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A Fourier]Bessel Expansion for Solving
Radial Schrodinger
Equation in Two
¨
Dimensions
H. TAŞELI AND A. ZAFER
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey; E-mail
(H.T.): taseli@rorqual.cc.metu.edu.tr
Received January 15, 1996; accepted March 1, 1996
ABSTRACT
The spectrum of the two-dimensional Schrodinger
equation for polynomial oscillators
¨
bounded by infinitely high potentials, where the eigenvalue problem is defined on a
finite interval r g w 0, L., is variationally studied. The wave function is expanded into a
Fourier]Bessel series, and matrix elements in terms of integrals involving Bessel functions
are evaluated analytically. Numerical results presented accurate to 30 digits show that,
by the time L approaches a critical value, the low-lying state energies behave almost as if
the potentials were unbounded. The method is applicable to multiwell oscillators as well.
Q 1997 John Wiley & Sons, Inc.
1. Introduction
I
n his recent articles w 1]3x , Taşeli modified the
usual requirement that the wave function
should tend to zero at infinity and showed that the
eigensolution of the Dirichlet boundary value
problem can be effectively used to find the spectrum of an unbounded problem in one dimension.
In these works, the eigenfunctions satisfying the
boundary value problem
d2 C
dx 2
q m2 C s 0,
C Ž a . s C Ž b . s 0 Ž 1.1.
International Journal of Quantum Chemistry, Vol. 61, 759]768 (1997)
Q 1997 John Wiley & Sons, Inc.
were employed as the basis set in the
Rayleigh]Ritz variational method. The approach
has a natural extension to the two-dimensional
Schrodinger
equation written in Cartesian coordi¨
nates. Actually, two-dimensional anharmonic oscillators can be treated in a similar fashion by
means of the boundary value problem defined by
d2 C
dx 2
q
d2 C
dy 2
q m2 C s 0,
C < G s 0, Ž 1.2.
where G denotes the boundary of a finite rectangular region in the xy-plane w 4x .
In this study, we examined the dimensionless
radial Schrodinger
equation in the cylindrical polar
¨
CCC 0020-7608 / 97 / 050759-10
TAŞELI AND ZAFER
coordinates:
y
d2
dr 2
y
1 d
r dr
q
l2
r2
q V Ž r . C Ž r . s EC Ž r . ,
special case of the harmonic oscillator, V Ž r . s r 2 ,
admits exact solutions in the unbounded domain
of r of the form
2
Fn l Ž r . s eyŽ1 r2. r LŽnl . Ž r 2 . ,
r g w 0, ` . ,
CŽ r . s OŽ r l .
Ž 1.4.
as r ª 0. The second condition, however, is replaced by
C Ž L. s 0
n, l s 0, 1, . . . ,
Ž 1.3.
where l s 0, 1, . . . , V Ž r . and E stand for the magnetic quantum number, the potential function, and
the energy eigenvalue, respectively. It is obvious
that the coupling might depend on the direction.
We have, however, omitted the angle dependence
of the potential for the sake of dealing with a
system which can be investigated via an ordinary
differential equation. Therefore, l characterizes the
angular dependence of the system in a global
sense, and the wave function C has been regarded
as a function of the single variable r.
The accompanying boundary conditions of Ž1.3.
are the regularity and the appropriately vanishing
behavior of the wave function specified as r ª 0
and r ª `, respectively. The regularity condition
implies that
Ž 1.5.
when the interval is truncated to w 0, L x . Such a
truncation is clearly motivated by the success of
the simple technique presented in the aforementioned articles w 1]4x . The question which now arises
is whether there exists a corresponding basis set,
preferably in terms of elementary or special functions of mathematical analysis for solving Ž1.3..
Fortunately, in the case of the radial Schrodinger
¨
equation, the Bessel functions of the first kind are
to be shown in Section 2 to play the same role with
the trigonometric basis.
The potential function V Ž r . in Ž1.3. is taken as a
general polynomial:
En , l s 2 Ž 2 n q l q 1 . ,
Ž 1.7.
LŽnl .
denotes the associated Laguerre polynowhere
mials.
In general, the asymptotic behavior of the wave
function as r ª ` completely depends on the
dominant coupling. As a result, it is rather difficult
to introduce a trial function reflecting the desired
properties of the solution for an arbitrary anharmonic interaction w 5x . The definition of the problem in a finite interval r g w 0, L x , however, makes
it possible to consider a general polynomial potential rather than a specific one. The idea is based
upon regarding the boundary value L as a nonlinear optimization parameter to be determined in
such a way that the spectrum fits to the spectrum
of the corresponding unbounded problem, where
L ª `, to any prescribed accuracy. Moreover, a
model of this kind, namely, an enclosed quantum
mechanical system, is of importance not only for
finding the spectrum of an unbounded one but
also its various applications in several fields w 6x
Žand the references cited therein..
Within these perspectives, Section 2 sets out the
basic variational formulation of the problem. Section 3 includes the evaluation of integrals containing Bessel functions. The last section presents the
applications of the method and concludes the article with a discussion of the results.
2. The Fourier]Bessel Expansion
In this section, we begin with solving the unperturbed Schrodinger
equation defined by
¨
ž
d2
dr 2
q
1 d
2
r dr
qm y
l2
r2
/
F Ž r . s 0,
r g w 0, L x ,
Ž 2.1.
M
VŽr. s
Ý v2 j r 2 j ,
v 2 M ) 0,
M s 1, 2, . . .
js1
Ž 1.6.
in r 2 . The positiveness of the dominant coupling
constant v 2 M is sufficient to make the potential
bounded below. Therefore, the operator being considered has now a purely discrete spectrum. The
760
subject to
FŽ r . s OŽ r l .
as r ª 0,
F Ž L . s 0, Ž 2.2.
where m is a constant, and we have assumed a
potential function of the form
VŽr. s
½
0,
`,
0Fr-L
r ) L.
Ž 2.3.
VOL. 61, NO. 5
A FOURIER ]BESSEL EXPANSION
It is readily shown that the two linearly independent solutions of Ž2.1. are Jl Ž m r . and Yl Ž m r .,
namely, the Bessel functions of the first and the
second kind, respectively. However, the Bessel
functions of the second kind do not remain finite
at r s 0 so that we may take
plying the result by r Fm Ž r ., and integrating from
zero to L, we obtain
F Ž r . s cJl Ž m r . ,
for m s 1, 2, . . . . If we define the variational matrix to be
Ž 2.4.
`
Ý
ns1
cml
2 l l!
rl
Ž 2.5.
Ž 2.6.
We know from the theory of Bessel functions that
there is an enumerable infinite set of roots m s
m 1 , m 2 , . . . m n , . . . satisfying Ž2.6. w 7x . Furthermore,
when c is properly chosen, the sequence of functions
LJlq1Ž m n L .
Jl Ž m n r . ,
forms an orthonormal set over the range of r,
r g w 0, L x , with respect to the weighting function
r. Thus, we have
L
n
m
Ž r . dr s dn m
Ž 2.8.
for all values of n and m.
The orthonormality property suggests the expansion of the wave function in the form
`
CŽ r . s
Ý a nFn Ž r . s
ns1
'2
L
`
Ý
ns1
Ž r . dr y Edn m a n
L
H0 V Ž r . r F Ž r . F
n
m
Ž 2.10.
Ž r . dr , Ž 2.11.
we then arrive at the secular equations of the form
Ý
Ž Hn m y Edn m . a n s 0,
m s 1, 2, . . . ,
ns1
Ž 2.12.
for determination of the coefficients a n . Using Ž1.6.
and making a simple change of variable, the variational matrix is expressible as
M
2
Hn m s Ž a nrL . dn m q
Ý v 2 j L2 j InŽ lm, j. ,
wherein InŽ l,m j. stands for the integral
InŽ lm, j. s c nŽ lm.
1
H0 j
2 jq1
Jl Ž a n j . Jl Ž a m j . d j Ž 2.14.
to be determined for each l. Hereafter, a1 ,
a 2 , . . . , a n , . . . denote the positive zeros of Jl Ž x .,
and the constant c nŽ lm. is given by
c nŽ lm. s
2
Jlq1Ž a n . Jlq1Ž a m .
.
an
Jl Ž m n r . ,
Ž 2.15.
It is apparent that InŽ l,m j. is symmetric in n and m
and, hence, that Hn m is symmetric.
In the coming section, we shall show that the
integrals in Ž2.14. may be evaluated recursively. In
particular,
InŽ lm, 0. s dn m
Jlq1Ž m n L .
Ž 2.13.
js1
n s 1, 2, . . .
Ž 2.7.
H0 r F Ž r . F
m
`
Jl Ž m L . s 0.
'2
n
Hn m s m 2n dn m q
as r ª 0. Imposing the second condition, we see
that Ž2.4. is the required solution if m is a positive
root of the equation
Fn Ž r . s
L
H0 V Ž r . r F Ž r . F
s 0,
which behaves correctly at the origin, where c is
some normalization constant. More specifically,
FŽ r . f
m2n dn m q
Ž 2.16.
provided that Jl Ž a n . s 0 for any l fixed.
Ž 2.9.
3. Evaluation of Matrix Elements
which is called the Fourier]Bessel expansion of
C Ž r .. The theory of such an expansion is given by
Watson in w 8x . Substituting Ž2.9. into Ž1.3., multi-
In this section, the integrals appearing in the
matrix elements are evaluated analytically for l s 0
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
761
TAŞELI AND ZAFER
and l s 1, which are representative for the other
cases. Since such integrals are not so trivial to deal
with, we feel that we should give a sketch of the
derivation steps. So let us consider Ž2.14. setting
l s 0,
InŽ0,m j. s c nŽ0.m
1
H0 j
2 jq1
J0 Ž a n j . J0 Ž a m j . d j ;
n, m s 1, 2, . . . ,
Ž 3.1.
tuting into Ž3.2., we find that
InŽ0,m j. s
s
Ž a n2 y a m2 .
a n2 y a m2
Ž am Am n y an Anm . ,
Ž 3.2.
H0 j
2j
J1 Ž a p j . J 0 Ž a q j . d j
Ž 3.3.
and A p q / A q p . Integration by parts of the last
integral leads to
jy1.
a m A n m q a n A m n s 2 y 2 Ž j y 1 . PnŽ1,
,
m
j ) 0,
Ž 3.4.
where a temporary quantity PnŽ1,m j.,
j.
Ž0.
PnŽ1,
m s cn m
1
H0 j
2 jq1
J1 Ž a n j . J1 Ž a m j . d j ,
j.
Ž1 , j.
PnŽ1,
m s Pm n ,
j.
PnŽ1,
m s
dj
j 2 j J0 Ž a n j . J0 Ž a m j .
y a m2 .
2
Ž 3.9.
for the calculation of PnŽ1,m j. recursively. The initial
conditions for these recursions are
InŽ0,m 0. s 0
Ž 3.10.
0.
PnŽ1,
m s 0,
Ž 3.11.
and
respectively. The first condition is a consequence
of the orthogonality of the Bessel functions, and
the second one can be deduced from the integral
Ž3.5. with j s 0.
Whenever m s n, InŽ0,n j. may be derived by
means of certain limit operations as a n ª a m using l’Hospital rule. It is, however, easier to follow
an alternative way. Indeed, noting that Ž3.4. and
Ž3.7. are valid for m s n as well, we show that
Ž j q 1 . InŽ0,n j. s 1 y jPnŽ1n, j.
Ž 3.12.
a n A n n s jInŽ0,n jy1. .
Ž 3.13.
and that
On integrating by parts, it is not difficult to prove
that PnŽ1,n j. satisfies also the equation
Therefore, substituting Ž3.13. into Ž3.14. and eliminating PnŽ1,n j. from Ž3.12. and Ž3.14., it follows that
y a n j 2 j J1 Ž a n j . J 0 Ž a m j .
y a m j 2 j J 0 Ž a n j . J1 Ž a m j . ;
Ž 3.6.
a n2 Ž 2 j q 1 . InŽ0,n j. s a n2 y 2 j 3 InŽ0n, jy1. ,
j s 1, 2, . . . , M.
it follows immediately that
j ) 0, Ž 3.7.
which is independent of Ž3.4.. Solving simultaneously Ž3.4. and Ž3.7. for A n m and A m n and substi-
762
4j
Ž a n2
j.
a n Ž j q 1 . PnŽ1,
s a n y ja n InŽ0n, j. q 2 jA n n . Ž 3.14.
n
s 2 j j 2 jy1 J0 Ž a n j . J0 Ž a m j .
a n A n m q a m A m n s 2 jInŽ0,m jy1. ,
Ž 3.8.
= Ž a n2 q a m2 . 1 y Ž j y 1 . PnŽ1m, jy1.
Ž 3.5.
has been introduced. It should be noticed that
PnŽ1,m j. is not a constant multiple of InŽ1,m j. defined by
Ž2.14. since a n’s are not the zeros of J1Ž x . in this
case of l s 0. If we now make use of the identity
d
1 y Ž j y 1 . PnŽ1m, jy1.
for j ) 0 and m / n. Likewise, we obtain the relation
when n / m, where
A p q s c Ž0.
pq
2 an am
y2 ja n a m InŽ0,m jy1. 4
2j
1
2
yj Ž a n2 q a m2 . InŽ0,m jy1. 4
with J0 Ž a n . s 0 and j s 1, 2, . . . , M. Integrating
by parts and using some well-known properties of
the Bessel functions, we see that Ž3.1. may be
written as
InŽ0,m j.
4j
Ž 3.15.
Initially, we have from Ž2.16.
InŽ0,n 0. s 1
Ž 3.16.
for all n.
VOL. 61, NO. 5
A FOURIER ]BESSEL EXPANSION
In a similar fashion, for l s 1, the integral
InŽ1m, j. s c nŽ1.m
1
H0 j
2 jq1
and the double-well potential of the form
V Ž r . s yr 2 q d 4 r 4
J1 Ž b n j . J1 Ž b m j . d j ;
n, m s 1, 2, . . .
Ž 3.17.
may be evaluated recursively, where the bn’s are
now the positive zeros of J1Ž x .. If m / n, we find
that
InŽ1,m j.
s
4j
Ž bn2 y bm2 .
2
2 bn bm
1y
for a wide range of the coupling constants. As
v 2 k ª ` in Ž4.1., we have the infinite-field limit
Hamiltonian described by the equation
ž
jQ nŽ0m, jy1.
y
d2
dr 2
s
4j
Ž bn2 y bm2 .
2
Ž
bn2
q
bm2 .
1y
Ž 3.18.
jQ nŽ0m, jy1.
y2 Ž j y 1 . bn bm InŽ1,m jy1. 4 ,
Q nŽ0,m0. s 0,
1
H0 j
2 jq1
J0 Ž bn j . J0 Ž bm j . d j ,
Ž0 , j.
Q nŽ0,m j. s Q m
n .
Ž 3.20.
If m s n, then
Ž 2 j q 1 . bn2 InŽ1n, j. s bn2 y 2 j Ž j 2 y 1 . InŽ1,n jy1. ,
InŽ1,n 0. s 1.
Ž 3.21.
In general, the problem of calculating integrals in
Ž2.14. for any l requires a treatment of this kind.
4. Applications and
Concluding Remarks
k s 2, 3, 4,
l2
r2
/
q r 2 k C Ž r . s lC Ž r . ,
Ž 4.4.
This asymptotic relation shows that the total enkq1.
ergy E of the system Ž1.3. grows like v 21rŽ
for
k
large values of v 2 k . For this reason, we also consider the potentials
VŽ r. s r2k,
k s 1, 2, 3, 4, and 10, Ž 4.5.
to cover the limiting case of the anharmonicity
constant, as v 2 k ª `. The potential in Ž4.5. with
k s 1, or Ž4.1. with v 2 k s 0, corresponds to the
harmonic oscillator whose exact eigenstates are
given by Ž1.7. for the unbounded interval of r,
r g w 0, `.. Therefore, this problem provides a very
good check on the accuracy of the present approach, which assumes a finite interval, r g w 0, L x .
As a specimen test, Table I demonstrates the
rate of convergence of the method as functions of
TABLE I
Convergence rate of the method as functions
of L and N for the ground-state energy E 0, 0
of the harmonic oscillator, V ( r ) = r 2.
L
The present technique is applied to some specific potentials in Ž1.6. to show its computational
performance. To this end, the infinite system Ž2.14.
has been truncated to a homogeneous system of a
finite number of equations, N say. The truncated
eigenvalues are then determined as the roots of the
so-called characteristic equation.
We examine numerically the generalized anharmonic oscillators
V Ž r . s r 2 q v2 k r 2 k ,
r dr
q
rŽ kq1.
l s vy1
E.
2k
Ž 3.19.
where
Q nŽ0,m j. s c Ž1.
nm
1 d
where l is connected with the energy eigenvalues
by the relation
and
Q nŽ0,m j.
y
Ž 4.3.
y Ž j y 1 .Ž bn2 q bm2 . InŽ1,m jy1. 4 ,
InŽ1,m 0. s 0
Ž 4.2.
N
E 0, 0
5
8
10
12
2.000 000 001 346
2.000 000 001 3308
2.000 000 001 33077
7
14
16
18
2.000 000 000 000 000 0306
2.000 000 000 000 000 000 1007
2.000 000 000 000 000 000 1006
9
20
22
24
2.000 000 000 000 000 000 005 952
2.000 000 000 000 000 000 000 188
2.000 000 000 000 000 000 000 000 00000
11
26
28
30
2.000 000 000 000 000 000 0168
2.000 000 000 000 000 000 000 000 0236
2.000 000 000 000 000 000 000 000 00000
Ž 4.1.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
763
TAŞELI AND ZAFER
the boundary parameter L and the truncation size
N in calculating the exact ground-state energy,
E0, 0 s 2, of the harmonic oscillator. It is seen that
if we restrict ourselves to about 10 digit accuracy,
it is sufficient to set L s 5. Increasing L from 5 to
7 and to 9, respectively, 20 and 30 significant
figures are obtained.
In our numerical tables, we report eigenvalues
to 30 digits. Table I shows clearly that this prescribed accuracy is first achieved at L s 9, and if
we take an L value beyond L s 9, for instance,
L s 11, we certainly get the same result again at
the cost of using a higher truncation size. Therefore, L s 9 is called the critical or optimum
boundary value, denoted by L cr , for this case. Such
a behavior of determining eigenvalues with respect to L is observed in all our calculations.
In general, we continue changing L until the
eigenvalues of the required accuracy, i.e., 30 digits,
are obtained. The definition of L cr implies the
statement that
E Ž`. y E Ž L cr . - e ,
Ž 4.6.
where e s 10y3 0 and EŽ L. and E Ž`. are the eigenvalues of the bounded and the corresponding unbounded problem, respectively. Also, the influence
of the finite boundary on the spectrum, numerically speaking, is greater than e when L - L cr . Of
course, L cr is not a unique value which has to be
estimated very precisely. Our computational search
shows that in the near vicinity of the reported L cr
values the rate of convergence is more or less the
same in finding the low-lying state energies. Furthermore, the algorithm employs a minimum
number of the basis functions in this neighborhood
of L cr . Actually, if L 4 L cr , then a considerable
slowing down of convergence may occur.
The existence of an optimum value of L, which
should not be very small or very large, may also
be deduced from the fact that the kinetic energy
TABLE II
Critical values L cr and the energy eigenvalues l n, 0 of the potential, V ( r ) = r 2 k , as a function of k.
k
l n, 0
n
Lcr
N
9.0
9.5
9.5
9.5
10.5
24
28
30
32
34
1
0
1
2
3
4
2.000
6.000
10.000
14.000
18.000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
00000
00000
0000
0000
0000
2
0
1
2
3
4
2.344
9.529
18.735
29.301
40.941
829
781
195
548
918
072
384
504
228
353
744
014
701
942
803
275
807
770
291
625
209
959
773
903
625
808
799
554
145
662
995
611
069
808
009
79643
30530
1439
4950
1641
5.0
5.0
5.0
5.0
5.25
28
30
30
32
34
3
0
1
2
3
4
2.609
11.946
25.463
42.067
61.261
388
863
626
859
386
463
508
993
920
349
253
851
869
619
562
714
368
977
309
942
006
705
140
008
201
877
043
819
621
185
033
604
711
746
703
41876
0993
2200
0328
1423
3.6
3.6
3.6
3.6
3.6
34
36
36
38
38
4
0
1
2
3
4
2.828
13.699
30.581
52.183
77.884
786
710
181
559
490
159
847
755
593
773
942
251
692
757
999
523
447
765
451
550
348
928
465
059
572
784
135
261
366
609
150
472
256
450
283
37593
2390
2851
9780
8675
2.9
2.9
2.9
2.9
2.9
42
42
42
44
44
10
0
1
2
3
4
3.651
19.019
45.936
83.598
131.346
024
015
257
817
976
848
732
303
125
425
669
785
451
768
067
465
642
846
624
960
833
006
037
847
776
849
588
295
870
224
211
600
055
170
366
68554
8668
3049
5885
561
1.72
1.72
1.72
1.72
1.72
84
84
84
84
84
764
VOL. 61, NO. 5
A FOURIER ]BESSEL EXPANSION
term in the variational matrix Ž2.13. grows unboundedly as L ª 0, whereas the potential energy
term becomes infinite as L ª `. On the other
hand, the full spectrum of an eigenvalue problem
cannot be calculated by estimating a fixed critical
distance. Naturally, the larger L cr values are
needed for the higher excited states. As a result of
these remarks, we may conclude that L cr depends
on the required accuracy e , the state number n,
and the potential function V Ž r . in question.
In Tables II and III, we report the first few state
energies of the potentials in Ž4.5. for l s 0 and
l s 1, respectively. It is clear that the number of
basis functions N we used increases from 24 to 84
as the potentials vary from harmonic oscillator to
the potential V Ž r . s r 20 . Owing to the contraction
of the potentials as k increases, however, L cr decreases.
The eigenvalues of the quartic, sextic and octic
anharmonic oscillators are tabulated in Tables
IV]VI for a very wide range of the coupling con-
stants. It is worth mentioning that there is no
accuracy loss in any regime of the eigenvalues. In
these tables, the eigenvalues for v 2 k ) 1 are rerŽ kq1.
placed by vy1
E to show how rapidly they
2k
converge to the v 2 k ª ` limit energies given in
Table II. The rate of convergence in each case is
consistent with that of the potential in Table II
having the same asymptotic behavior. The numerical data are presented only for l s 0 in order not
to overfill the content of the article with tabular
material anymore. Further results are available
from the authors.
Finally, the method is applied to an eigenvalue
problem of a different nature, where the potential
has two minima in Cartesian coordinates. We see
from Table VII that the method has the capability
of yielding accurate results in this case as well. In
general, the recorded eigenvalues to 30 digits in
our tables are in good agreement with the previously published results to the accuracy quoted w 5x
Žand the references therein..
TABLE III
Critical values L cr and the energy eigenvalues l n, 1 of the potential, V ( r ) = r 2 k , as a function of k.
k
l n, 1
n
Lcr
N
9
9.5
9.5
9.5
10.5
26
28
30
32
34
1
0
1
2
3
4
4.000
8.000
12.000
16.000
20.000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
00000
00000
0000
0000
0000
2
0
1
2
3
4
5.394
13.811
23.775
34.922
47.042
227
109
788
189
438
164
536
766
824
982
172
873
400
920
602
288
734
460
427
590
035
556
908
180
421
827
089
084
557
409
128
177
504
215
773
08911
2372
4638
2974
5228
5
5
5
5
5.25
28
30
30
32
34
3
0
1
2
3
4
6.298
18.042
33.226
51.197
71.579
495
624
111
907
036
901
963
031
527
686
483
215
283
921
497
604
149
994
898
045
243
284
491
817
242
475
883
360
904
774
867
606
615
695
490
95748
1618
7976
5520
7208
3.6
3.6
3.6
3.6
3.6
34
36
36
38
38
4
0
1
2
3
4
6.973
21.172
40.562
64.303
91.915
963
924
306
564
659
604
759
275
967
881
062
544
335
276
269
474
699
065
949
497
495
545
280
092
907
964
598
118
416
582
177
219
837
203
925
5380
3823
1528
5681
9112
2.9
2.9
2.9
2.9
2.9
42
42
42
44
44
10
0
1
2
3
4
9.234
30.496
62.906
105.705
158.326
548
392
669
424
821
141
239
978
592
543
850
950
234
282
865
486
681
303
686
973
506
357
678
348
806
709
948
956
235
795
132
495
781
155
382
62479
1962
7182
655
263
1.72
1.72
1.72
1.72
1.72
80
80
80
80
82
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
765
TAŞELI AND ZAFER
TABLE IV
Critical values L cr and the energy eigenvalues E n, 0 of the quartic oscillator, V ( r ) = r 2 + v4 r 4 ,
as a function of v4 .
v4a
n
10 y 3
0
1
2
3
4
2.001
6.013
10.037
14.073
18.120
995
936
726
268
467
522
098
447
901
854
094
189
540
848
849
708
653
990
599
535
533
073
997
025
519
684
559
168
546
253
913
642
029
920
608
20578
74371
2097
3647
5835
1
0
1
2
3
4
2.952
10.882
20.661
31.725
43.816
050
435
082
128
823
091
576
690
191
442
962
819
597
323
711
874
807
886
846
769
287
243
008
000
292
056
980
613
694
920
570
305
860
493
245
38705
2898
6236
6603
9907
5.25
10 3
0
1
2
3
4
2.351
9.543
18.754
29.326
40.971
338
744
903
237
122
918
980
714
058
304
312
405
202
380
391
985
963
645
066
642
396
422
896
510
340
323
344
830
244
873
609
031
091
951
918
83486
57957
2136
0992
4630
1.6
10 6
0
1
2
3
4
2.344 894 220 027 885 068 182 968 20759
9.529 921 064 696 036 065 423 201 49701
18.735 392 632 619 681 244 508 244 2998
29.301 795 163 271 981 703 005 355 0114
40.94 221 043 942 410 630 883 781 62603
0.5
a
E n, 0
Lcr
10.5
Eigenvalues for v4 ) 1 are v4y 1/ 3 E n, 0 .
TABLE V
Critical values L cr and the energy eigenvalues E n, 0 of the sextic oscillator, V ( r ) = r 2 + v6 r 6 ,
as a function of v6 .
v 6a
n
10 y 4
0
1
2
3
4
2.000
6.007
10.032
14.086
18.178
598
751
542
013
227
762
220
763
194
767
132
149
227
042
416
326
997
701
651
675
882
804
804
079
616
320
295
824
974
115
348
328
865
348
325
64889
80437
7460
5427
6809
10
1
0
1
2
3
4
3.121
12.914
26.720
43.558
62.954
935
938
687
836
081
474
793
689
621
100
246
084
389
235
886
425
835
101
999
758
991
743
123
594
921
126
473
735
757
519
392
718
627
325
461
39363
5394
4174
0745
6064
3.6
10 4
0
1
2
3
4
2.614
11.956
25.476
42.082
61.278
732
628
265
827
364
045
506
711
572
418
811
631
477
804
969
295
557
599
005
761
321
144
711
058
486
263
562
336
067
805
037
373
570
242
218
39260
9480
5449
6942
6638
1.4
a
E n, 0
Lcr
Eigenvalues for v6 ) 1 are v6y 1/ 4 E n, 0 .
766
VOL. 61, NO. 5
A FOURIER ]BESSEL EXPANSION
TABLE VI
Critical values L cr and the energy eigenvalues E n, 0 of the octic oscillator, V ( r ) = r 2 + v8 r 8 ,
as a function of v8 .
v 8a
n
10 y 5
0
1
2
3
4
2.000
6.004
10.030
14.106
18.271
239
995
661
953
189
435
387
401
931
889
330
244
804
490
023
015
901
310
152
522
489
355
454
999
434
836
807
756
688
922
799
670
728
231
736
61699
65716
4019
7547
7644
9.2
1
0
1
2
3
4
3.287
14.491
31.551
53.294
79.113
880
330
234
182
049
426
259
380
462
969
306
511
201
089
649
474
367
413
064
292
147
660
588
595
429
366
816
563
699
073
316
113
682
742
078
46427
6330
1136
6777
1720
2.9
10 5
0
1
2
3
4
2.833
13.707
30.590
52.194
77.896
519
644
894
675
784
160
868
905
854
653
868
712
108
570
678
379
494
705
619
648
169
398
724
011
629
539
947
012
260
907
757
958
094
569
144
15041
8335
2824
8965
5488
0.9
a
E n, 0
Lcr
Eigenvalues for v8 ) 1 are v8y 1/ 5 E n, 0 .
In this article, to the best of our knowledge, a
Fourier]Bessel expansion for the wave function is
utilized for the first time for solving eigenvalue
problems of this kind. The accuracy of the method
is quite impressive. The evaluation of the matrix
elements given in Section 3 requires only the de-
termination of the zeros of Bessel functions accurately. To this end, making use of Mathematica w 9x ,
we calculated the zeros to 40 digits.
It is well known, from the variational principle,
that the Rayleigh]Ritz method provides an upper
bound for the eigenvalues. Therefore, better ap-
TABLE VII
Critical values L cr and the energy eigenvalues E n, 0 of the two-well potential, V ( r ) = yr 2 + d4 r 4 ,
as a function of d4 .
E n, 0 + 1 / (4 d4 )
d4
n
10 y 2
0
1
2
3
4
1.398
4.164
6.863
9.491
12.040
819
445
695
359
996
606
401
271
674
051
958
438
722
405
530
560
967
825
070
290
193
973
702
730
407
588
159
783
025
545
327
747
019
119
701
78011
89411
62531
44685
1565
15
1
0
1
2
3
4
1.637
8.087
16.716
26.785
37.973
487
207
860
032
893
952
576
324
000
745
723
543
044
948
187
690
170
199
397
635
820
457
941
536
988
759
679
511
055
564
675
953
949
018
044
40693
48886
9881
3172
5983
5.25
10 2
0
1
2
3
4
10.745
43.934
86.538
135.475
189.408
379
440
410
768
304
344
817
698
778
514
538
669
064
152
324
852
132
907
347
643
199
038
492
645
004
155
133
909
925
961
496
141
114
981
139
0807
7197
7669
316
847
2.35
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
Lcr
767
TAŞELI AND ZAFER
proximations are sure to be achieved for successive values of N. On the other hand, it is not
surprising that the larger region has smaller eigenvalues for such a boundary value problem. Actually, Nunez w 10x showed theoretically that the
eigenfunctions of an unbounded system can be
approximated by means of the numerical solutions
of the Dirichlet problem in the Hilbert space L2 Ž V .
with sufficiently large V, where V is a bounded
region. This makes our strategy plausible in obtaining the eigenvalues of unbounded oscillators
by increasing the boundary parameter L. However, our studies on the estimation of error bounds
and the generalization of the method to higherdimensional spaces are in progress and will be
reported in the near future.
768
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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H. Taşeli, Int. J. Quantum Chem. 46, 319 Ž1993..
H. Taşeli, Int. J. Quantum Chem. 60, 641 Ž1996..
H. Taşeli and R. Eid, Int. J. Quantum Chem. 59, 183 Ž1996..
H. Taşeli, Int. J. Quantum Chem. 57, 63 Ž1996..
F. M. Fernandez and E. A. Castro, Int. J. Quantum Chem.
19, 521 Ž1981..
I. N. Sneddon, Special Functions of Mathematical Physics and
Chemistry ŽOliver and Boyd, Edinburgh, 1966., p. 133.
G. N. Watson, A Treatise on the Theory of Bessel Functions
ŽCambridge University Press, Cambridge, 1962., p. 59.
S. Wolfram, Mathematica ŽAddison-Wesley, New York,
1991., p. 576.
M. A. Nunez, Int. J. Quantum Chem. 50, 113 Ž1994..
VOL. 61, NO. 5
Agacik Zafer