New Appr~oach to Calculate Bound State Eigenvalues - Sociedade

Revista Brasileira de Flsica, Volume 13, n? 1, 1983

New Appr~oachto Calculate Bound State Eigenvalues EDGARDO GERCK Instituto de Estudos Avançados, CTA, São José dos Campos, 12.200, SP, Brasil and Max Planck Institut fur Ouanlenoptik, 8046, Garching, M s t Germany

JASON A. C. GALLAS Departamento de Física, Universidade Federal de Santa &tarina, Florianbpolis, 88.000, SC, Brasil

and Max Planck Institut fur Quantenoptik, 8046, Garching, West Gerrnany and AUGUSTO E.d'OLIVEIRA

Instituto de Ffsica, Universidade Federal Fluminense, Niterbi, 24.000, RJ, Brasil Recebido em 10 de dezembro de 1982

Abstract A method o f s o l v i n g t h e r a d i a l schrgdinger equat i o n f o r bound s t a t e s i s discussed. The method i s based on a new piecewise represent a t i o n o f t h e second d e r i v a t i v e o p e r a t o r on a s e t o f f u n c t i o n s t h a t obey t h e boundary c o n d i t i o n s . T h i s r e p r e s e n t a t i o n i s t r i v i a l l y diagon a l i s e d and leads t o c l o s e d f o r m e x p r e s s i o n s o f t h e t y p e = ~ ( a+ b b + c / n + ...) f o r the eigenvalues. Examples a r e g i v e n f oEnr t h e= power-law and l o g a r i t h m i c p o t e n t i a l s .

1. INTRODUCTION

This paper discusses a method o f c a l c u l a t i n g t h e eigenvalues o f t h e r a d i a l ~ c h r g d i n c j e requation ( i n atomic u n i t s )

s u b j e c t tci t h e boundary c o n d i t i o n s '?(O) =

Y(m)

= 0, where

~ ( rmay ) in-

c l u d e t h e usual c e n t r i f u g a l terrn. A b r i e f v e r s i o n o f t h i s work appeared recent l y1 . The proposed approach i s based on a new piecewise t a t i o n of t h e d i f f e r e n t i a l o p e r a t o r

represen-

d2/&2 from eq. ( I ) on t h e s e t

Using t h i s piecewise r e p r e s e n t a t i o n

for

d 2 / d r 2 i n the pro-

posed appi-oach i s e q u i v a l e n t t o c o n s i d e r i n g a c o n t i n o u s l y and i n f i n i -

t e l y piecew i s e approximation Y ( r ) =

Yk ( r ) t o t h e e i g e n f u n c t i o n $ ( r )

( I ) , where i n each k - p i e c e t h e Yk

o f eq.

the set U

%

( r ) i s a linearcornbinationon

with k 6 N :

For any ~ ( r t)h a t i s a l i n e a r combination on t h e s e t ,U ,

the

unique two- point r e p r e s e n t a t i o n o f t h e o p e r a t o r d2/dr2 i s g i v e n by

as shown i n s e c t í o n 2. The i d e a l r e p r e s e n t a t i o n

of

the

operator

d2/dr2, i .e. t h e exact r e p r e s e n t a t i o n , would be g i v e n by i t s

expán-

s i o n on t h e complete s e t o f s o l u t i o n s o f eq. ( 1 ) . T h i s s e t i s o f couris U , c l e a r l y a good piecewise approxirnation t o t h e i d e a l s e t i f t h e parase i n f i n i t e and, a p r i o r i , unknown. Wenote t h a t t h e b a s i s s e t

ba-

meter a i s l a r g e r than zero. T h i s i s due t o t h e f a c t t h a t t h e Ua s i s s a t i s f i e s t h e boundary c o n d i t i o n o f eq. (1) a t i n f i n i t y .

The r e p r e s e n t a t i o n of t h e second d e r i v a t i v e o p e r a t o r

in

g i v e n b y e q . ( 3 ) has severa1 u n i q u e p r o p e r t i e s . T h e m a i n p r o p e r t y t h a t t h i s r e p r e s e n t a t i o n can be t r i v i a l l y d i a g o n a l i z e d . Another

U , is cha-

r a c t e r i s t i c aspect i s t h a t parameter a.

U (and hence Y ( r ) ) depends on t h e f r e e a This v a r i a t i o n a l parameter wi I 1 be determined by the me-

thod i t s e l f , through the s t a t i o n a r y c o n d i t i o n f o r

En

.

T h i s procedu-

r e w i l l enable us t o o b t a i n a n a l y t i c a l expressions f o r t h e lues o f eq. (1 ) , o f t h e form

En = E ( a n

+ b+

c/n

eigenva-

+ . . .) .

T h i s paper i s organized as f o l l o w s . S e c t i o n 2 presents method f o r a general p o t e n t i a l ~ ( r ) .S e c t i o n

the

3 deals w i t h t h e appl i -

c a t i o n o f the method t o power law and l o g a r i t h m i c p o t e n t i a l s ,

toge-

t h e r w i t h a comparison of known r e s u l t s f o r some p o t e n t i a l s now b e i n g s t u d i e d as p o s s i b l e models o f quark conf inement (/2/ and

references

t h e r e i n ) . The eigenvalues a r e g i v e n as a f u n c t i o n o f two parameters,a and b

,

i n each case. The d e t e r m i n a t i o n , o f these parameters i n

r a l i s t h e s u b j e c t o f s e c t i o n 4. i 84

I n s e c t i o n 5 t h e numerical

gene-

accuracy

o f t h e eigenvalues, f o r t h e same examples as i n s e c t i o n sed and compared w i t h known r e s u l t s .

3, i s d i s c u s -

The l a s t s e c t i o n c o n t a i n s a sum-

mary o f the important f e a t u r e s o f t h e method. 2. THE METHOD

The method proposed here i s based on the r e p r e s e n t n t i o n o f t h e second d e r i v a t i v e o p e r a t o r = (exp(-ar),

r.exp(-ar),

r2.exp(-w)),

with

two- point on

the

0 0 . The

piecewise set

main

Ua

--

reasons

f o r using t h i s p a r t i c u l a r set are:

( i ) Any element of U s a t i s f i e s t h e boundary c o n d i t i o n i n eq. a infinity.

(1) a t

( i i ) The s e t L ( u ~ )o f a11 t h e l i n e a r combinations o f t h e elements

of

U i r i closed t o t h e o p e r a t i o n d2/dr2. T h i s means t h a t t h e second a d e r i v a t i v e o f any l i n e a r cornbination o f t h e elements o f Ua i s i t s e ' l f a l i n e a r cornbination o f t h e elements o f U ., Foi- a f u n c t i o n

Y

i n L ( u a ) t h e two- point

d2/dr2 cai7 be found as t h e unique s o l u t i o n o f t h e

representation desired

of

identity

gi ven by

where p , q and t a r e c o n s t a n t s . These c o n s t a n t s a r e u n i q u e l y c a l c u l a t e d by r e q u i r i n g t h a t eq.(4) be v a l i d f o r each o f t h e t h r e e f u n c t i o n s , o f .U

The constantsare

Equation (6) means t h a t t h e d i s t a n c e between t h e must be

two

points

T h i s r e p r e s e n t a t i o n o f d2/dr20n U

C1

i s the cornerstone

of this

paper and i s g i v e n by 2

d2

-Y(r1 dr2 I n t h i s case,

=

2a Y(r-l/a)

-

a2 ~ ( r )

(9)

e

eq.

(9)

means

that

t h e second d e r i v a t i v e o f

Y E L ( u ~ ) i s g i v e n e x a c t l y a t any p o i n t r by a 1 i n e a r

combination of

t h e v a l u e s o f Y i t s e l f a t t h e two p o i n t s r - l / u and r , c o r r e s p o n d i n g t o p r o p e r t y ( i i ) above. For a f u r t h e r d i s c u s s i o n a b o u t

representa-

d 2 / d r 2 on Ua t h e r e a d e r i s r e f e r r e d t o t h e Append i x .

t ion o f

Introducing a g r i d r wri t e r

the

-

k - l - rk

w i t h f(O)

k

-

I/%,

on t h e space c o o r d i n a t e 0 s r rk-lf o r any k . N o t e t h a t $ i s an

preserving function,

may

i.e.

k > j irnplies t h a t f ( k ) > f ( j ) .

order

This g r i d

w i l l be d e t e r m i n e d by t h e method i t s e l f .

The w i d t h o f each k - p i e c e g i v e n by A r f i e s eq.(8),

%%=

i.e.

k

= ]f(k)-f(k-1)

1 , and may change w i t h k

satis-

s i n c e eq.

(9)

is

a two- point formula. From eqs.

Since f ( k )

(8) and (10) i t

, f(k-1)

o1 lows t h a t

one can r e a d i y show t h a t (see Append i x )

$(k

=ak

+

b+c/k

+

...

f ( k ) - f (k-1) where t h e severa1 a ,

b, c , e t c . a r e c o n s t a n t s and k = 1,2,

E q u a t i o n (12) shows t h a t f o r any f u n c t i o n f ( k ) i s true:

( i ) The h i g h e r

k - o r d e r a p p e a r i n g i n eq.

(12 ) i s one.

the

... . following

(i i ) As point number k increases, eq. (12) is approxirnately in k.

I

inear

For these reasons we rnay neglect the 0(l/k) contribution ineq. (12) and wr i te

This expression will play an important role in the

following

discussion. One should note that, in what follows, higher-order terms could have been used in eq. (13). Armed wi th eqs'. (9) and (13) we now turn to the problem o f calculating the n-th eigenvalue of a given potential ~ ( r )in eq. (1). To this end we use the two-point representation of d2/dr2 given by to consider the piecewise approximation

eq.

(9) in eq. (1)

and ak is yet a variational parameter, obtaining

for i< = 1,2,... where if r,,= O then Y(")(r0) = O or if ro#O is connected to the origin by eq. (~.5)(see ~p~endix).Equation (14) may also be written in rnatrix forrn: M = EY with Y T and M a bidiagonal matrix given by \

=

(Y(rl),

Y(ri),

w i t h no l i m i t imposed on i t s o r d e r . T h i s m a t r i x i s t h e r e p r e s e n t a t i o n o f t h e Hami l t o n o p e r a t o r o f eq. (1 ) and depends on t h e v a r i a t i o n a l parameters

a , , a,,

...

with constraints. the

n-

and t h a t a11 the eigenvalues of t h e eq. (14) a r e g i v e n t r i v i a l l y

by

We now n o t e t h a t , owing t o t h e b i d i a g o n a l form o f M, -eigenvector o f M has t h e f o l l o w i n g form

t h e main diagona 1 o f

3

:

M, w i t h no l i m i t imposed on i t s o r d e r .

Using the e i g e n f u n c t i o n form I(") eq. (14) and r e c a l l i n g eq.

( r k ) o f eq. (16) f o r k=n

in

(13) we g e t

where n i s t h e r a d i a l quantum number.

ince

Y (n) ( r ) i s a f u n c t i o n o f t h e v a r i a t i o n a l parameter the i s a ~ ' / a a = O and can be d i r e c t l y a?.n

v a r i a t i o n a l p r i n c i p l e f o r En p l ied t o eq. (17) g i v "'g

The above equation a l s o determines t h e parameters sis

Q i n any

k - p i e c e by means o f eq.

(A.8) from t h e Appendix

t h e r e f o r e corresponds t o t h e v a r i a t i o n a l p r i n c i p l e t o t h e whol e

a o f the ba-

aE/aa

=

O

Y (n) ( r ) .

The general procedure be i n g proposed here t? o b t a i n s t a t e s o f eq. (1) begins by s o l v i n g eq. the best

\

and

applied

(18) i n o r d e r

to

as d e f i n e d above. By s u b s t i t u t i n g t h i s an i n eq.

the bound determine

(17) one

o b t a i n s t h e eigenvalues as a known f u n c t i o n o f the constants a and b , name 1y

Thetieterminationofaand b sha 11 now compare eq.

isdiscussed i n s e c t i o n b .

( 1 9 ) f o r t h e power law and l o g a r i t h m i c

We poten-

t i a 1s w i t h Itnown r e s u l t s . 3. APPLICATION TO THE POWER-LOW AND LOGARITHMIC POTENTIALS

To f u r t h e r c l a r i f y t h e proposed method we now a p p l y i t t o some w e l l -known eigenproblems. The r e s u l t s o b t a i n e d a r e compared w i t h t h e exact ones, whenever possi b l e , o r t o approximations. For we o n l y analyse

s- states.

For t h e general power law p o t e n t i a l ~ ( r =) K and p # O,

r,

with

eq. (17) g i v e s (dropping the s u b s c r i p t n o f

From the s t a t i o n a r y c o n d i t i o n , pare w i t h eq.

simpl i c i t y

aEn/aa

n

= O,

p > -2

a )

i t follows

(com-

(18)) t h a t

S u b s t i t u t i n g back i n eq.

(201, one o b t a i n s

as t h e bound s t a t e eigenvalues o f t h e power law p o t e n t i a l , o r

i n terms o f ct o f eq. ( 2 1 ) . Using t h e same procedure f o r the l o g a r i t h m i c p o t e n t i a l = K R n ( r ) , one o b t a i n s the eigenvalues

En =

r

k n( e / K q

+

K !Ln ( a n

+

b)

~ ( r =)

I n T a b l e 1 we g i v e n as f u n c t i o n s o f a and b , t h e s o l u t i o n s f o r some p o t e n t i a l s w h i c h a r e nowadays o f i n t e r e s t f o r a q u a r k - q u a r k con-

f i n i n g model t o g e t h e r w i t h t h e g e n e r a l power law. The parameter a and

b a p p e a r i n g i n t h i s t a b l e a r e c a l c u l a t e d i n s e c t i o n 4 . However, as an example, t h e r e a d e r may r e c a l l t h a t f o r a = 1 and

b

= O

t h e Coulomb

e i g e n v a l u e s i n T a b l e 1 r e p r e s e n t t h e c o r r e c t b o u n d - s t a t e spectrum f o r any o r d e r n. I n T a b l e 1 two i m p o r t a n t f u n c t i o n a l r e l a t i o n s h i p s f o r t h e e i genvalues can be seen: w i t h t h e s c a l i n g parameter K and w i t h

(an+b).

The r e m a i n i n g p a r t o f t h i s s e c t i o n i s d e v o t e d t o a comparison o f t h e s e two a s p e c t s o f t h e f u n c t i o n E

=

n

~ ( a n + bw) i t h t h e known r e s u l t s . As a

s o u r c e o f known r e s u l t s we use t h e work o f Qu i g g and ~ o s n e r ~ . The K dependences o f

En g i v e n i n T a b l e 1 a r e a l l c o r r e c t ,

as

can v e r y e a s i l y be v e r i f i e d by r e s c a l i n g t h e ~chr:d i n g e r e q u a t i o n ( 2 ) . The f u n c t i o n a l dependence o f t h e e i g e n v a l u e s w i t h t h e quantum number a s g i v e n i n T a b l e ! i s e x a c t f o r t h e Coulomb, l i n e a r and

har-

monic p o t e n t i a l s . F o r t h e o t h e r cases no e x a c t s o l u t i o n s a r e known. However, a s can be seen f r o m eqs.

(4.33) and (4.59) f r o m

',

indeed show t h e same quantum number dependence a s t h e WKB

our resul t s ones. From

t h e above comparison one sees t h a t t h e proposed method r e p r o d u c e s t h e known f u n c t i o n a l dependences f o r t h e power l a w and l o g a r i t h m i c p o t e n t i a l s w i t h K and

n.

For o t h e r p o t e n t i a l s ,

the constants

a and b c o u l d

depend

on

t h e scaZing c o n s t a n t K. However, and h e r e we emphazise t h i s p o i n t , t h e f u n c t i o n a l dependence En = t ion

E(an + b ) i s a l w a y s t h e same i f t h e

func-

~ ( r does ) n o t change. T h i s can eas i ly be v e r i f i e d f r o m eqs. (17)

and ( 1 8 ) . T h i s means t h a t f o r any g i v e n p o t e n t ia1 t h e p o t e n t ia1 may o n l y change

a and b b u t n o t t h e

~ ( ra ) s c a l i n g o f f u n c t i o n En=E(an+b).

I n t h e n e x t s e c t i o n we d i s c u s s t h e c a l c u l a t i o n o f a and b .

4. CALCULATIONS OF A AND B

The p r o c e d u r e

described

i n the previous sections

ctosed-form expressions of the type

En

=

leads t o

E(an+b) f o r t h e e i g e n v a l u e s .

To d e t e r m i n e a and b , we s t u d y t h e b e h a v i o u r o f two e i g e n f u n c t i o n s a f t e r t h e o u t e r t u r n i n g - p o i n t .

the

first

I n o r d e r t o genera-

t e a c c u r a t e e i g e n f u n c t i o n s one u s u a l l y needs A r n c o r r e s p o n d s t o t h e

decaying

part.

4. E.Gerck and A.B.d l Ol i v e i r a , J.Comp.App1 .Math., Vol .6, 81 (1980). 5. The s q u a r e - r o o t p o t e n t i a l has a l s o been proposed t o model t h e quark -quark

intera~tion~'~.

6 . H.F.Carvalho, R.Chanda and A . B . d l O l i v e i r a , 679 (1978). See a l s o H.F.Carvalho, v e i r a , Proc.

I ENFPC,

R.Chanda,

Carnbuquira, MG,

L e t t . N u o v o Cimento, 22, E.Gerck and A.B.

Soc.Bras.de

F í s i c a , p.49

dlOli-

(1979).

7. H.F.Carvalho and A.B.d l Ol i v e i r a , L e t t . Nuovo Cimento, 33, 572(1982). RESUMO

Um método p a r a a s o l u ç ã o da equação r a d i a l de s c h r 8 d i n g e r par a e s t a d o s 1 igados 6 d i s c u t i d o . Baseia- se em uma nova r e p r e s e n t a ç ã o s e c c i o n a l ("piecewise") do o p e r a d o r d e r i v a d a segunda s o b r e um c o n j u n t o de funções que s a t i s f a z e m as c o n d i ç õ e s de c o n t o r n o . Essa r e p r e s e n t a ç ã o 6 d i a g o n a l i z a d a t r i v i a l m e n t e e conduz a expressões fechadas do t i p o = E ( a n + b + c/n + . . ) p a r a o s a u t o v a l o r e s . P o t e n c i a i s que são pot e n c i a s e l o g a r i tmos são usados como exemplos.

5

.