑖) 𝜑𝑛𝑙𝑚

=

(𝑖) (0) 𝜗𝑛𝑙𝑚 (𝜆, 𝜃, 𝜙)𝑟2 𝑅𝑛𝑙 (𝜆, 𝑟),

(21)

where 𝐶𝑖 and 𝐷𝑖 are arbitrary constants. The corrected wave functions are given by Eqs. (17)–(19). Let us take nondegenerate case 𝑖 = 1, 𝑚 = 0. From Eqs. (18)–(20) we perform the calculations ∫︁ ∞ [︁ 𝜆 − 1 𝛾 2 2 2 ]︁ (1) (1) − 𝑟 sin 𝜃 𝜀𝑛𝑙0 (𝜆, 𝜃) = 2 𝐸𝑛𝑙0 − 𝑟 8 0 (0)

(0)

· 𝜑𝑛𝑙0 (𝜆, 𝑟, 𝜃)𝑅𝑛𝑙 (𝜆, 𝑟)𝑟2 𝑑𝑟,

Combining Eqs. (25),(28) and (29) with Eqs. (17)– (21), the second-order corrected energy is (2)

𝐸100 =

(−9𝛾 4 − 4𝛾 2 𝜆3 + 4𝛾 2 𝜆4 ) 128𝜆6

with variational parameter 𝜆. For 2𝑝−1 states, 𝑛 = 2, 𝑙 = 1, 𝑚 = −1, the zeroth-order wave function (0) 𝜑21−1 (𝜆, 𝑟, 𝜃, 𝜙) is

(22) (0)

∫︁

𝜑21−1 (𝜆, 𝑟, 𝜃, 𝜙) =

𝜋

(0)

(1)

𝜗𝑙0 (𝜃)𝜀𝑛𝑙0 (𝜆, 𝜃) sin 𝜃𝑑𝜃 = 0,

0

(1) 𝜗𝑛𝑙𝑚 (𝜆, 𝜃)

(0) = 𝜗˜𝑙0

∫︁

𝜃

𝐶𝑖

(23)

∫︁ 𝜃 (0) (0) (1) − 𝜗𝑙0 𝜗˜𝑙0 𝜀𝑛𝑙0 (𝜆, 𝜃) sin 𝜃𝑑𝜃 𝐷𝑖 ∫︁ ∞ (0) (1) = 𝑅𝑛𝑙 (𝜆, 𝑟)𝜑𝑛𝑙0 (𝜆, 𝑟, 𝜃)𝑑𝑟. Combining Eq. (22) with Eq. (23), we can obtain the (1) first-order corrected energies 𝐸𝑛𝑙𝑚 . Inserting Eq. (24) into Eq. (21), we can construct the first-order cor(1) rected wave functions 𝜑𝑛𝑙𝑚 (𝜆, 𝑟, 𝜃). For 𝑖 = 2, 3, · · ·, applying Eqs. (17)–(21), we can obtain the secondorder, third-order and higher-order corrected energies and corrected wave functions. We only consider the nondegenerate cases of the hydrogen atom in a uniform magnetic field. For the simple case 𝑛 = 1, 𝑙 = 0, 𝑚 = 0 with 1 𝑆 states, the (0) zeroth-order wave function 𝜑100 (𝜆, 𝑟, 𝜃, 𝜙) is[36]

1 (0) 𝜗00 = √ , 4𝜋 (0) 𝜗˜00

= =

(0) −𝜗00

√

∫︁

(25)

(1)

𝐸21−1 =

(1)

𝜑21−1 =

(33)

3 √ 𝛾 2 𝜆1/2 𝑟3 𝑒−(𝜆𝑟/2+𝑖𝑚𝜙) (𝐾1 + 𝐾2 ); 256 𝜋 (34)

(35) 𝐾2 = sin 𝜃{4 cos(2𝜃) + 4 cos(4𝜃) +8 cos(𝜃)[−ln(cos(𝜃/2))+ln(sin(𝜃/2))] sin4 𝜃}. (36)

(26)

The second-order corrected energy is given by inserting Eqs. (33)–(36) into Eqs. (17)–(21),

𝑑𝜃 (0)

(𝜗00 )2 sin 𝜃

(2)

𝐸21−1 = − (27)

Inserting Eqs. (25) and (27) into Eqs. (17)–(21), the (1) first-order corrected energy 𝐸100 and the first-order (1) corrected wave function are 𝜑100 , (1)

12𝛾 2 + 2𝑚𝛾𝜆2 − (1 − 𝜆)𝜆3 , 4𝜆2

[︁ 𝐾1 = 2 cos(𝜃) − csc(𝜃/2)2 − 4ln[cos(𝜃/2)] ]︁ + 4ln[sin(𝜃/2)] + sec(𝜃/2)2 sin5 𝜃,

4𝜋{ln[sin(𝜃/2)] − ln[cos(𝜃/2)]}.

𝐸100

(32)

Inserting Eqs. (31) and (32) into Eqs. (17)–(21), the (1) first-order corrected energy 𝐸21−1 and the first-order (1) corrected wave function are 𝜑21−1 , (24)

2 (0) 𝜑100 (𝜆, 𝑟, 𝜃, 𝜙) = √ 𝜆3/2 𝑒−𝜆𝑟 , 4𝜋

(︁ 𝜆 )︁3/2 (︁ 𝜆𝑟 )︁ √ 𝑒−𝜆𝑟/2 𝑒−𝑖𝑚𝜙 sin 𝜃, 2 8𝜋 (31)

√ 3 (0) 𝜗1−1 = √ sin 𝜃. 8𝜋

(0) (1) 𝜗𝑙0 (𝜃)𝜀𝑛𝑙0 (𝜆, 𝜃) sin 𝜃𝑑𝜃

0

(30)

𝛾 2 − 4(1 − 𝜆)𝜆3 = , 4𝜆2

3(372𝛾 4 + 7𝛾 2 𝜆3 − 7𝛾 2 𝜆4 ) . 64𝜆6

(37)

For 3𝑑2 states, 𝑛 = 3, 𝑙 = 2, 𝑚 = −2, the zeroth(0) order wave function 𝜑32−2 (𝜆, 𝑟, 𝜃, 𝜙) is (0)

(28) 013101-3

𝜑32−2 (𝜆, 𝑟, 𝜃, 𝜙) =

𝑟2 𝜆7/2 −( 𝜆𝑟 +𝑖𝑚𝜙) 2 √ 𝑒 3 sin 𝜃, 162 𝜋

(38)


√︂

(0)

𝜗2−2 =

15 sin2 𝜃𝑒−𝑖𝑚𝜙 . 32𝜋

(39)

Inserting Eqs. (40) and (41) into Eqs. (17)–(21), the (1) (2) corrected energies 𝐸43−3 and 𝐸43−3 are

Inserting Eqs. (38) and (39) into Eqs. (17)–(21), we (1) (2) obtain the corrected energies 𝐸32−2 and 𝐸32−2 , (1)

𝐸32−2 =

243𝛾 2 + 9𝑚𝛾𝜆2 − 2(1 − 𝜆)𝜆3 , 18𝜆2

(1)

𝐸43−3 =

(2)

𝐸43−3 = −

27(292𝛾 4 + 74𝛾 2 𝜆3 − 74𝛾 2 𝜆4 ) . 1024𝜆6

(41)

For 4𝑓3 states, 𝑛 = 4, 𝑙 = 3, 𝑚 = −3, the zeroth-order (0) wave function 𝜑43−3 (𝜆, 𝑟, 𝜃, 𝜙) is 𝑟3 𝜆9/2 −( 𝜆𝑟 +𝑖𝑚𝜙) √ 𝑒 4 × sin3 𝜃, 6144 𝜋 (42)

(0)

𝜑43−3 (𝜆, 𝑟, 𝜃, 𝜙) =

√︂

(0)

𝜗3−3 =

(44)

(40) (2)

𝐸32−2 = −

640𝛾 2 + 8𝑚𝛾𝜆2 − (1 − 𝜆)𝜆3 , 16𝜆2

35 sin3 𝜃𝑒−𝑖𝑚𝜙 . 64𝜋

(43)

25(991616𝛾 4 + 547𝛾 2 𝜆3 − 547𝛾 2 𝜆4 ) . 2048𝜆6 (45)

For 1𝑠, 2𝑠, 2𝑝0 , . . ., the total energies 𝐸 are given in Tables 1–2. They also include the results of the simple basis variational method, the perturbation method, the variation-perturbation method and an elaborate (0) (1) (2) variational calculation. Here 𝐸 = 𝐸𝑛𝑙𝑚 +𝐸𝑛𝑙𝑚 +𝐸𝑛𝑙𝑚 , 𝐸VIPM denotes the results of the total energies by using the VIPM, 𝐸𝑣𝑝 the result of the simple basis variation method, 𝐸𝑓 𝑝 the result of the perturbation method, and 𝜆 the variational parameter.

Table 1. Some energies (in atomic units) of 𝐵 = 2.35 × 108 G (𝛾 = 0.1), and comparison of the results of the VIPM and the other methods. (0)

State

𝜆

𝐸𝑓 𝑝

𝐸𝑣𝑝 [28]

Killingbeak[27]

Praddaude[26]

Kravchenko[29]

[36]

1𝑠 2𝑠 2𝑝0

1.00493 1.17334 1.09212

−0.497500 −0.090000 −0.100000

−0.497512 −0.095822 −0.111363

−0.4975217 −0.0956530 −0.1117526

−0.497525 −0.098085 −0.112410

0.54752648 0.148089155 0.162410078

−0.497518 −0.09760 −0.112112

Table 2. Comparison of the energy (in Rydberg units(= State 2𝑠

2𝑝0

4𝑝0 4𝑝−1 4𝑝1 4𝑑0 4𝑑−1 4𝑑−2 4𝑓−1 4𝑓−2 4𝑓−3

Authors Jason[28] Smith[24] Kravchenko[29] Yuan[36] Jason Smith Kravchenko[29] Yuan[36] 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM 𝐸VIPM

𝛾 = 0.005 −0.2498 −0.2498 0.2548 −0.24983 −0.2499 −0.2499 0.2549 −0.24993 −0.06112 −0.06463 −0.05463 −0.06096 −0.06577 −0.07055 −0.0.06632 −0.07104 −0.07556

1 a.u.)) 2

spectrum in uniform magnetic field for some excited states.

𝛾 = 0.02 −0.2472 −0.2472 0.2672 −0.2473 −0.2488 −0.2488 0.2688 −0.24881 −0.05079 −0.04889 −0.0089 −0.03662 −0.06166 −0.07790 −0.06641 −0.08389 −0.09841

A number of results have been obtained by studying the effect of a magnetic field on the spectrum of hydrogen. We construct the corrected wave functions and calculate the energy corrections for the ground state and some excited states by applying the VIPM to the hydrogen atom in a strong and uniform magnetic field. Because it is quite hard to estimate the error that occurred in this calculation, we try to ensure the validity of the present calculation by systematic comparison with that obtained by others.

𝛾 = 0.05 −0.2335 −0.2340 0.2840 −0.2342 −0.2427 −0.2429 0.2928 −0.2428 −0.03834 0.01809 0.11809 0.07014 −0.03068 −0.06174 −0.03137 −0.0555 −0.11473

𝛾 = 0.1 −0.1916 0.2961 −0.1972 −0.2227 0.3248 −0.2242 −0.03774 0.16243 0.36240 0.31825 0.04180 −0.00472 0.02784 −0.06451 −0.11347

𝛾 = 0.2 −0.06715 −0.09785 0.2978 −0.09761 −0.1575 −0.1702 0.3703 −0.1684 0.49307

0.21338 0.14846 0.20325 0.00869 0.07394

Table 1 shows some energy values for 𝛾 = 0.1. The first column shows the variational parameter, the second column the result of the first-order perturbation theory 𝜆, the second column the result of the firstorder perturbation theory (𝜆 = 1), the third column the result obtained by using the simple parameter variational method,[28] and the last column the result obtained by using the variational-integral perturbation method. Table 2 shows the field evolution of some excited

013101-4


states. The calculations of Smith and Jason[8,28] have also contained positive energies, and the positive energies are expected to lose accuracy quickly. We have obtained the energies of all states, which are negative by using the VIPM. In this table our results are compared with the ones obtained by Smith et al.[24] and Jason. Applying an elaborate variational calculation, Smith used an eigenfunction made of up to 12 Slater-type orbits. These same states in the same field range were investigated by Jason and Brandi. Jason used the simple basis for hydrogen atoms in magnetic fields and his results are in good agreement with those from the elaborate calculation, but in better agreement with our VIPM results. We calculate 4𝑝, 4𝑑 and 4𝑓 energy levels and find a very interesting result. In 4𝑓 energy levels, when magnetic fields increase, the energy level trap appears as shown in Figs. 1–3. Because magnetic fields increase, the ionic appear displacement polarization makes the role of electronic potential energy reduce. 0.01

γ 0.05

0.1

0.15

0.2

Ε43-3

-0.01 -0.02 -0.03 -0.04

Fig. 1. Plots of the profiles of the 4𝑓−3 energy level in atomic units for uniform magnetic field.

0.05

0.1

0.15

γ 0.2

Ε43-2

-0.01 -0.02 -0.03 -0.04

Fig. 2. Plots of the profiles of the 4𝑓−2 energy level in atomic units for uniform magnetic field. 0.1

Ε43-1

0.08 0.06 0.04 0.02

γ 0.05

0.1

0.15

0.2

-0.02

Fig. 3. Plots of the profiles of the 4𝑓−1 energy level in atomic units for uniform magnetic field.

The variational-integral perturbation method (VIPM) has been established by combining the variational perturbation with the integral perturbation. We present the results obtained by application of the VIPM to the ground and excited states associated with the Zeeman effect on a hydrogen atom. In this case, the wave equation is not separable. We show that the VIPM works very well for a strong and uniform magnetic field. We obtain convergence of the correction wave function.[33−37]

References [1] Benjamin L L et al 2006 Phys. Rev. A 74 061402 [2] Adam K et al 2012 Phys. Rev. A 85 043418 [3] Horsewill A J and Abuokhumra S M M 2012 Phys. Rev. B 85 064512 [4] Rebecca C et al 2010 Astrophys. J. 708 1579 [5] Rosner W et al 1984 J. Phys. B 17 1301 [6] Machet B and Vysotsky M I 2011 Phys. Rev. D 83 025022 [7] Efstathiou K et al 2008 Phys. Rev. Lett. 101 253003 [8] Smith E R et al 1972 Phys. Rev. D 6 3700 [9] Brandi H S 1975 Phys. Rev. A 11 1835 [10] Peek J M and Katriel J 1980 Phys. Rev. A 21 413 [11] Cohen M and Herman G 1981 J. Phys. B 14 2761 [12] Caswell W E and lepage G P 1986 Phys. Lett. B 167 437 [13] Lozovik Y E and Volkov S Y 2004 Phys. Rev. A 70 023410 [14] Claudio L C 2001 Phys. Rev. A 64 023418 [15] Rosato J et al 2012 J. Phys. B: At. Mol. Opt. Phys. 45 165701 [16] Garstang R H and Kemic S B 1974 Astrophys. Space Sci. 31 103 [17] Zhao J J et al 2011 Chin. Phys. B 20 053101 [18] Adams B G et al 1988 Adv. Quantum Chem. 19 1 [19] Silverstone H J and Moats R k 1981 Phys. Rev. A 23 1645 [20] Čížek J and Vrscay E R 1982 Int. J. Quantum Chem. 21 27 [21] Turbiner A V 1982 Z. Phys. A 308 111 [22] Guillou J C L and Justin J Z 1983 Ann. Phys. 147 57 [23] Turbiner A V 1984 J. Phys. A 17 859 [24] Guth E 1929 Z. Phys. 58 368 [25] Schiff L I and Snyder H 1939 Phys. Rev. 55 59 [26] Praddaude H C 1972 Phys. Rev. A 6 1321 [27] Killingbeck J 1981 J. Phys. B 14 L461 [28] Jason A C G 1984 Phys. Rev. A 29 132 [29] Kravchenko Y P and Liberman M A 1996 Phys. Rev. A 54 287 [30] Bayrak O, Soylu A and Boztosun J 2011 Phys. Lett. 107 [31] Li H et al 2007 Chin. Phys. Lett. 24 851 [32] Duan Y S et al 2004 Chin. Phys. Lett. 21 1714 [33] Zhao Y H et al 2008 Chin. Phys. B 17 1720 [34] Hai W H 1998 Chin. Phys. Lett. 15 472 [35] Zhao Y H et al 2009 Acta Phys. Sin. 58 734 [36] Yuan L et al 2012 Chin. Phys. B 21 103103 [37] Hai W H et al 2000 Phys. Rev. A 61 052105

013101-5

Chinese Physics Letters Volume 30

Number 1

January 2013

GENERAL 010201 A Method of Choosing the Optimal Number of Singular Values in the Inverse Laplace Transform for the Two-Dimensional NMR Distribution Function JIANG Zhi-Min, WANG Wei-Min 010301 Double Barrier Resonant Tunneling in Spin-Orbit Coupled Bose–Einstein Condensates LI Zhi, WANG Jian-Zhong, FU Li-Bin 010302 An Alternative Approach to Construct the Initial Hamiltonian of the Adiabatic Quantum Computation DUAN Qian-Heng, ZHANG Shuo, WU Wei, CHEN Ping-Xing 010303 One-Way Quantum Computation with Cluster State and Probabilistic Gate DIAO Da-Sheng 010304 Computation of Quantum Bound States on a Singly Punctured Two-Torus CHAN Kar-Tim, Hishamuddin Zainuddin, Saeid Molladavoudi 010305 Four-State Modulation Continuous Variable Quantum Key Distribution over a 30-km Fiber and Analysis of Excess Noise WANG Xu-Yang, BAI Zeng-Liang, WANG Shao-Feng, LI Yong-Min, PENG Kun-Chi 010306 Fractals in Quantum Information Process BI Feng, LI Chuan-Feng 010501 Efficiency at Maximum Power of a Quantum Dot Heat Engine in an External Magnetic Field ZHANG Yan-Chao, HE Ji-Zhou 010502 Nonergodic Brownian Motion in a Collinear Particle-Coupled Harmonic Chain Model LU Hong, BAO Jing-Dong 010503 Single-Hopf Bursting in Periodic Perturbed Belousov–Zhabotinsky Reaction with Two Time Scales LI Xiang-Hong, BI Qin-Sheng 010601 Accuracy Evaluation of NIM5 Cesium Fountain Clock LIU Nian-Feng, FANG Fang, CHEN Wei-Liang, LIN Ping-Wei, WANG Ping, LIU Kun, SUO Rui, LI Tian-Chu 010701 Research of Infrared Imaging at Atmospheric Pressure Using a Substrate-Free Focal Plane Array WU Jian-Xiong, CHENG Teng, ZHANG Qing-Chuan, ZHANG Yong, MAO Liang, GAO Jie, CHEN Da-Peng, WU Xiao-Ping

THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS 011201 Photoproduction of Large Transverse Momentum Dimuonium (µ+ µ− ) in Relativistic Heavy Ion Collisions YU Gong-Ming, LI Yun-De

NUCLEAR PHYSICS 012101 Cluster Structure in Be Isotopes within Point-Coupling Covariant Density Functional TANG Zhong-Hua, LI Jia-Xing, JI Juan-Xia, ZHOU Tao

ATOMIC AND MOLECULAR PHYSICS 013101 Variational-Integral Perturbation Corrections for Hydrogen Atoms in Magnetic Fields ZHAO Yun-Hui, PAN Yi-Qing, LI Wen-Juan, DENG Xia, HAI Wen-Hua 013201 Visible Light Emission in Highly Charged Kr17+ Ions Colliding with an Al Surface YANG Zhi-Hu, XU Qiu-Mei, GUO Yi-Pan, WU Ye-Hong, SONG Zhang-Yong

013701 Trapping, Transporting, and Splitting Cold Molecules Employing a Spatial Liquid Crystal Modulator GONG Tian-Lin, HUANG Yun-Xia, MU Ren-Wang, JI Xian-Ming, YANG Xiao-Hua 013702 Demonstration of Cold 40 Ca+ Ions Confined in a Microscopic Surface-Electrode Ion Trap CHEN Liang, WAN Wei, XIE Yi, WU Hao-Yu, ZHOU Fei, FENG Mang 013703 Cooling and Crystallization of Trapped 113 Cd+ Ions for Atomic Clock WANG Shi-Guang, ZHANG Jian-Wei, MIAO Kai, WANG Zheng-Bo, WANG Li-Jun

FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) 014201 Birefringence Optical Feedback with a Folded Cavity in HeNe Laser WU Yun, TAN Yi-Dong 014202 Research on High-Intensity Picosecond Pump Laser in Short Pulse Optical Parametric Amplification PAN Xue, PENG Yu-Jie, WANG Jiang-Feng, LU Xing-Hua, OUYANG Xiao-Ping, CHEN Jia-Lin, JIANG You-En, FAN Wei, LI Xue-Chun 014203 Effect of Laser Pulse Width on the Laser Lift-off Process of GaN Films CHEN Ming, ZHANG Jiang-Yong, LV Xue-Qin, YING Lei-Ying, ZHANG Bao-Ping 014204 Spectral Characteristic Based on Fabry–P´ erot Laser Diode with Two-Stage Optical Feedback WU Jian-Wei, Bikash NAKARMI 014205 Principal State Analysis for a Compact in-Line Fiber Polarization Controller LI Zheng-Yong, WU Chong-Qing, WANG Zhi-Hao, QIN Tao, WANG Yi-Xu 014206 Magnetic Field Induced Spectroscopy of 88 Sr Atoms Probed with a 10 Hz Linewidth Laser LIN Yi-Ge, WANG Qiang, LI Ye, LIN Bai-Ke, WANG Shao-Kai, MENG Fei, ZHAO Yang, CAO Jian-Ping, ZANG Er-Jun, LI Tian-Chu, FANG Zhan-Jun 014207 High Efficiency Grating Coupler for Coupling between Single-Mode Fiber and SOI Waveguides ZHANG Can, SUN Jing-Hua, XIAO Xi, SUN Wei-Min, ZHANG Xiao-Jun, CHU Tao, YU Jin-Zhong, YU Yu-De 014208 Two-Crystal Design and Numerical Simulations for High-Average-Power Second-Harmonic Generation ZHONG Hai-Zhe, YUAN Peng, ZHU He-Yuan, QIAN Lie-Jia 014209 Stop Band Gap in Periodic Layers of Confined Atomic Vapor/Dielectric Medium LI Yuan-Yuan, LI Li, LU Yi-Xin, ZHANG Yan-Peng, XU Ke-Wei 014501 Effect of Interstitial Media on Segregation in Vertically Vibrated Granular Mixtures YUAN Xiao-Xian, LI Liang-Sheng, WEN Ping-Ping, SHI Qing-Fan, ZHENG Ning 014701 Dynamic Characteristics of Gas Transport in Nanoporous Media SONG Hong-Qing, YU Ming-Xu, ZHU Wei-Yao, ZHANG Yu, JIANG Shan-Xue

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES 015201 Super-X Divertor Simulation for HCSB-DEMO Conception Design ZHENG Guo-Yao, PAN Yu-Dong, FENG Kai-Ming, HE Hong-Da, CUI Xue-Wu 015202 Influence of Discharge Voltage on Charged Particles in a Multi-Dipole Device in the Presence of an Ion Collecting Surface M. K. Mishra, A. Phukan, M. Chakraborty

CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES 016101 Formation of Co-implanted Silicon Ultra-Shallow Junctions for Low Thermal Budget Applications Rehana Mustafa, S. Ahmed, E. U. Khan

016202 Temperature Effects of Electrorheological Fluids Based on One-Dimensional Calcium and Titanium Precipitate YAN Ren-Jie, WU Jing-Hua, LI Cong, XU Gao-Jie, ZHOU Lu-Wei 016203 Dynamic Mechanical Behavior and Failure Mechanism of Polymer Composites Embedded with Tetraneedle-Shaped ZnO Whiskers RONG Ji-Li, WANG Dan, WANG Xi, LI Jian, XU Tian-Fu, LU Ming-Ming, CAO Mao-Sheng 016801 Electric-Field Switching of Bright and Dark Excitons in Semiconductor Crossed Nanowires LI Xiao-Jing, K. S. Chan

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 017101 Electronic Structure, Lattice Dynamics and Thermoelectric Properties of PbTe from First-Principles Calculation LU Peng-Xian, QU Ling-Bo 017102 Ultrafast and Broadband Terahertz Switching Based on Photo-Induced Phase Transition in Vanadium Dioxide Films CHEN Zhi, WEN Qi-Ye, DONG Kai, SUN Dan-Dan, QIU Dong-Hong, ZHANG Huai-Wu 017103 First-Principles Calculation of Lithium Adsorption and Diffusion on Silicene HUANG Juan, CHEN Hong-Jin, WU Mu-Sheng, LIU Gang, OUYANG Chu-Ying, XU Bo 017201 Spin-Dependent Electron Transport in an Armchair Graphene Nanoribbon Subject to Charge and Spin Biases ZHANG Xiao-Wei, ZHAO Hua, SANG Tian, LIU Xiao-Chun, CAI Tuo 017202 Efficiency Enhancement of MEH-PPV:PCBM Solar Cells by Addition of Ditertutyl Peroxide as an Additive LI Yan-Fang, YANG Li-Ying, QIN Wen-Jing, YIN Shou-Gen, ZHANG Feng-Ling 017301 On the Voltage and Frequency Distribution of Dielectric Properties and ac Electrical Conductivity in Al/SiO2 /p-Si (MOS) Capacitors Ahmet Kaya, S¸emsettin Altındal, Yasemin S¸afak Asar, Zekayi Sönmez 017302 High Deep-Ultraviolet Quantum Efficiency GaN P–I–N Photodetectors with Thin P-GaN Contact Layer LIAN Hai-Feng, WANG Guo-Sheng, LU Hai, REN Fang-Fang, CHEN Dun-Jun, ZHANG Rong, ZHENG You-Dou 017303 Electronic Properties of a Phenylacetylene Molecular Junction with Dithiocarboxylate Anchoring Group LIU Wen, XIA Cai-Juan, LIU De-Sheng 017401 Intra-Valley Spin-Triplet p + ip Superconducting Pairing in Lightly Doped Graphene ZHOU Jian-Hui, QIN Tao, SHI Jun-Ren 017402 Experimental Investigation of the Electronic Structure of Ca0.83 La0.17 Fe2 As2 HUANG Yao-Bo, RICHARD Pierre, WANG Ji-Hui, WANG Xiao-Ping, SHI Xun, XU Nan, WU Zheng, LI Ang, YIN Jia-Xin, QIAN Tian, LV Bing, CHU Ching-Wu, PAN Shu-Heng, SHI Ming, DING Hong 017501 Magnetoelastic Anisotropy of FeSiB Glass-Coated Amorphous Microwires LIU Kai-Huang, LU Zhi-Chao, LIU Tian-Cheng, LI De-Ren 017601 Detecting Larmor Precession of a Single Spin with a Spin-Polarized Tunneling Current GUO Xiao-Dong, DONG Li, GUO Yang, SHAN Xin-Yan, ZHAO Ji-Min, LU Xing-Hua 017801 Dosimetric Characteristics of a LKB:Cu,Mg Solid Thermoluminescence Detector Yasser Saleh Mustafa Alajerami, Suhairul Hashim, Ahmad Termizi Ramli, Muneer Aziz Saleh, Ahmad Bazlie Bin Abdul Kadir, Mohd. Iqbal Saripan 017802 Phase Shift of Polarized Light after Transmission through a Biaxial Anisotropic Thin Film HOU Yong-Qiang, LI Xu, HE Kai, QI Hong-Ji, YI Kui, SHAO Jian-Da 017901 Current Density-Sensitive Welding of a Semiconductor Nanowire to a Metal Electrode TAN Yu, WANG Yan-Guo

017902 Elimination of the Schottky Barrier at an Au-ZnSe Nanowire Nanocontact via in Situ Joule Heating TAN Yu, WANG Yan-Guo

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 018101 Fabrication of Thin Graphene Layers on a Stacked 6H-SiC Surface in a Graphite Enclosure DENG Peng-Fei, LEI Tian-Min, LU Jin-Jun, LIU Fu-Yan, ZHANG Yu-Ming, GUO Hui, ZHANG Yi-Men, WANG Yue-Hu, TANG Xiao-Yan 018102 Phase Structure and Electrical Conduction of CaTi1−x Scx O3−δ Ceramics ZHANG Qi-Long, LIU Yang, YANG Hui 018103 Characterization of Modified Tapioca Starch in Atmospheric Argon Plasma under Diverse Humidity by FTIR Spectroscopy P. Deeyai, M. Suphantharika, R. Wongsagonsup, S. Dangtip 018501 A SQUID Bootstrap Circuit with a Large Parameter Tolerance ZHANG Guo-Feng, ZHANG Yi, Hans-Joachim Krause, KONG Xiang-Yan, Andreas Offenhäusser, XIE Xiao-Ming 018701 Enhanced Response to Subthreshold Signals by Phase Noise in a Hodgkin–Huxley Neuron ¨ Hua-Ping KANG Xiao-Sha, LIANG Xiao-Ming, LU 018901 A Micro-Community Structure Merging Model Using a Community Sample Matrix LI Lin, PENG Hao, LU Song-Nian, TIAN Ying

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS 019401 Large Bi-Polar Signature in a Perpendicular Electric Field of Two-Dimensional Electrostatic Solitary Waves Associated with Magnetic Reconnection: Statistics and Discussion LI Shi-You, ZHANG Shi-Feng, DENG Xiao-Hua, CAI Hong 019601 Spatial Distribution and Anisotropy of Energetic Particles Accelerated by Shock Waves: Focused Transport Model ZUO Ping-Bing, FENG Xue-Shang