Different approach to the Stark effect: Application to the hydrogen

VOLUME 42, NUMBER 7

PHYSICAL REVIEW A

1

OCTOBER 1990

Different approach to the Stark effect: Application to the hydrogen ground state O. L. S. Filho, A. L. A. Fonseca, H. N. Nazareno, and P. H. A. Guimaraes Departamento

de Fssiea da Uni Uersidade de Brasilia, 709IO Brasilia, Distrito Federal, Brazil (Received 31 January 1990)

A method is proposed to evaluate the energy levels and wave functions of atomic hydrogen in a uniform electric field. A self-consistent criterion is used and the method proves to be sufficiently accurate and easy to implement even for high electric-field strengths. We present the results obtained for the ground-state level and compare them with those obtained by other authors.

I. INTRODUCTION

=g 5. 14 X 10 V/cm .

ao

The Stark effect on atoms, or spectral line splitting in atoms placed in an external electric field, has been studied ever since Stark pioneered this concept. The major difficulty in its treatment arises because of the potential energy form associated with the field, no matter how weak the field is. Thus, the effect of the "perturbation" is that it tends to strip the electron from the atom. In this work we have undertaken the calculation of the discrete levels associated with a hydrogen atom placed under a static electric field. We propose a self-consistent method that provides the energy levels and also a welldefined wave function that can be easily evaluated. At the same time it is worth mentioning that we were able to obtain the electron stripping degree as a function of the intensity of the applied field 8. In our model the dependence of the wave function and the energies on 6 is easily implemented. In Sec. II we present the method which basically consists of writing the Schrodinger equation in paraboliccylindric coordinates, in which the equation is separable, and express the wave-function components in terms of polynomials whose order is determined self-consistently. By doing so we arrive at a transcendental equation for the energy levels that constitutes our fundamental result. In Sec. III, we apply the method to the ground state of the H atom in the presence of a uniform electric field. Then, we discuss and compare our results with other works.

'

In the parabolic-cylindrical write (1) as

+a

„Bq'

a

Bu

Bu

1

+=E% . with the electric-field

strength

by

)

(3)

4= U(u) V(v)F($),

(4)

where 4

=T+2

v=f

(5)

Z

part of the wave function

and the angular pressed as

can be ex-

F(P)=e™, considering the cylindrical symmetry. From Eqs. (3) and (4) we can write the following ordinary differential equations:

d (udU) du

+Eu

2

du

dv

+

b

Ev

2

m

gu

4

4u m

4v

gv

4

U=O,

(8)

V=O,

where the two separation constants a and b satisfy

a+b =1

.

(10)

Guided by the situation encountered in the field-free case, where the hydrogen radial wave function is of the form R (r) =e " "p„(r), where p„ is a polynomial, we propose for the wavefunction components the following structure: m ~~e —yu~2P U (u )— (y u ) (u )

and 42

8

Now we write the wave function as the product

1

where g is connected the equation

we can

v

4

2

dv

T

u

— + 1+E(u+v) + g(u v

d (vdV)

— + —+gz 2

4

Bv

system

J

II. METHOD We adopt the parabolic system to deal with the problem of the hydrogen atom in a uniform electric field of strength 6' applied along the z axis, since in this system the Schrodinger equation is separable in three ordinary differential equations. The Schrodinger equation in atomic units is given by

+1 1+1

Bq'

Bv

coordinate

4008

l

DIFFERENT APPROACH TO THE STARK EFFECT:

42

V(u)=(yu)'

'e r" P, (v),

'

(13)

E}'/ . According to Eqs. (12) and (13) we where y=( 2— shall have the following equations for the unknown P, (u) and P2(u): u

d P1 dQ

+(m+1 —u)

du

u

+(m+1 —v)

+

dU

dU

b,'

—1+ lml 2

=a/y,

bi

=bly,

=g

g&

(14}

=0,

(15)

j

(16)

In the field-free case, on the other hand, the connection between the separation constants and the integers n, and n2 is given by the following relations: n

1

=a

1+ fmf 1

=b—1+lml

n2

(17)

1

we have assumed for our problem

Consequently

a( =n,

~

2

+ 1+ Iml +f, (g) ),

1+ Im

b, =n2+

I

+f, (g, ), (18)

where the unknown are such that

field-dependent

functions

f,

and

f (0) =f2(0) =0 .

f

2

(19)

1

To proceed with the presentation of the method we write

U(u)=u

lml

/2e

u/2L

Iml

nl

(ti)Q (u)

(20)

and

y ( v ) —v

I

m

I

/2e

—v /2L

I

m

n2

I

—~ [&u]lml/2e

(21)

( v )Q 2 ( v )

where L„ is the associate Laguerre function which in the free-field case is the associated Laguerre polynomial. The equations for the last introduced functions are

—(u +v)/2eimg (

u)Q, (u)Q

(u),

(24)

or in terms of the parabolic-cylindrical

&y Iml /2plml XL

where we have introduced dimensionless variables by replacing u and u by (yu) and (yu), respectively, and a&

y)

qi(p p z )

+g,'4 P2

u

XL„' l(u)L„I

u' + a, —1+ lml +g1'4 2 P, =0,

d P2

qi(&

'

I

(r

coordinates: — /2e rre™ 4L Iml ( r +z )

—z)Q, (r +z)Q2(r —z),

(25)

where X is a normalization constant, making explicit the cylindrical symmetry of the present problem. We see that the physical significance of the function Q, (u)Q2(u) is that it modulates the zero-field function u, P).

P„„(u, 1

2

By taking into account Eqs. (10) and (18) together with the field-free relation between the quantum numbers n

=n, +n +Iml+1,

(26)

and using Eqs. (10) and (16) we get for the energy the following transcendental equation 1

2f.

1+f1(gl }+f2(gt }]'

Equation (27) is our fundamental result and makes the desired connection between the energy levels of the hydrogen atom when placed under an electric field and its intensity. Notice that this equation can be applied to all energy levels. The way to solve Eq. (27) is as follows. We first choose the level that will be studied by giving a value for the main quantum number n. The degenerate sublevel will be determined by the choice of the values for n1, n2, and Iml. Then, we introduce polynomials in the expressions for Q, and Q2 and truncate the expansion in a previously determined order, say This will provide two di6'erent equations in terms of 2, and g, for each jth coefFicient. By making these two equations equal, for the sake of consistency, we obtain two nonlinear equations in f2, g, , one for each jth coefficient. We can then solve this set of nonlinear equations for an initial value of the and 2 and use Eq. (27) to get energy (or g& ) to obtain the next value of the energy. Repeated applications of these steps will allow the energy to be determined through a self-consistent procedure.

j.

f„ f

f„

f,

f

III. APPI. ICATION GF THE METHOD uLI

nl

Q1

+ (Iml+1)LInl

I

u

AND DISCUSSION OF THE RESULTS

nl

I

L

vL lmlQ

+ (Iml+1}L ml +

v

f2 —g,

lml

Q

L

(22)

()

Q2

L'

Q

=0.

(23)

We are now in position to write the final wave function as

We have applied our method to evaluate the eigenfunctions and eigenvalues associated with the ground state of the H atom under electric fields of various strengths. In Fig. 1 we have plotted the eA'ective potential energy for two of the electrical field. We were able to accomplish our goal and covered cases from weak to strong fields. Of course, when we increase the value of g it is necessary to include more terms in the wave-function expansion. The expansion convergence can be studied from Table I where we quote the energy values as a function of the field strength and the number of terms included in the wave-function expansion.

FILHO, FONSECA, NAZARENO, AND GUIMARAES

4010

TABLE I. Values of the energies of the Stark effect of the field strength.

42

of the polynomials and various values

in the ground state for different degrees

Field (a.u. )

8 functions

9 functions

Energy (a.u. ) 11 functions

14 functions

15 functions

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0. 100

—0.500 000 —0.500056 —0.500 226 —0.500 509 —0.500 909 —0.501 429 —0.502 072 —0.502 845 —0.503 753 —0.504 807 —0.506 016 —0.507 393 —0.508 953 —0.510 717 —0.512 705 —0.514 945 —0.517 468 —0.520 309 —0.523 506 —0.527 101 —0.531 132

—0.500000 —0.500056 —0.500 226 —0.500 509 —0.500 909 —0.501 428 —0.502 071 —0.502 844 —0.503 752 —0.504 804 —0.506 012 —0.507 387 —0.508 945 —0.510 706 —0.512 691 —0.514 927 —0.517 446 —0.520 281 —0.523 473 —0.527 062 —0.531 086

—0.500000 —0.500056 —0.500 226 —0.500 509 —0.500 909 —0.501 428 —0.502 071 —0.502 844 —0.503 752 —0.504 804 —0.506 012 —0.507 388 —0.508 946 —0.510 707 —0.512 693 —0.514 930 —0.517 449 —0.520 286 —0.523 478 —0.527 068 —0.531 093

—0.500 000 —0.500056 —0.500 226 —0.500 509 —0.500 909 —0.501 429 —0.502 072 —0.502 844 —0.503 752 —0.504 805 —0.506 012 —0.507 387 —0.508 946 —0.510 707 —0.512 693 —0.514 929 —0.517 448 —0.520 285 —0.523 477 —0.527 068 —0.531 092

—0.500000 —0.500056 —0.500226 —0.500 509 —0.500 909 —0.501 429 —0.502 072 —0.502 844 —0.503 752 —0.504 805 —0.506012 —0.507 388 —0.508 946 —0.510 707 —0.512 693 —0.514 930 —0.517 449 —0.502 285 —0.523 477 —0.527 067 —0.531 092

-0.

We notice that for inoderate fields (g 025) we get, a quite good convergence with 8 function while for intense fields (g 1) we have to consider up to 15 functions to reach the desired precision of 10 in the eigenvalues. This shows that the proposed model is rapidly convergent since not many terms are needed in the wave-function expansion in order to reach self-consistency. We have plotted in Figs. 2, 3, and 4 the wave-function contours for four difT'erent values of the probability am-

-0.

plitude for g=0.04, 0.07, and 0.09 a. u. , respectively, on the relevant xz plane. We can see that the anisotropy introduced by the field applied in the positive z direction is represented in the wave function. We can also see the appearance of the classical region predicted by an a priori analysis of the form of the field. Increasing the value of the field we can see that the lobe-shaped central contour

6.01.01—

5.0-

0, 50-

I.O-

0.00

20 O

~ -1.52-

v l.o-

& -2.02-

lK

0.0-

UJ

QJ

2.53

-3.03-

- I.O-

-3.54-

- 2.0-

-4.04-

- 3.0 -S.O

-4.54-5,05

I

-10.00

-5.00

5.00

0.00 Z

-6.0

1

- 3.0

I

0.0 X (a. u.)

I

3.0

6.0

S.O

10.00

(a.u. }

FIG. 2. Wave-function contours for the following values of the probability amplitude (not normalized): 0.05 ( ———), 0.09 ( ———), 0. 13 (- ). The applied ), and 0.20 ( field is g=0.04 a.u. ~

FIG. 1. Effective potential plied field.

,

energy for two values

g=0.01 a.u, .

-

~,

g=0. 10 a. u.

of the ap-

~

~

DIFFERENT APPROACH TO THE STARK EFFECT:

4011

S.O-

4.70 5.0-

3.70-

4.0-

2.70-

3.0~

2.0-

I

O

N I.o-

N

.70-

0.70-

0.0-

-0.30-

1.0-

-1.30-

2.03,0

-9.0

I

-S.O

-3.0

0, 0 X (o.a)

3.0

-I.o

FIG. 3. Same as Fig. 2 for the case g=0.07 a.u. 0.09 ( ———. ), 0. 13 ( ———), 0. 17 ( ~

and ampli), and 0.20

).

is stretched along the direction of the applied field, at the same time that the opening of the central contour occurs earlier than for weaker fields. This can be interpreted so because of the probability enhancement in the classical region. In Figs. 5(a)—5(c) we show the polar projection in the xz plane of the angular behavior of the probability density (not normalized), using for each one various values of fixed radial distances r. It is easily seen that for smaller

TABLE II. Normalization constants, probability points for various values of the field.

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0. 100

1

-3.0

Normalization constant

0. 177 245 385 1 x 10' 0. 177 279 984 1 X 10' 0. 177 384050 8 X 10' 0. 177 559 479 4 x 10' 0. 177 809 6502 x 10' 0. 178 139 838 6 X 10' 0. 178 558 913 6 x 10' 0. 179 084 830 7 x 10' 0. 179 758 558 9 x 10' 0. 180 674431 8 x 10' 0. 182047 8901 x 10' 0. 184 327 812 1 X 10' 0. 188 387 525 OX 10' 0. 195 780 087 5 X 10' 0.208 953 602 5 x10' 0.231 284 7544 x 10' 0.266 681009 1 x 10' 0.318 931 767 7 X 10' 0.391 267 966 5 X 10' 0.486 206 570 6 x 10' 0.605 938 468 6 X 10'

3.0

0.0

{o.u.)

S.O

9.0

FIG. 4. Same as Fig. 2 for the case g=0.09 a.u. and ampli—.—), 0. 14 ( ———), 0. 17 ( . . ), and 0.23 0.09 ( —.

tudes (

Field (a.u. )

-S.O

X

tudes (

-2.' 30

9.0

S.O

).

values [Figs. 5(a) and 5(b)] of this distance the wave function is stretched in the positive z direction while for intermediate values of r the stretching [Fig. 5(c)] is in opposite direction and finally for greater values of r we come back to the situation, described in Fig. 5(b). We have calculated on the other hand a parameter that measures the stripping of the electron, namely, the ratio between the integral of the square modulus of the wave function on the classical region and the total probability.

of being

in the continuum,

Continuum probability

and classical turning

Turning u,

0.8774947 X 10 0. 165 9606 x 10-" 0.4140761 x 10-" 0.309002 9 x 10-" 0. 124086 7 x10-" 0.431 837 7 x 10-" 0.448 681 3 x 10-" 0.243 779 1 x 10-" 0.913 5406x10 0.277 324 9 x 10-" 0.728 165 0 x 10-" 0. 170479 2 x 10-" 0.358 3461x10-" 0.670 526 3 x 10-" 0. 110 515 9 x 100. 160004 3 x 100.206221 6X 10 0.242 793 9 x 10-~ 0.268 497 6 x 10 0.285 231 5 X 10

"

v

points: (a.u. )

200.00, 100.10, 66.78, 50. 16, 40.20, 33.58, 28. 86, 25.33, 22. 60, 20.43, 18.66, 17.20, 15.97, 14.93, 14.05, 13.28, 12.62, 12.05, 11.56, 11.13,

1.002 1.005 1.007 1.009 1.011 1.013 1.015 1.016 1.018 1.019 1.021 1.022 1.023 1.024 1.026 1.027 1.028 1.028 1.029 1.030

FILHO, FONSECA, NAZARENO, AND GUIMARAES

4012

42

We have also calculated the average values of the coordinates r and z (Table III). We can see that as the field increases, the average distances also increase, making the electron more delocalized; these average distances are greater than the classical turning points calculated by For stronger fields this dislocation is mostly Alexander. due to the change in the z direction because z and r are closer to each other. It is important to observe that those average distances increase rapidly for medium fields but in the stronger field region they begin to increase at a lower rate. To understand it, we first have to note that in the stronger field region the electron average distances

We can call it the probability of being in the continuum and it gives us the probability of tunneling of the electron. The results agree quite well with the physics of the problem. In fact this is approximately the behavior we expect for the transmission coeScient of a particle passing through a potential barrier, except for the fact that now we have a bound initial state and the barrier changes its shape with the field. The initially bound state and the static perturbation are responsible for the disappearance of the resonance points and this makes the probability of being in the continuum approach one asymptotically. We quote these results in Table II.

3.0I.O. 2.0-

~

/ I.O-

5.0-

essgg

0.0-

0.0-

X

/

X

I

I

~

- 2.0-

~

l

-50.

/

I

r

I

r

I

-3.0I

- 2.0

- 4.0

I

l

- I.5 -s.o

I

4.0

2.0

0.0

8.0

- 4.0

0.0

(o.u.l

Z

/

l

- I.O-

-4.0

r

/

4.0

/

e.o

X (o. u. )

!

3.0-

2.0I.O-

o X

~

0 4IRllggy ~

L

r

/

0.0-

- I.O-

-2.0" ~

- 5.0-

-4.0

- 2.l

-0.8 Z

FIG. 5.

(

~

~

distances: 4 ( dial distances: 6

O. S

plot of the probability density (not normalized) for the following values of the radial distance r: 3.5 ). The applied field is g=0.4 a.u. (b) Same as in (a), for the case g=0.06 a.u. and radial ), 5 ( ), 4.5 ( - - ), 5 ( — ), and 5.5 ( ———). Notice different scales on the x and z axes. (c) Same as (b) for ra— — —), and 7.2 ( ———). Notice different scales on the x and z axes. . (6. 6. 5 8 ( ), ), ~

~

~

(

0.0

to. Q

(a) Contour

———), 4 ( ———), 4.5 ( ~

8

~

—.—.

~

~

~

DIFFERENT APPROACH TO THE STARK EFFECT:

42

TABLE III. Average distances and classical turning points

-5, 000

for various values of the field. Field (a.u. )

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0. 100

r, „(a.u. )

z, „(a.u. )

1.5000 1.5006 1.5023 1.5052 1.5094 1.5151 1.5227 1.5340 1.5538 1.5944 1.6852 1.8861 2.3029 3.0865 4.3697 6. 1360 8. 1252 9.9583 11.3813

0.0000 0.0225 0.0452 0.0682 0.0917 0. 1161 0. 1420 0. 1710 0.2083 0.2667 0.3764 0.5986

12.3502 12.9453

Turning points: r, z (a.u. )

1.0421 1.8612 3. 1914 5.0151 7.0654 8.9539 10.4212 11.4228 12.0416

4013

-5.050-

-5.100-

100.51, 99.51 50.53, 49.52 33.87, 32.87 25. 56, 24. 55 20.58, 19.57 17.26, 16.25 14.89, 13.88 13.12, 12. 11 11.72, 10.74 10.66, 9.64 9.77, 8.75 9.03, 8.02 8.41, 7.40 7.88, 6.87 7.43, 6.42 7.03, 6.02 6.69, 5.68 6.39, 5 39 6. 13, 5 13 5.90, 4.90

-5.150-5. 200-

-5. 250-

-5.300-5.350

0.00

0.02

0.06

0.04 g

0.08

~

FIG. 6. Ground-state strength (in atomic units).

~

are inside the classical region, as shown in Table III, and a competition between bound and free states arises because of the type of potential considered. A brief look at Fig. 1 shows us that as the field grows the potential barrier becomes thinner and smaller and, therefore, easier for the electron to be tunneled, but it approaches higher values near the origin, giving a tendency to localize the

energy

as a function

TABLE IV. Comparison between the results of various theories applied to the Stark effect in the Energies (a.u. ) Hehenberger,

0.005 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0. 100

McIntosh, Silverstone'

—0.500 056 284 794 —0.500 225 556 05 —0.500 909 225 254 —0.502 074 263 6 —0.5D3 771 46 —0.5D6 099 —0.509 18 —0.512 94 —0.516 7 —0.521 9 —O. S28 1

'Perturbation theory (see Ref. 3). Reference 4. 'Reference S. Values for polynomials of degree 15.

of the

field

electron. The values in Table III give the net behavior of the electron average distances and the values of the classical turning points. Finally we give in Fig. 6 a plot of the ground-state energy eigenvalue as a function of the field strength. As a concluding remark we would like to refer to Table IV where we have compared our results with the previous

ground state.

Field (a.u. )

0.10

(e.u.)

Alexander

—0.500 056 28 —0.500 225 56 —0.500 909 22 —0.502 074 27 —0.503 771 5 —0.506 105 —0.509 20 —0.513 0 —0.517 5 —0.522 2 —0.527

and Brandas'

—0.502 092 5 —0.503 771 8 —0.506 105 4 —0.509 203 5 —0.513 075 —0.517 56 —0.522 4 —0.527 4S

Present work

—0.500056 —0.500 226 —0.500 909 —0.502 072 —0.503 752 —0.506012 —O. S08 946 —0.512 693 —0.517 449 —0.523 477 —0.531 092

4014

FII HO, FONSECA, NAZARENO, AND GUIMARAES

and HehenAlexander, calculations by Silverstone, which and Brandas, happen to show a berger, McIntosh, good agreement with our results. Calculations for the excited states based on this new proposed method are in progress.

~J. Stark, Ann. Phys. 43, 965 (1914). For a treatise on the Stark effect see N. Ride, Atoms and Molecules in Electric Fields (Almqvist and %'iksells, Stockholm,

1976).

42

ACKNOWLEDGMENTS

The authors wish to thank the Conselho Nacional de Cientifico e Technologico (CNPq) for Desenvolvimento partial financial support during the course of this work.

H. J. Silverstone, Phys. Rev. A 1S, 1853 (1978). 4M. H. Alexander, Phys. Rev. 178, 34 (1969). 5M. Hehenberger, H. McIntosh, and E. Brandas, Phys. Rev. A 10, 1494 (1974).