[L NUOVO CIMENTO

VOL. 34 B, lq. 1

11 Luglio 1976

Hydrogen Atom in a Strong Magnetic Field. /~, GALIND0 Faculty o! Physics - Madrid

•.

PASCUAL

Faculty o] Physics

.

Barcelona

(ricevuto il 20 Febhraio 1976)

Summary. - - Taking into account the perturbative expansion useful for weak magnetic fields as well as some rcsults for very strong ones, we are able through the use of rational approximants to give an analytic expression for the lowest energy levels for a hydrogen atom valid for any value of the magnetic field.

I.

-

Introduction.

I n the last few years great effort has been devoted to the s t u d y of the h y d r o g e n a t o m (without spin) in the presence of strong magnetic fields. This problem presents considerable m a t h e m a t i c a l difficulties, ~nd a good understanding of it seems f u n d a m e n t a l for several problems arising in astrophysics, plasma physics and solid-state physics (x.z). I t is r a t h e r c o m m o n to measure the intensity of the magnetic field b y the dimensionless p a r a m e t e r 7 defined as

(i)

__ p ~RB ~'--

4.258980(18)" 10-1~ G -1

where / ~ stands for the Bohr m a g n e t o n and R fox" the r y d b e r g (both with the reduced mass of ttte electron}. (I) H. S. BRANDI: Phys. Rev. A, 11, 1835 (1975). (2) D. CABIn, E. F).BRI and G. FIORIO: Nuavo Cimen$o, 10 B, 185 (1972). 155

156

A. G A L I N I ) O a l l d

P. I ' A S C U A L

W e shall not review the existing calculations for the g r o u n d - s t a t e energy a n d for the energies of the first excited states, since this is done in a fairly comp l e t e w a y in (1.2). Nevertheless we would like to p o i n t out t h a t m a n y of those calculations cover only a restricted r a n g e of values of ~. F o r instance in (*) a n a c c u r a t e numerical calculation is carried out for the ground energy for several values of y between 0.1 a n d 5, as well as for the energy of the first excited s t a t e for values of 7 in the r a n g e 0.1 to 2. F u r t h e r m o r e m o s t of the available calculations rest upon one of the following m e t h o d s : i) numerical, ii) variational, iii) adiabatic, a n d t h e y are unable to provide analytical expressions for the energy in t e r m s of y. I n this p a p e r we present a c o m p l e t e l y different a p p r o a c h to the p r o b l e m : we c a r r y out a p e r t u r b a t i v e calculation which provides i n f o r m a t i o n on the energy levels for y<
2.

-

Properties of the ground state.

The Sehr6dinger equation for a hydrogenic s y s t e m in a c o n s t a n t m a g n e t i c field along the z-axis can be written, in spherical co-ordinates and in the gauge where A = (B x r)/2, as

(2)

~-0

2 ~ + ~

4-7L=+~o'~

3V5

o ~

where ~ ~ r/ao, L is the a n g u l a r - m o m e n t u m o p e r a t o r in units of ~ and the energy is m e a s u r e d in rydbergs. I t is d e a r f r o m t h e s t r u c t u r e of H t h a t the p a r i t y and t h e maguaetic q u a n t u m n u m b e r m are constants of motion. :In order to c a r r y out a p e r t u r b a t i v e calculation we split the above-given H a m i l t o n i a n in the following w a y : H =Ho§

(3)

Ho--

~ ~q~

2 ~ L2 9aq-4-~'

2 q'

H1 = r L = + gy~.o--- 3 C ~ 7 - q ~ Y,(e) 9

I[YDROGEN

ATOM

IN

A

flTRONG

MAf~NETIC

FIELD

l ~

We shall write t h e R a y l e i g h - S c h r S d i n g e r p e r t u r b a t i o n series for t h e g r o u n d state ~nerg'y a.s E = E (~ : E ~) t - E('J-[- ...,

(1)

where the unperi.urbed e n e r g y is E (o) = - - 1 ealeulated in t h e usual m a n n e r (a):

a n d t h e correction t e r m s can be

E (~ = <~(') H~-- El~)Iv;(')>, (5)

. . . . . . . .

, ~ ....

~ . . . . .

in t e r m s of t h e u n p e r t u r b e d au(t its c o r r e c t i o n s

(eJ)

9 . . . . . . . . . . . . . . . . . .

normalized

~ ....

eigenfunction

, , . . o ~

yt ~ -- exp [-- ~]/V/-~

~ ( p ) = V ~ ~ v/('~-i- ~ ' ~ - i - - . . .

Ttm,~c cort'eetion.~ can be r e c u r r e n t l y o b t a i n e d b y solving tile e q u a t i o n s (7)

( H o -- E f~ ~f~"~ = ( E ~ -- H , ) r

Et-'~f :''-~' -= ... i- Ec,,,) ~0)

for m = 1, 2, 3, ... u n d e r t h e r e q u i r e m e n t t h a t ~v~''~ be o r t h o g o n a l to ~co~. A s t r a i g h t f o r w a r d calculation gives for t h e g r o m t d - s t ~ t e e n e r g y

(8)

E(7) . . . .

1+

1 53 ~72--9~y'+

5 581 21577 397 2--~-~78-- k3_05920 7 s ~-

-~-

3 1 2 8 3 298 283 132 710 4 0 0

r'o+ o(v':)

w h i c h seems to be at m o s t a s y m p t o t i c . This expression r e p r o d u c e s the num e r i c a l result.s given in (2) for small ~r up t o ~, : 0.2.

(~) A. DAI.(~AaNO: Stationary perturbation theory, in Quantum Theory, edited by I ). R. B.~TES, Vol. 1 {.New York, Y.Y., 1961).

158

A. GALIKDOand P. I'ASCI:AL

I n order to obtain the d o m i n a n t behaviour of E(y) for very large magnetic fields we shall write our Hamiltonian in the following w a y :

1

H'~ = - - A 4 -

(9)

7 L , § ~y~(x2-~ .V:)

2

Izl + a 2

2

H;--: ] z ~ _ a - - r , where all lengths are measured in units of ao, and a is a positive c o n s t a n t which will be determined later on. The unperturbed-eigenvalue problem corresponding to H~ can be easily solved since it can be separated in cylindrical coo r d i n a t ~ and the problem reduces basically to a bidimensionaI armonic oscillator and a one-dimensional (, CouIomb potential ,)(q. We find for the unperturbed wave function ~2'(~ and the corresponding eigenvalue E '(~ W,(o,(~, 0, z ) = N exp [4- im O] M(--n~, 1 4- Iml, ~). exp [-- ~/2] $1,,tn W~,i(z ~) ,

z~0,

(10) 1

E'(') .... ~,[2n~+ m-t-- I ~ 1 § 1 ] - - - -

n,==0, 1 , 2 , . . . , m = 0 ,

4-1, -L2 .... ,

where Q, 0, z are the cylindrical co-ordinates. N is ~ normalization c o n s t a n t ; M and Wa,t are, respectively, the K u m m e r ' s and W h i t t a k e r ' s functions as defined in (~). F u r t h e r m o r e ~ = ~o~]2 and z~ == 2(a + z)/fl for z ~ 0 . Finally fl is obtained by solving either the equation

[

dz

::0

J,f2aJ#

for the even-parity eigenstates, or the equation

W~,i(2a/fl ) : 0

(12)

for the o d d - p a r i t y eigenstates. The unperturbed ground-state energy ~(o) in the sector m = O of interest is obtained for nq 0 and m = 0, in which case the cylindrically s y m m e t r i c probability density *,~,,'~r ]2 a.s a function of ~ presents a well-defiued m a x i m u m :=

(4) L. K. HA~N~S and D. H. ROBEua'S: Amer. Jaurn. Phys., 37, 1145 (1969). (~) M. AnRAMOWITZand I. A. STEou~', Editors: tla~book o] Mathematical Functions (New York, N.Y., 1965).

HYDROGEN

ATOM

IN

A

STRONG

MAGNETIC

FIELD

159

for 0 - ; 1 / ~ / ~ . If 7>>1 we expect t h a t this situation will be quite insensitive to the presence of the p e r t u r b a t i o n H~ a n d t h a t a possible w a y to minimize the effect of H~ is to a s s u m e t h a t a == ).Iv/7, where ~t is a c o n s t a n t of order one. W h e n e v e r 29,/V'~ < 10 -2 eq. (11) can be a p p r o x i m a t e d b y in ~ 22

(13)

~- 2~ + 27,~-4- ,,o(1-- fl) -- O,

where 7z= 0.57721... is Euler's c o n s t a n t a n d v/ the psi-function as defined in (5). H e n c e fl-~ ~ In 7 as ~ -~ ~ . F u r t h e r m o r e it c~n be easily p r o v e d t h a t for v e r y high m a g n e t i c fields ! the first-order H~ correction to E Iq~ vanishes when 2 is chosen such t h a t |

(14)

dy ~-:~.)i/~2 - - ~

e x p [y~] erie (y) == 0 ,

Q

where eric (y) is the error function (6). This implies t h a t 2 ~ 0.53. F r o m all these results we obtain E(y) y ~ y -- (ln ~)2.

(15)

Therefore all our information for the g r o u n d - s t a t e e n e r g y E(?) is c o n t a i n e d in eqs. (8) a n d (15).

3. - Rational

approximants.

If, using only the result (8), we t r y to construct the sequence of Pad6 app r o x i m a n t s ~ . , . m ( y ) , _N == 1, 2, ... with fixed n, it is quite possible that. this will i m p r o v e the convergence b u t the m e t h o d will clearly fail for 7 large enough since E"v*-.n~(7).-~72- for 7--~ oo in contradiction to (15). F o r instance b o t h i~3,~](7 ) a n d ~ 2 m ( 7 ) reproduce the results of (2) with an accuracy b e t t e r t h a n 1 % h)r small 7 up to 7 = 0.8. I n order to use all t h e available information we will proceed as follows: let us ,sTite the p e r t u r b a t i v e series (8) as

(16)

E(7) = -- 1 + ~ ~. ~,2..

The ionization energy B(7 ) will be

(17)

B(7 ) ~- ~,-- E(r ) = 1 -~- ~ , - ~ e . 7 2. .

160

~. GALIND0 and P. PASCUA~

On the other h a n d , eq. (15) shows that B(y)~--(lny) 2 as y - ~ c ~ . introduce a new variable T defined a.s

(18)

Let us now

T= in(l-~-~),

where ~ is a positive c o n s t a n t to be determined h~ter. B y expressing B in terms of this varia.ble v, (17) will lead formally to cO

(19)

~r

--: 1 + ~ b.(~)T".

Now we can construct the series of rational .~pproximants V/B-~ c:~+~'N~ a n d for each value of N we will determine the constant ~ in such a w a y t h a t v~E~+~'~'J~-.v for v -> cr (i.e. for y -~ oo). m order to see how the m e t h o d works, let us assume t h a t the p e r t u r b a t i v e calculation of E has been carried out only up to first order, i.e. (20)

B(y) = 1 + y - ~ y-" + o ( y ' ) .

Then a straightforward calculation allows us to write

(.21)

VB(-~ = 1 + bt(a)r-~- b~(~)r-"+ ba(~)r3+ O(T'), b~ = ~ ~,

b~= ~ ( 2 - - 3 ~ ) ,

b~ = ~:r (4--18:r

9:r

So t h a t we m a y construct the rational a p p r o x i m a n t (22)

V~-(~:2,1~ ~_ b2 ~t_ (b,b2 - - b3) v ~_(b_~ - - thba) r ~ b~ - - b3~

In order to fix ~ we impose the above-given condition which t u r n s out to be, in this case, b~ + (1--bl}ba = O, i.e. ~ must be a root of the equation 9 ~ 3 + + 36a ~ - 6 8 a - [ - 1 6 = 0. This equation has only two positive roots b u t only for the sma.llest one (~--= 0.279 562 068 ...)b3/b2 < O, which is a v e r y i m p o r t a n t condition to be m e t whenever is possible, since it garantees t h a t the rational ~pproximant (22) has no poles for v > 0. We obtain therefore ~./)~E~,~J (23)

=

1 + 0.18695787v.-]- 0.047176838v'1-i- 0.047176838~ '

r = In [ 1 + 0.279562 068"] " I n table I we give (columm B :2nJ) the values for the ionization energy obtained b y using (23) for different values of y. I n the same table the results of some available older calculations are also quoted. Up to y : 5 the best calculation is the numerical one b y CAU~ et al. (3); for these authors a result as 1.09508(5)

IIYDROGEN

ATOM

IN

A

STRONG

MAGNETIC

FIELD

161

,..A

":" t'.-

g.-,

v

v

,ff

r

e~

I 9

9

~

~

9

.

~

,..=1

e~

9

o

9

.

u',i e~

.-4 e~

e~

~4 e~

v 9

~

,..-i

,....i

9.=1

@:1

@,J

~o o

v 9

9

~

~

.

~4 ,d

I N

v

,--t

.~

el

M

~-,i N

I

e4

9

o~

e~

,~.

~

~

,-~ M ' N t~ ~

~

..

te

9~

~4 el

N~

..,~

~

'~ ~

~ .~g ~ 9

~

9

,

~

9

I ll

,Vttovr

,

o

~

I

11

e~

g..

,.=i J

e~

U~'tl~ettto B .

o

,

9.4

N

e~

t'-.

162

A. GALII','DO and P. PASCUA:L

m e a n s t h a t t h e actual v a l u e lies on t h e interval (1.09503, 1.09513). W e c a n see t h a t w i t h v e r y little information w e obtain quite satisfactory results for a n y v a l u e of ~. W i t h the explicit terms entering (8) we can construct the rational approxi m a n t s u p t o ~r a n d for each order w e determine u b y i m p o s i n g t h e condition v / B ~ t ~ + l z ~ v w h e n v--> oo. I n general t h e e q u a t i o n to fix ~ has several positive roots a n d w e c h o o s e as before t h e smallest one. Our calculation s h o w s t h a t the sequence %/B(~ tt'l~, %/~(~)E,m, v,B~E6.sJ, ... s e e m s to c o n v e r g e rather quickly a n d furthermore t h e v a l u e of ~ does n o t change considerably. The s e q u e n c e B ~ / B ~ ca'~J, B V ~ E~'4~,... s e e m s t o c o n v e r g e m o r e slowly. For

20

B

15

,.

/

7" t~ et 10 /gS SI / o~

9 o.t ~ /

/i 4,"

9

a a" /,~

9

.."

. ."

#

.s

s"

1

0

1

tog

I

I

2

3

Fig. 1 . - Our results from B(7) [6,5] compared with some previous calculations: - our calculation, - - - - - SmTH et al. ( 1 ~ ) , _ . . . . ComsN et al. (13), X CABIB et al. (2), o LARSEN (~).

ItYDROGEN ATOM IN

163

A STRONG MAGNETIC FIELD

t h e useful r a t i o n a l a p p r o x i m a n t s we o b t a i n ~/B--~ r~''v+~--- N:'*"v+~/D~'~+~, w h e r e N~*,a~- 1 ~- 0 . 0 5 9 4 6 7 6 5 3 ~ 4 - 0 . 6 0 7 6 4 0 0 7 7 ~ ~-, -J- 0 . 1 1 3 2 7 9 2 1 2 ~ ~-]- 0.023816753~* , (21)

D ~*,a~= 1 - - 0 . 1 1 6 5 6 2 8 9 1 v ~ 0.586623591v'-q- 0 . 0 2 3 8 1 6 7 5 3 v a, -:ln

1+(}.352061266

~t6,5~ = 1 - - 0 . 0 7 2 8 4 9 4 7 4 ~ - ] -

' 2.500855394~2q - 0.348154255T3-~ -

1 . 0 2 7 1 8 8 2 6 4 ~ q - 0.189203517~5q - 0 . 0 3 5 6 9 5 1 7 7 v 6 , D :6,51= 1 - - 0.265011248T ~- 2.511088 758~ ~ - - 0.110878699v3 q-

(25)

0.948839056~4 ~ 0 . 0 3 5 6 9 5 1 7 7 z ~ , r

= i n 1-t- 0 . 3 8 4 3 2 3 5 4 9

"

I n t a b l e I w e g i v e t h e v a l u e s of t h e i o n i z a t i o n e n e r g y o b t a i n e d f r o m t h e s e r a t i o n a l a p p r o x i m a n t s . I n fig. 1 we c o m p a r e o u r b e s t r e s u l t s t h r o u g h B~/B-~ ce'~ w i t h s o m e p r e v i o u s c a l c u l a t i o n s u p t o ~ ' ~ 4 " 1 0 s. O u r a n a l y t i c a l r e s u l t s , o b t a i n e d f r o m B ~ - ~ ~~ a r e a b l e t o r e p r o d u c e e x t r a o r d i n a r i l y w e l l t h e numerica.1 r e s u l t s of CABIB et a~. (5) for m a g n e t i c fields u p t o ~ = 5, as w e l l as t h o s e o b t a i n e d w i t h ,~ v a r i a t i o n a l m e t h o d b y S ~ T H et al. (~2) f o r h i g h e r v a l u e s of 7. Unfortunately we have no rigorous arguments to understand completely the n u m e r i c a l success of o u r m e t h o d , w h i c h q u i t e p r e s u m a b l y w o u l d p e r f o r m e v e n b e t t e r b y w o r k i n g o u t t h e n e x t t e r m s i n (8) n e c e s s a r y for h i g h e r P a d ~ ' s . I n s o m e s e n s e t h e m e t h o d w e h a v e f o l l o w e d in t h i s p a p e r is t h e d u a l ~ n a l o g u e to t h a t p r e s e n t e d in (14) for c o m p u t i n g b o u n d - s t a t e e n e r g i e s f o r ~ class of central potentials.

4. -

First

excited

levels.

I f w e u s e t h e o r d i n a r y p e r t u r b a t i v e m e t h o d i t is e a s y t o o b t a i n t h e e x pansions E~oo(r) = - - ~ (26)

E,,o(7)

=--1

1[

3824

1--287~--~-7

'

1620608

9

]

7~ ~O(r') '

[ 1 _ 1 2 7 . , ~_ 3 3 6 7 , _ 33 92076_~_~ 0(Ts)] '

E21:,(7) = - - 41-[1 ~ 47 - - 2472 -t- 9 2 8 7 ' - - 129 5367 s § 0 ( 7 8 ) ] ,

(1,z) A. GALINDO and P. PASCUAL: _Nuovo Cimento, 3 0 A , 111 (1975).

164

A. GAL~Nr)O and r'. PASCtT,tr,

which are p r o b a b l y a s y m p t o t i c series. E,,~,,(~) s t a n d s for t h e e n e r g y of t h e s t a t e w h i c h for y - + 0 goes to t h e s t a t e (n/m) of t h e hyr a t o m . On t h e o t h e r h ~ n d , t h e r e s u l t s obt~dned in sect. 2 s u g g e s t t h a t for v e r y high m a g n e t i c fields t h e b o u n d levels b e h a v e like ~,[2ne -4- m -4- Iml + 1 ] - l/fl 2, t h e l a s t t e r m b e i n g d o m i n a t e d b y the first one for s u f f i c i e n t l y large ~,. T h e a p p r o x i m a t e e i g e n s t a t e s ~v'(0) will h a v e as n o d a l surfaces 'n~ c y l i n d e r s w i t h axis Oz t o g e t h e r w i t h -~ set of p l a n e s ortho~'onal to Oz a n d d e t e r m i n e d b y t h e n o d e s of W~.~. To a s c e r t a i n wtfich are t h e a s y m p t o t i c q u a n t t t m n u m b e r s n~ for t h e 2s, 2p h y d r o g e n s t a t e s poses ~ difficult q u e s t i o n , a n d a c o n c l u s i v e a n s w e r is still n e e d e d i n o u r o p i n i o n . F o l l o w i n g (z), we will h o w e v e r ~ c c e p t t h a t nQ = 0 for t h e s e state% so t h a t

(2;)

L',~(),) ,~:. 3),,

s

~r ~'.

N o w we will a,pply a g a i n t,he m e t h o d of rationa~l a p p r o x i m ~ n t s t o o b t a i n t h e vah~e of

(2s) k=l

TABLE II. -- Energies ]or the (2, 0, O)-level. ~,

E [2.,~

E (~')

E (7)

E (11)

E (8)

E (1)

E (s)

0.1

- - 0.198730

- - 0 . 1 9 6 2 (1)

--0.19614

--0.19617

--0.1530

--

0.1930

--0.1950

0.2

- - 0.109325

- - 0.0979 (l)

--0.0732

- - 0.0511

- - 0.0934

0.3

-

-}-0.0033 (1)

-{.0.0196

4- 0.1723

-{-0.0067

0.4

4

0.084579

+ 0 . 1 0 1 7 (1)

+0.1165

4- 0.4809

+0.1040

0.5

+

0.183317

~-0.1984 (1)

4- 0.2164

+

t0.2006

0.6

-t- 0.282470

-4- 0.2945 (l)

+ 0.3189

4- 1.358

+ 0.2967

0.013344

0.8762

0.7

.'-- 0.381863

~-0.3905 (1)

+0.4235

--

1.929

--0.3927

0.8

-~ 0.481407

-~-0.4865 (2)

-{,0.5301

+

2.586

-{-0.4889

0.581051

+ 0 . 5 8 2 8 (2)

+0.6385

+

3.332

+0.5854

4- 0.680766

7-0.6793 (3)

-{-0.67897

-{-0.7483

4- 4.165

-t-0.6821

4- 1.314

+

4- 1.651 94

~- 1.902

4- 17.32

+2.503

-{.27.18

-4- 3.116

4- 39.24

4- 4.363

4- 69.94

0.9 1.0

4-

1.5

-i- 1.179911

+ 1.1636 (5)

2.0

-]- 1.679482

4- 1.651

2.5

~.- 2.179225

3.0

4- 2.679053

4.0

4- 3.678839

5.0

4- 4.678710

25

4- 24.67830

I00

~- 99.67822

(2)

-{-0.683

-4- 2.63463

9,647

+ 1.169 -~- 1.657

IIYDROGEN

165

A T O M 1N A S T R O N G M A G N E T I C F I E L D

I n order t o b e a b l e to u s e b o t h (26) a n d (27) we will i n t r o d u c e for each one of the levels a n e w v ~ r i a b l e

= 89[V&~ + ~-'- ~],

(29)

where a is a p o s i t i v e c o n s t a n t . :Note t h a t r , - - ~,0- as y --> 0 a n d r ,-~ ~, as ~ -> c ~ . Since y 2 = r ( r + a) we c a n ~Tite i m m e d i a t e l y

k--X

The m e t h o d is n o w to d e t e r m i n e t h e r a t i o n a l a p p r o x i m a n t s S,z ~ '~+ l , m(~,j ., to the series (30) b y f i x i n g l a t e r t h e p o s i t i v e c o n s t a n t ~ i n such a w a y t h a t t h e a s y m p t o t i c b e h a v i o u r is c o n s i s t e n t w i t h (27). W i t h t h e i n f o r m a t i o n c o n t a i n e d i n (26) we c a n c a l c u l a t e o n l y Sl(a), $2(~) and $3(~), a n d t h e o n l y i n t e r e s t i n g r a t i o n a l a p p r o x i m a n t to b e d e t e r m i n e d is ,gt~,~J T h e e q u a t i o n t o fix :r t u r n s o u t to b e of f o u r t h orde r a n d i t has o n l y o n e p o s i t i v e s o l u t i o n for t h e l e v e l (2, 0, 0) a n d t h r e e p o s i t i v e s o l u t i o n s for t h e o t h e r l e v e l s ; i n this l a s t case we choose t h e l a r g e s t o n e t h a t g u a r a n t e e s $3(~)/$2(o0<0. TAI~LE I l l . - Energies ]or the (2, 1, O)-level. ,~,

E ~.1~

B (11)

E (6)

B (1)

0.1

--

0.224457

--0.22482

--0.2098

--

0.2234

0.2

--

0.165172

0.3

--

0.088133

0.4

.... 0.002084

0.5

+

+0.47999

+0.4813

+

2.070

+1.40461

-]-1.413

+

9.005

+ 2.35996

+ 2.385

+ 20.56

0.088750

o.6

+

0.182334

0.7

+

0.277616

0.8

+

0.374011

0.9

+

0.471171

1.0

+

0.568879

1.5

+

1.061 909

2.0

+

1.558382

2.5

+

2.056256

3.0

-" 2.554835

4.0

+

3.553055

5.0

+

4.551987

25

~- 24.54856

100

+ 99.54792

166

A.

We obtain

P.

PASCUAL

for the energies + 3 8 0 . 8 9 5 761"r z , 1 -- 95.223 940V .......

1 1 -- 99.242289v

(31)

and

GALINDO

T~'~1](7) :

-- 4

77~.[2.1] [ ,~ -a~210 ~ f / ~

. . . . .

1 1 "- 41.801190v-4

187.128354~ 2

1 + 46.782088v 1 1 + 53.881332v--

Et•.•11^,•

21~1~'J --- • 7 - - ~

In tables II, III

and

IV we compare

If we take into account

513.283884v ~

agreement

with

existing

these results

results

,

~--=- 0 . 4 2 8 2 9 8 0 7 7

with some available

the little effort put is a m a z i n g l y

,

=: 0 . 4 1 5 0 7 4 8 8 5 ,

'

1 + 64.160 486z ~

putations.

previous

~ : 0.143512438

.

com-

in ottr calculation

the

good.

~ o t i c e t h a t w e h a v e n o t b e e n a b l e t o a v o i d a p o l e f o r 1,~2o o~t~'u'W)' i n t i l e p h y s i c a l region.

~-evertheless

pole appears

t h i s is n o t ~ s e r i o u s d r a w b a c k

at y----0.04021675,

7 --: 0 . 0 4 0 2 1 2 4 4 a n d t h e r e f o r e region around

while

there

the region

where

asymptotic

since the

numerator

-"-'~oo (~'J f a i l s is a v e r y ~t2,~,

t h i s v a l u e of },, w h i c h , f u r t h e r m o r e ,

to use for it the

of the method

is a z e r o o f t h e

at

small

is s u f f i c i e n t l y c l o s e t o 7 ---- 0

s e r i e s (26).

TA~LF. IV. - Ene~yies ]or the (2, l, 1)-level. r

E t2.~!

E {1~}

E (*)

0.1

---

0.100387

- - 0.101 69

--

0.0723

--

0.1229

0.2

+

0.115154

0.3

--'

0.366952

0.4

f-

0.637476

+ 2.08682

+

2.0920

+

3.067

J- 4.80083

+

4.804

+ 10.99

+ 7.59297

+

7.597

+ 23.54

0.5

+

0.918042

0.6

+

1.204409

0.7

+

1.494372

0.8

+

1.786696

0.9

-~-

2.080646

1.0

+

2.375760

1.5

+

3.860896

2.0

~

5.353371

2.5

J-

6.848833 8.345800

E (7)

3.0

+

4.0

+

ll.34200

5.0

+

14.33972

+

13.15

+

13.28

25

+ 74.33241

+

71.98

+

71.81

100

+ 299.331 0

+ 300.2

+ 294.7

E (t)

HYDROGEN

ATOM

IN A S T R O N G

MAGNETIC

I~

FIELD

~PF:E!NDIX

We p r e s e n t h e r e t h e w a v e functions n e e d e d for our p e r t u r b u t i v e calculations. i) G r o u n d state. The g r o u n d - s t a t e w a v e function up to t e r m s of order 7 ' is

,p(P) = [~*o(9) :[o~ + u,(e) ~o(~) + u,(9) Y~(4)] exp [-- @], 1

uo(9) = 2 + ~66(33--6@2+ 2~s)72+ 1 + i7~

U2(@) -

-

(--33495+ 186092+ 6209aH- 3908'nu 12095 ~- 16@')7',

1 36 C ~ (3@'+ 2e~)W + 1 -~- 60480 %/5 (- 1743@ 2 --11629'--1 3389,-- 516o ~ -- 80@')?' , 1

u,(@) ----7-56-0( 5 @ ' 4

4@~4 9')7' 9

ii) L e v e l 200. U p to t e r m s of order 7* we h a v e

~(p) = [uo(q)~(4) + u,(@) ]~,~(4)]exp [- @/2], uo(e)=~ i--

0 -~-~(984--492@--42@2+7@'+q')7',

1 u2(@)---- 36%/10(2492+ 4@'H-9')72 . iii) Levels (21 m), m = -}- 1, 0. U p to t e r m s of order 74 we h a v e

~(p)

= [u,.(e) :Y~(e) + u,-(e) :[. (q)] exp [-- q/2], i

u..(q) = ~

u~.(e) =

i+ m 2 e + -6-o-.~ (48~ -- 9 9 ~ - q')7~ '

360 V ' ~

(4q,+

e')7' 9

A. GALINDO and P. PASCUAL

168

@

RIASSUNTO

(*)

Tenendo confa~ sia dello sviluppo perturbativo utile per eampi magnetiei deboli ehe alcuni risultati per eampi molto forti, si pub mediante l'uso di approssimanti razionali dare un'espressione analitiea per i livelli energetiei inferiori di un atomo d'idrogeno v a l i d a per qualsiasi valore del eampo magnetico.

(*)

Traduzione a eura della Redaz~ne.

ATOM BO,~OpO,~a B C]HI,.rl~HOM M.al"HRTHOM HoHe.

Pe,~loMe (*). - - HciIon~,3ys pe3ynr~TaT1,i Teopm~ BoaMyl~eHrr~ ~.ag caa6I,LX I~al"HHTHLLX r~o.ael~, a Tax~xe HexoTop]ae pe3y~,TaT]a ~sm o~eH~, CHnbn],~X Marrm'rmax noze~i, ~,x MOdeM l l o n ~ - b anannannecxoe B~apa~erme ~n~ nH3nmx yponHe~ 3Heprml BonoponHoro aTOMR, ci/paBe~[JEl~oe ~[3i~i n~OEI3BO31I~H/~IX 3 H a ~ / e ~

(*)

HepeeeOeno pee)ar~ue~.

M a r m 4 T H o r o rrong.