Revista Brasileira de Física, Vol. 10, NP 2, 1980
Some Comments on the Hydrogen Atom in a Spherical
Enclosure*
V. C. AGUILERA-NAVARRO, E. LEY KOO** and A. H. ZIMERMAN
Instituto de Física Tedrica, São Paulo
Recebido
ein 22 de novembro de 1979
We discuss some p r o p e r t i e s
o f t h e ground s t a t e energy s o l u -
t i o n s f o r t h e hydrogen atom i n a s p h e r i c a l enclosure. We
consider a l s o
the a p p l i c a t i o n o f the many-point Padé approximants t o t h i s k i n d of systems i n s ide a box.
Discutimos algumas propriedades das soluções que
correspon-
dem ao estado fundamental de um átomo de h i d r o g ê n i o d e n t r o de uma c a i x a
e s f é r i c a . Consideramos também a a p l icação dos aproximantes de
Padé
de
muitos pontos para e s t e t i p o de sistemas d e n t r o de uma c a i x a .
The hydrogen atom w i t h i n a s p h e r i c a l box w i t h u n p e n e t r a b l e
w a l l s has been i n v e s t i g a t e d i n many o c ~ a s i o n s ~ -More
~ . recently
i t has
been a l s o considered the hydrogen atom w i t h i n s p h e r i c a l boxes w i t h
pe-
5
n e t r a b l e wal l s .
The a p p l i c a t i o r i o f the Rayleigh -schr;dinger
theory t o the hydrogen atom i n a s p h e r i c a l box has
sed6-9 and,
in particular,
also
the ground- state energy up t o
perturbation
been
discus-
the f i f t h o r -
der i n e 2 has been o b t a i n e d l O , namely,
*
**
Work supported by FINEP, Rio de Janeiro, under c o n t r a c t 522/CT.
On leave o f absence from I n s t i t u t o de ~ i s i c a ,U n i v e r s i t y
w i t h f i n a n t i a l support o f FAPESP, B r a s i l .
of
Mexico
i n atomic u n i t s .
guedll
R
i s t h e r a d i u s of the s p h e r i c a l box. I t
t h a t t h i s formula can be very u s e f u l f o r t h e
has
been a r -
obtent ion of
the
ground- state energy f o r s u f f i c i e n t l y m a l l R, where t h e p r e v i o u s c a l c u lations3
have presented numerical e r r o r s ( f o r
R
f
1.4 a . u . ) ;
therefore
t h e p e r t u r b a t i o n expansion would be u s e f u l f o r t h e H atom s u b j e c t t o h i g h
pressure.
Ley-Koo and ~ u b i n s t e i n ' have a l s o improved
c a l c u l a t i o n s by de Groot and Ten seldam3 even f o r small
s t a t e energy o f t h e hydrogen atom i n s p h e r i c a l
box
the
numer i c a l
R, f o r
t h e ground
with unpenetrable
walls.
The hydrogen atom enclosed i n a s p h e r i c a l box has a l s o been
s t u d i e d i n an a ~ p l i c a t i o no f a v a r i a t i o n a l approach t o p e r t u r b a t i o n theo r y 12.
I n s e c t i o n 2, we discuss some p r o p e r t i e s o f the
t h e hydrogen atom w i t h i n a s p h e r i c a l boi< and i n s e c t i o n
solutions o f
3 we discuss t h e
a p p l i c a t i o n o f t h e many-point Padé approximants t o t h i s k i n d o f problem.
2. PROPERTIES OF THE SOLUTIONS
The r a d i a l equation f o r the s - s t a t e reads 5
ao
=?T2/me2 being
t h e Bohr r a d i u s . The s o l u t i o n t o Eq. (2)
is
where M ( a , b , z ) i s a Kummer o r c o n f l u e n t hypergeometric f u n c t i o n . For l a r ge
r
and small 6, w i t h
V
=
1+B,
i t behaves l i k e 1 3
As f o r r = R,
= 0,
the erpression
(5)
gives,
by
keeping
o n l y terms i n fir s t o r d e r i n 6,
and, t h e r e f o r e , i n t r o d u c i n g t h i s v a l u e o f
6 i n (3) , we have f o r t h e e-
nergy t h a t
( T h i s expression i s very s i m i l a r t o t h a t one c o n j e c t u r e d by wigner6, b u t
i t i s val i d only f o r large R ) .
The second term i n expression (7) decays e x p o n e n t i a l l y
R and e x p l a i n s why w i t h n o t so l a r g e R, l i k e R
reach t h e v a l u e -0.5
- 5 or
6ao,
we
( i n u n i t s o f e2/a0), as t h e exact numerical
with
already
calcu-
l a t i o n s heve i n d i c a t e d (see Table I ) .
L e t us observe t h a t
which f o r s m 1 1 B gives
M(-B, 2, Z R / W ~ ) = i
-
6
-1
k=O
'
(ZR/W~)
k(k
+
l)!
Now,. t h e c o n d i t i o n M(R) = 0, c ~ i v e s ~
I t i s n o t d i f f i c u l t t o see t h a t expression (7)
with
(6)
f,or l a r g e R.
i s c-~sistent
I n order t o study t h e s o l u t i o n around E=O,
rewrite
$V(x)
o f Eq.
(4)
it
is
useful t o
i n t h e form
where
and so on.
We see t h a t f o r E=O, t h e r a d i u s R o f our box corresponds
to
t h e zero o f the Bessel f u n c t i o n o f t h e r i g h t hand s i d e o f expression (10).
As t h e f i r s t zero occurs a t 3.8317,
= 1.8353400
Eq.
2
.
we o b t a i n t h e
known
v a l u e R,, =
I t i s easy t o se= t h a t expression (10) i s a
solution o f
(2) f o r E=O.
3. APPLICATION OF THE MANY-POINT PADÉ APPROXIMANTS
The purpose o f t h i s s e c t i o n i s t o discuss t h e a p p l i c a t i o n o f
the many-poirit Padê-approximants t o systems i n s i d e a box.
observed i n Ref.
It
has been
(10) t h a t i f we c o n s t r u c t the one- point Padê a p p r o x i
-
mants14 w i t h t h e h e l p o f expression ( I ) , then we o b t a i n v e r y bad r e s u l t s
f o r the ground- state energy,
f o r s u f f i c i e n t l y l a r g e R. But we can consi-
der t h e two- point Padê approximants14 by means o f * t h e use o f
the
beha-
v i o u r given i n expression (1) around R = O and t h e behaviour around R-,
given i n expression
(7). Ir t a b l e I, we reproduce ~ r 4 / 2 1aod ~ [ 5 / 3 ] w h i c h
have been o b t a i n e d by consider;ng
for
the
f u n c t i o n S ( R ) = R ~ E ( R ) For comparison, we a l s o reproduce the exact
nu-
.
two- point Padé approximants
m e r i c a l values f o r E(R) as g i v e n i n Ref.5 and t h e p e r t u r b a t i v e valuesobt a i n e d from expression ( I ) .
We see t h a t f o r R 6 3ao, the p e r t u r b a t i o n expansion
(1) g i -
ves very good r e s u l t s compared t o t h e exact ones, w h i l e f o r l a r g e r R the
two- point Padê approximants g i v e a somewhat b e t t e r tendency ( f o r R + = ,
t h e p e r t u r b a t i v e expansion diverges, w h i l e t h e two- point Padé
mants tend t o the exact r e s u l t
-
0.5,
by c o n s t r u c t i o n )
approxi-
.
The two- point ~ a d éapproximant f o r t h e ground- state
energy
o f t h e hydrogen atom i n s i d e a s p h e r i c a l box o f r a d i u s R w r i t e s
i n atomic u n i t s . This formula reproduces reasonably w e l l (Table I)
ground- state energy o f the H atom over t h e e n t i r e range o f values
thc
f o r R.
E v i d e n t l y , i f we want a b e t t e r f i t t i n g , we can use a t h r e e - p o i n t Padé ap
Table 1
-
R i s given i n u n i t s o f ao. The energies are given i n u n i t s o f
proximant14, which uses a l s o an expansion o f E(R) around some o t h e r p o i n t
besides t h e ones used. For instance, w i t h t h e h e l p o f expression
can o b t a i n t h e expansion o f E(R) around R = Ro, where E(R,)
(91, we
= 0.
The pressure c a l c u l a t e d w i t h t h e h e l p o f the p e r t u r b a t i v e expansion
(11,
Eq.
c o i n c i d e s w i t h i n few percent w i t h t h e
exact
one f o r
R 6 3ao. For t h e h y p e r f i n e s p l i t t i , n g constant A, t h e p e r t u r b a t i v e expans i o n g i v e s r e s u l t s which a r e comparable w i t h t h e exact ones
for R
within
2a0. For l a r g e r values o f R, the p e r t u r b a t i v e expansion
for
10%
A
g i v e s very bad r e s u l t s , w h i l e t h e two- points ~ a d éapproxirnants
~ [ 4 / 1]
c o n s t r u c t e d from the behaviour o f A around R = O and R = -,gives
results
which a r e comparable t o the exact ones w i t h i n 10%.
Recently
15,16
,
exact numerical c a l c u l a t i o n s have
been
done
f o r t h e energy l e v e l s o f quantum o s c i l a t o r systems i n s i d e a box. We
show t h a t p e r t u r b a t i v e expansions 1 i ke expression ( I ) a r e v e r y
easy
can
to
be o b t a i n e d and a r e very useful f o r t h i s k i n d o f problems.Specially when,
w i t h t h e h e l p o f the behaviour o f E ( R ) f o r l a r g e R also,we c o n s t r u c t t h e
two- point Padé approximants.
I n t h i s way we o b t a i n i n t e r p o l a t i n g
l a e s i m i l a r t o expression (12) which reproduce very w e l l t h e
formu-
energy l e -
v e l s f o r t h e e n t i r e range o f values o f R.
Other problems o f systems i n s i d e a box a r e being
t r e a t e d by
t h e very same methods.
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