the Schrodinger Equation with Nonseparable Potentials

Eigenvalues of the Two-Dimensional Schrodinger Equation with Nonseparable Potentials

H. TASELI" AND R. EID Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Received June 6, 1995; accepted August 9, 1995

ABSTRACT m The energy eigenvalues of coupled oscillators in two dimensions with quartic and sextic couplings have been calculated to a high accuracy. For this purpose, unbounded domain of the wave function has been truncated and various combination of trigonometric functions are employed as the basis sets in a Rayleigh-Ritz variational method. The method is applicable to the multiwell oscillators as well. 0 1996 John Wiley & Sons, Inc.

1. Introduction The particular example which we consider in

this study is that of the Schrodinger equation with the potential

e examine the two-dimensional Schrodi- W nger equation H q = E q in Cartesian coor- dinates with a Hamiltonian of the perturbed oscillator form

M i / . \

M = 1,2, ... (1.3)

where the wave function Y'( x , y ) usually satisfies the condition

lim q( r ) = 0, r = (x, y ) , x, y E (-a,m).

(1 2) llrll+ m

*To whom correspondence should be addressed.

International Journal of Quantum Chemistry, Vol. 59, 183-201 (1996) 0 1996 John Wiley & Sons, Inc.

in which the coupling constants q i and will be freely chosen so that, in general, we deal with a nonseparable problem. A separable equation, namely, a circularly symmetric oscillator problem is under discussion when

ai-, , j = ai; i = 1,2 ,..., M , j = 0,1, ..., i. (1.4)

CCC 0020-7608 I 96 I0301 83-1 9

TASELI AND EID

Numerical evaluations will be carried out for M = 1,2, and 3. For M = 1, the potential (1.3) reduces to the trivial case of the harmonic oscillator, where

which admits exact solution in the form of expc- nentially weighted Hermite’s polynomials. We take into account this case only for testing the accuracy of our method. The case of M = 2, in which

corresponds to the quartic oscillator that has been studied from different points of view [l-81. The main interest is in the case of investigating bound states where the system parameters in pure quartic coupling satisfy the inequalities

If, in addition, the harmonicity constants are all positive, we have a single-well potential with only a minimum located at the origin. However, negative values of v2, v2 < 0, enable us to consider two-well quartic oscillators (Fig. 1). Numerical results for a typical example of such oscillators are also included in this work. As far as we know, eigenvalues of two-well potentials in two dimensions have not been reported previously.

X

FIGURE 1. A two-well oscillator in two dimensions.

The sextic oscillator, where

(1.8)

has received less attention in two dimensions. A nonnegative sextic anharmonicity is at hand if we assume that

making the full potential bounded below. More- over, the potential function V3( x, y) with nonnegative harmonic and quartic terms has an obvious single minimum at the origin, and otherwise is positive with no extrema. On the other hand, it may be shown, analogues to the sextic oscillators in one dimension [9], that the same potential with a strictly negative quartic coupling, v, < 0, pos- sesses three minima provided that vi > 3v2v6. The investigation of eigenvalue problems of this kind is, however, left to a future study.

Energy eigenvalues of the aforementioned non- trivial systems are determined by using a two- dimensional Rayleigh-Ritz variational method in which the function to be determined depends upon two independent variables. The crucial point of the method lies in the consideration of a truncated domain of the independent variables x and y such that

and the modification of the usual boundary condition given in (1.2). Therefore, we encounter the mathematical problem which consists of finding the solution of H 9 = ElIr subject to Dirichlet boundary conditions

for all values of x and y on the surfaces bounding the finite rectangular region being considered.

The motivation stems from the success of the similar approximation for solving one-dimensional

184 VOL. 59, NO. 3

NONSEPARABLE POTENTIALS IN TWO DIMENSIONS

E [ En, ] =

Schrodinger equation [9,101. Indeed, the accuracy of the simple technique presented by TaSeli [9] was very impressive in one dimension and sug- gests evidently the introduction of trigonometric basis in two dimensions as well. Hence we utilize the exact solutions of the unperturbed Schrodinger equation of the form

El, Ell El2 El3

E20 E21 E 2 2 E23 , (2.7) E30 E31 E32 E33

-V2W = AW (1.12)

satisfying the boundary conditions in (l.ll), where V2 is the well-known Laplace’s operator.

In other words, the eigenvalue problem in which the potential is in the form of a rectangular box with impenetrable walls has been considered. In this way, we derive four different enumerable infinite sets of eigensolutions. Such solutions in terms of circular functions and the variational for- mulations are given in Section 2. In Section 3, we introduce the numerical applications. The rest of the work contains some analytical results, to make the numerical procedure plausible, and the discussion of the energy levels crossings with further concluding remarks.

2. Trigonometric Basis

Separating the variables x and y, it is a simple matter to construct the eigenfunctions and the eigenvalues of the boundary value problem defined in (1.12) and (1.11). Actually, we obtain, for k , I = O,l,. . . , the following four normalized se- quences of orthogonal eigenfunctions:

xcos I + - -y , [i :I; I

xsin ( I + 1)-y , [ p ” 1 2 r 2 + ( I + 1) 2, (2.2)

P

xcos[(i + ijfy],

and

2 r2 2 A,, = ( k + 1) - + ( I + 1) 7, (2.4) a2 P

where the A,, are the corresponding eigenvalues. Thus we have

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 185

TASELI AND EID

it is then possible to split [ E n , ] into four matrices

(2.8)

owing to the reflection symmetry of the wave function

w x , y) = W - X , y) = w x , -y) = ? P ( - x , -y) (2.9)

being considered in this study. According to the above decomposition of the spectrum, we conjec- ture that sets 0- @ can be used separately to determine the four blocks of eigenvalues in (2.8), respectively.

Now the wave function is postulated to be of the form

r a m

where fkl are the linear combination coefficients to be determined. We assume that the double-infinite series on the right-hand side converges uniformly to the sum W x , y). If we substitute W x , y) into the Schrodinger equation H? = E 9 , we may multiply both sides by +,Jx, y) and integrate term by term with respect to x and y over their intervals to obtain the relation

which, on evaluating the double integrals, yields the system of algebraic equations

m m

m, n = 0,1, ..., (2.12)

where Elklrnn is defined by

+ < [ ( k + F) 2

01

In this definition RP) stands for the definite inte- gral

1 7 T RV) = ; x2'cos kxdx, (2.14)

which, after integration by parts 2r times, is ex- pressible explicitly as

1 r - 1 ( - & ) j (2.15) ' (2r - 2i - i = O

for k > 0 and r = 1,2,. . ., M . In particular, we find that

Alternatively, R r ) may be evaluated recursively by the relation

k2R'," = 2r(-l)k7r2r-2 - 2r(2r - l)RC,'-'),

r = 1,2 ,..., M (2.17)

for any fixed k > 0, with the initial condition that R(kO) = 0.

On the other hand, the integer parameters sl, s2, pl, and p 2 in Hklrnn are either zero or T1,

186 VOL.59, NO. 3


depending on the basis set under consideration. We must take

altered to the standard matrix eigenvalue problem of the form

s1 = s2 = 1, p1 = p 2 = 0, (2.18) N 2

C ( A,, - E6i j )g j = 0, j = 1

i = 1,2, . . . , N 2 s1 = 1, s2 = -1 , p1 = 0, p 2 = 1,

(2.26) (2.19)

s1 = -1, s2 = 1, p1 = 1, p 2 = 0 , to determine numerically the energy spectrum of the Schrodinger equation using available routines such as TRED2 and TQL2 [ll].

(2*20)

and

s1 = s2 = -1, p1 = p 2 = 1, (2.21)

for the sets 0, 0, 0, and @, respectively. Notice also the block symmetry of H k l m n

H k l m n = H m n k / (2.22)

in the first two and the last two indices. At the computational side of this work, we

consider a truncated wave function so that the resulting infinite system of algebraic equations in (2.12) is replaced by the finite system

N-1 N-1

C C ( H k l m n - E'km' /n) fk / = 0, k = O l = O

m , n = 0 , 1 , ..., N - 1 (2.23)

of order N 2 , where N is the size of truncation. It is interesting to note that if we now introduce an integer transformation T, T: N i + N2 defined by

T = {(i, j ) E N2: i = mN + n + 1 and

j = kN + 1 + 1 V(k, I , m, n ) E Ni) (2.24)

Hklmn and 6,,S,, are reduced to Ai j and S,,, respectively, where N = {1,2,. . .} is the set of natural numbers and No = N U {O). Here the matrix [ Aii] of order N2 may be called the reduced variational matrix, and [ a i j l is the identity matrix of the same order since the transformed indices i and j differ from 1 to N 2 for k, I , m, n =

0,1,. . . , N - 1. The block symmetry of Hklmn implies immediately the symmetry [ A i j ] = [ A j i ] of the reduced variational matrix.

Similarly, the transformation S, S: No X No + N

S = { j e N : j = k N + I + l V ( k , l ) ~ N , x N , } (2.25)

3. Application

The present method is applied to calculate bound-state energies of anharmonic oscillators for a wide range of coupling constants. We first consider the harmonic oscillator (1.5) whose exact eigenvalues are known as

E = E n , = &[(2n + l)& + (21 + 1)&] (3.1)

in the unbounded domain of x and y. The problem is interesting from the viewpoint of testing our approximation for each basis set. Results accu- rate to 20 digits are given in Table I for a,, = aol =

v2 = 1. In the numerical tables, n and 1 show the quan-

tum numbers of the state and N stands for the truncation size which is sufficient to reach the desired accuracy. In each case, we report eigenvalues to 20 digits and set p = a. Therefore, the sole boundary parameter a, for which the presented accuracy is achieved, is defined as the critical value a,, similar to that of the one-dimensional Schrodinger equation [9].

It should be noted that a systematic investigation of the potentials considered here requires too many numerical tables since the number of coupling parameters is rather large with infinitely many combinations. So we present extensive data only for potentials with the interchange symmetry

V(X, y ) = V(y, X I , (3.2)

which explains why we solve the problem on a square domain putting p = a. For such an oscillator the coefficients in (1.8) satisfy the relations

allJ = a O l , 4.0 = a21 = a 1 2 r a30 = a 0 3 ~ transforms fkl with k, 2 = 0,1,. . ., N - 1 into g, with j = 1,2,. . . , N 2 . Hence the system in (2.23) is (3.3)

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 87

TASELI AND EID

TABLE I Eigenvalues of the harmonic oscillator, V ( x , y) = x2 + y2.

Basis set aCr k I N

7.5 0 0 17 7.5 0 2 18 7.5 2 0 18 7.5 0 4 18 8.0 2 2 20 8.0 4 0 20 8.0 0 6 20 8.0 2 4 20 8.5 4 2 22 8.5 6 0 22

7.5 0 1 17 7.5 0 3 18 8.0 2 1 19 8.0 0 5 19 8.0 2 3 20 8.5 4 1 21

7.5 1 0 17 7.5 1 2 18 8.0 3 0 19 8.0 1 4 19 8.0 3 2 20 8.5 5 0 21

7.5 1 1 17 8.0 1 3 19 8.0 3 1 19 8.0 1 5 19 8.5 3 3 21 8.5 5 1 21

2.0000000000000000000 6.0000000000000000000 6.0000000000000000000

10.000000000000000000 10.000000000000000000 10.000000000000000000 14.000000000000000000 14.000000000000000000 14.000000000000000000 14.000 000 000 000 000 000

4.0000000000000000000 8.000 000 000 000 000 0000 8.0000000000000000000

12.000000000000000000 12.000000000000000000 12.000 000 000 000 000 000

4.0000000000000000000 8.0000000000000000000 8.0000000000000000000

12.000000000000000000 12.000000000000000000 12.000 000 000 000 000 000

6.0000000000000000000 10.000000000000000000 10.000000000000000000 14.000000000000000000 14.000000000000000000 14.000000000000000000

and the corresponding potential, on making a simple scaling transformation, is equivalent to

V ( x, y ) = x2 + y2 + c4( x4 + 2ax2y2 + y4>

+ c6(x6 + 3bx4y2 + 3bx2y4 + y6), (3.4)

where a, b, c4, and c6 are the new system parameters. In the case of the quartic coupling, when c6 = 0, we see from (1.7) that

c4 > 0, -1 I a 5 1. (3.5)

Numerical results are included in Tables II-VI for the values of c4 = 1, lo3, lo6, and 00 corresponding to each a = - l , O , and 1, covering the

range of a. Notice that c4 -, 00 limit represents the pure quartic oscillator problem

( - V 2 + x 4 + 2ax2y2 + y 4 ) q = 89, (3.6)

where E = c:138. For this reason eigenvalues for cq > 1 have been scaled by cq1I3 to infer how rapidly the c4 -, m limit values of energies are achieved as c4 increases.

Tables VII-X are devoted to the more general problem of the sextic oscillators in two dimensions by taking into account the potential in (3.4) with c4 = 0. It may be shown that the conditions in (1.9) now imply the relations

c6 > 0, (3b + l ) (b - l I3 I 0 (3.7)

so that the parameter b varies between - I b I 1. Thus the entire range of b is covered by setting b = - i , O , i, 5, and 1 while c6 changes from to m. Here, c6 -, 00 limit Hamiltonian corresponds to the pure sextic oscillator

( - V 2 + x6 + 3bx4y2 + 3bx2y4 + y6)'P = 23' (3.8)

with E = c;I42F". As in the case of the quartic coupling eigenvalues in the numerical tables for c6 > 1 are those scaled by ct114.

4. Discussion

In this work, the spectra of unbounded oscillators in two dimensions are obtained by way of increasing systematically the boundary value a, regarding it as a nonlinear optimization parameter. It was shown numerically that there exists a critical value acr at which the low-lying state energies are equal to those of a -+ 00 limit to a prescribed accuracy. In order to justify the procedure analyti- cally, we first prove the domain monotonicity of the eigenvalues.

Let us reconsider the Schrodinger equation

[ - V 2 + V ( X , y) - E ] ~ ( x , y) = 0,

X , y E l. -a, aI (4.1)

in which the potential, as does the wave function, satisfies both reflection and interchange symmetries with the accompanying Dirichlet boundary conditions

188 VOL.59, NO. 3


TABLE II Eigenvalues of the quartic oscillator for c, = 10 -3 as a function of a.

a %r N n I E", Basis set

- 1 7.0

7.5

7.5

8.0

8.0

7.5

8.0

8.0

7.5

7.5

0 7.0

7.5

7.5

7.5

7.5

8.0

7.5

8.0

8.0

8.0

1 7.0

7.5

7.5

7.5

8.0

8.0

8.0

8.0

8.0

8.0

17

18

17

19

19

18

19

20

18

19

17

17

17

18

18

19

18

19

20

19

17

17

18

18

19

19

20

20

20

20

2.000 998 505 469 810 4735

4.002 992 541 671 302 8329

6.0029940290428079650

6.006 970 242 132 821 7062

6.008 968 736 025 175 0823

8.0056893858679641592

8.01 6 209 589 31 0 761 3409

10.006398993925805298

10.008948049206928896

10.01 4 940 522 380 068 574

2.001 497 385 346 371 3991

4.004 488 440 841 91 4 81 60

6.0074794963374582330

6.01 0 460 565 461 293 1866

8.013 451 620 956 836 6035

8.019 401 284 730 701 9984

10.01 9 423 745 576 21 4 974

10.022 392 340 226 245 41 5

10.031 298 258 747 896 51 5

12.028364464845623786

2.001 995 522 094 708 5337

4.005 980 633 720 156 0486

6.01 1 949 449 045 707 0830

6.01 3 936 098 189 653 0736

8.01 9 896 128 373 71 6 731 4

8.023 860 639 292 485 4844

10.029 81 4 877 549 869 265

10.035 748 547 202 91 2 722

10.037726447540990997

12.041 699947383956422


TASELI AND EID

TABLE 111 Eigenvalues of the quartic oscillator for c, = 1 as a function of a.

a %r N n I E", Basis set

- 1 5.5

5.5

5.5

5.5

5.5

5.5

5.5

5.5

5.5

6.0

0 4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

1 4.0

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

24

24

24

25

25

25

25

26

25

28

19

20

19

20

20

20

20

20

22

20

20

19

20

20

20

20

21

21

21

21

2.561 626 575 640 031 6393

5.396 803 983 194 131 3950

7.5491560629022652232

8.435 987 322 348 467 4428

9.617 587 764 733 419 3257

10.366020696842355027

12.81 8 706 298 776 658 594

12.886 584 502 841 939 840

13.406 949 149 052 421 685

15.336126803517827432

2.784 703 283 060 583 71 13

6.041 164 345 742 369 3920

9.2976254084241550728

10.047 401 599 289 601 544

13.303 862 661 971 387 225

14.549155539580166935

17.310 099 915 518 619 376

17.805 616 602 261 952 616

19.449909077833544750

21.81 1 853 855 809 184 767

2.952 050 091 962 874 2871

6.4629059998638721377

10.390 627 295 503 782 127

10.882 435 576 81 9 807 244

14.658513813565503097

15.482 771 577 251 666 477

19.217523495888984907

20.293829707535892571

20.661 082 690 597 886 009

24.033 166 193 470 850 31 7

1 90 VOL. 59, NO. 3


TABLE IV Eigenvalues of the quartic oscillator for ca = lo3 as a function of a.

a a c r N n I c, "3E ", Basis set

- 1 2.25

2.30

2.45

2.53

2.55

2.65

2.25

2.70

2.75

2.87

0 1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1 1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

31

34

38

39

41

44

33

45 47

49

21

21

21

21

21 22

21

22

22

22

21

21

22

22

21

21

22

22

22

22

1.768 690 935 077 571 7644

4.886 a37 91 o 059 663 6584 3.796 544 01 0 966 257 7256

5.621 924 276 31 9 793 31 24

6.554 700 407 71 7 1 15 9352 7.624 249 447 422 198 2538

7.9726698208795028150

7.9927690262545928550

8.828 600 108 158 501 4158

9.883 663 1 84 61 1 307 61 08

4.872 662 217 071 031 0149

7.61 7 366 691 876 452 81 70

8.532 119 291 149 285 9325

12.724 298 764 862 1 i 5 750

14.936280840032962652

17.344 21 6 290 830 327 624

19.128 460 313 745 792 470

2.351 338 918 312 985 3963

5.405 485 579 551 943 9394

8.9433434033749367277

9.543 7449804059634223 12.861 961 618 091 473 035

13.828 303 842 944 239 989

17.099 777 828 093 744 127

18.330 633 81 o 778 597 643 I 8.754 903 71 4 202 645 a97 21.61 5 194 758 663 360 719

2.1279577422656092127

1 1.276 823 765 954 707 735

15.469 003 239 667 537 552


TA$ELI AND EID

TABLE V Eigenvalues of the quartic oscillator for c, = lo6 as a function of a.

a %I N n I

- 1 0.70

0.75

0.78

0.80

0.83

0.85

0.86

0.65

0.90

0.92

0 0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

1 0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

31

35

37

40

43

45

45

30

48

49

20

20

20

21

21

21

21

21

21

21

20

20

20

21

20

20

21

21

21

21

c, "3E ", 1.759 289 793 41 1 603 6752

3.776 059 993 838 585 81 27

4.851 584 349 400 640 3507

5.5800598644623280793

6.498 549 188 437 034 8450

7.5453598456056280755

7.91 3 941 507 199 668 3826

7.9550877330791376552

8.7300925248249975341

9.755 601 597 151 036 0650

2.1207965848079286706

4.860 161 482 220 11 5 0495

7.599 526 379 632 301 4283

8.516 220 701 336 840 0245

11.255 585 598 749 026 403

12.705299594230404304

14.91 1 644 81 7 865 751 378

15.444 664 491 642 590 683

17.322 408 471 698 891 729

19.100 723 710 759 315 658

2.3448942200278850682

5.394 339 81 0 182 01 1 0501

8.9282348795923658359

9.529 921 064 696 036 0654

12.843 322 871 678 291 494

13.81 1 281 529 191 623 350

17.077 902 477 951 329 145

18.31 0 477 090 468 21 5 166

18.735 392 632 61 9 681 245

21.590299679556297524

Basis set

192 VOL. 59, NO. 3


TABLE VI Eigenvalues of the quartic oscillator for ca + ~0 as a function of a.

a %r N

- 1 7.0

7.5

7.7

7.9

8.3

8.5

8.7

7.0

9.0

9.1

0 4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

1 4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

4.5

32

35

38

39

43

45

46

32

48

49

20

20

20

21

21

21

21

20

21

21

20

20

20

21

20

20

21

21

21

21

I c, "3E ", Basis set

1.759 194 606 175 532 8805

3.775 852 192 553 403 3973

4.851 226 439 802 121 0076

5.579 633 81 2 123 752 71 02

6.497 977 435 474 221 7622

7.544 555 91 0 429 662 3294

7.91 3 136 442 782 998 9634

7.954 910 003 220 516 0493

8.7290858666290958645

9.754 292 581 190 71 1 0350

2.120 724 180 968 365 7993

4.860 035 120 285 577 0684

7.5993460596027883376

8.51 6 060 028 470 921 291 8

1 1.255 370 967 788 132 561

12.705 107 601 862 344 921

14.91 1 395 875 973 476 784

15.444 41 8 541 179 556 190

17.322 188 109 334 408 838

19.100 443 449 364 900 413

2.3448290727442752098

5.394 227 164 172 288 0358

8.928082199849951 1796

9.529 781 384 014 807 9598

12.843 134 529 795 971 61 0

13.81 1 109 536 873 734 556

17.077 681 440 978 31 9 570

18.31 0 273 432 403 605 41 2

18.735 195 504 701 770 774

21.590 048 138 799 549 057


TA!$ELI AND EID

TABLE VII Eigenvalues of the sextic oscillator for c, = 10 - 4 as a function of b.

b %r N n I E”, Basis set

0

1 3 -

2 3 -

1

1 3 7.5 _ -

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

7.5

17

17

17

18

18

18

17

17

17

18

18

18

17

17

17

18

18

18

17

17

17

18

18

18

17

17

18

18

18

18

2.0002995843269869073

4.001 196 882 188 188 1348

6.001 496 387 585 537 4414

6.003 880 744 71 4 01 2 391 2

6.004 479 567 062 021 7761

8.0035190431683775975

2.000 374 456 307 361 5366

4.001 496 071 783 281 0674

6.002 61 7 687 259 200 5982

6.004 851 939 453 658 0155

8.005 973 554 929 577 5463

8.01 1 907 751 874 106 071 1

2.0004492764891242568

4.001 794 873 150 229 1097

6.003736442045601 1795

6.0052238677284844739

6.0058207367287196935

8.0082557751325872247

2.000 524 045 041 502 4289

4.002 093 288 336 146 7761

6.004 852 673 034 787 5965

6.0055953541496350391

6.006 787 156 944 01 2 7888

8.01 0 257 378 793 795 41 49

2.000 598 762 132 326 8823

4.002 391 31 9 363 885 3068

6.005 966 400 953 823 8332

6.007 751 220 149 997 8043

8.01 1 903 984 229 91 3 9729

8.016 641 198 747 866 0570

194 VOL. 59, -NO. 3


TABLE Vlll Eigenvalues of the sextic oscillator for c6 = 1 as a function of b.

b %r N n i E", Basis set -

1 3 4.25 _ _

4.25

4.25

4.25

4.25

4.25

0 3.3

3.3

3.3

3.3

3.3

3.3

1 - 3 3.3

3.3

3.3

3.3

3.3

3.3

2 ~

3 3.3

3.3

3.3

3.3

3.3

3.3

1 3.3

3.3

3.3

3.3

3.3

3.3

32

32

32

32

32

33

23

23

22

23

23

23

23

22

22

23

23

23

22

23

22

22

23

23

23

22

23

23

23

23

2.737 044 026 929 101 9539

6.033 484 650 760 71 6 1587

8.747 981 386 479 224 8633

10.086324578132956927

11.162953553999709351

12.551 942 789 464 750 847

2.871 249 238 006 784 6315

6.469 020 556 723 658 7926

10.066 791 875 440 532 954

1 1.402 246 61 8 721 502 597

15.000 01 7 937 438 376 758

17.425 065 406 829 123 373

2.969 508 952 31 1 923 9630

6.7469883327354985277

10.846142578598203954

11.602 009 41 1 398 876 921

12.063 494 193 071 301 714

16.121 361 61 4 430 563 557

3.0509955009759625904

6.965 532 860 574 152 1337

1 1.442 609 682 526 850 828

11.777 971 944 480 521 001

12.537 333 598 045 554 51 9

16.856 072 965 577 102 558

3.121 935 474 246 425 991 1

7.149 928 601 438 550 745

11.937 202 695 862 041 726

12.91 4 938 793 084 835 744

17.387 207 807 460 190 623

19.186717422068084624


TASELl AND EID

TABLE IX Eigenvalues of the sextic oscillator for c6 = lo4 as a function of b.

b %-r N n I c , "4E,,

1 3 1.375 _ _

1.425

1.425

1.425

1.425

1.425

0 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o

1 - 3 1 .o

1 .o 1 .o 1 .o 1 .o 1 .o

2 - 3 1 .o

1 .o 1 .o 1 .o 1 .o 1 .o

1 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o

32

35

36

36

36

36

22

22

22

22

22

22

22

22

21

22

22

22

22

22

21

22

23

22

22

22

22

22

23

23

2.0975808799120669993

4.870 540 61 9 061 649 71 94

6.931 646 924 801 588 1548

8.335 31 6 355 492 296 4653

9.9477224374698557932

10.21 7 708 598 378 632 420

2.2957596084529087923

5.493 658 278 494 498 7590

8.691 556 948 536 088 7258

10.230007695397338966

13.427906365438928933

16.093 676 835 860 058 990

2.4248503751522079053

5.836 1 12 695 396 163 4754

9.6045287820752772240

10.458 234 530 760 772 432

1 1.01 0 253 362 71 9 965 524

14.683 406 877 259 461 638

2.527 522 134 914 928 3046

6.0942950372799287802

10.281 500 502 1 15 144 972

10.656 109 793 305 061 175

11.542 886 291 648 062 198

15.486804503684603506

2.61 4 732 045 81 1 295 3213

6.307 243 065 747 81 1 2548

10.833360928259358595

11.956 628 506 631 557 145

16.064420709675920747

18.054 166 693 936 036 202

Basis set

196 VOL. 59, NO. 3


TABLE X Eigenvalues of the sextic oscillator for c6 as a function of b.

b %r N n I c6- "4E,, Basis set

1 3 4.4 _ _

4.5

4.6

4.6

4.3

4.6

0 3.2

3.2

3.2

3.2

3.2

3.2

1 - 3 3.2

3.2

3.2

3.2

3.2

3.2

2 - 3 3.2

3.2

3.2

3.2

3.2

3.2

1 3.2

3.2

3.2

3.2

3.2

3.2

35

35

37

39

32

38

22

22

23

22

22

22

22

22

22

22

22

23

21

22

22

22

22

23

22

22

22

23

22

23

2.090 555 813 41 1 345 8286

4.8577428470977704618

6.911 638 107392863 1624

8.315 353 833 801 998 91 29

9.935 11 1 771 797 729 51 00

10.191 755 372 544 930 578

2.2896049075941055275

5.483 401 165 31 1 033 9554

8.6771974230279623833

10.21 7 887 01 4 71 8 486 620

13.41 1 683 272 435 41 5 048

16.079 972 088 707 788 780

2.41 9 069 938 706 81 1 2236

5.826 605 081 974 832 9375

9.591 659 11 0 784 036 9904

10.446 427 342 171 923 296

10.999 474 090 184 501 288

14.668 616 182 761 143 176

2.521 990 692 361 91 6 9774

6.085 232 707 418 081 6496

10.269 508 149 945 561 882

10.644 540 491 467 883 697

11.532 741 742 823 300 098

15.472 738 954 447 459 989

2.609 388 463 253 714 0069

6.298 495 901 483 604 2435

10.821 985 609 888 21 0 291

11.946 863 508 851 368 705

16.050 846 880 370 264 51 4

18.042 634 963 21 5 149 285


TASELI AND EID

for all y and

W ( x , - a ) = 0 , W x , a ) = 0 (4.3)

for all x . It is obvious that the eigenvalues and the eigenfunctions in (4.1) depend on the boundary parameter a. Therefore, we may denote

W = W( x , y, a ) , E = € ( a ) . (4.4)

Now, differentiating (4.1) with respect to a, multi- plying by W and integrating the result over the range of the independent variables, we obtain

d E d dW d a

T(x, y, a ) dxdy . (4.5) d a

We may assume, without any loss of generality, that the wave function is normalized, (W, W) = 1, so that

d a

dE da - = ([ - V 2 + V ( x , y) -

where the derivative operators with respect to a and the Laplace operator in (4.5) have been inter- changed. On integrating by parts, the right-hand side of (4.6) can be put into a form

( L 2, W) = boundary terms + (g, L * Y )

(4.7)

wherein the last inner product vanishes from (4.1) since the operator L = -0' + V ( x , y) - E is for- mally self-adjoint, L* = L. Hence, writing explicitly the boundary terms we arrive at the relation

d E dW d Y - ."jla dy

dW d Y - ?" ) la dx. (4.8)

da d x aa d x d a X = - a

d y d a Y = - a

To derive a more neat expression for d E / d a , we consider the total differential of W. From (4.41, we have

dW dW dW d X JY d a

d Y = -dx + -dy + -da. (4.9)

If x is a function of a, x = x ( a ) say, then dx =

( d x / d a ) d a so that

where the meaning of d W / d a + [ ( d x / d a ) ( d Y / d x ) ] is surely that of the partial derivative of W with respect to a when y is kept constant, dy = 0. Therefore, for x = d a ) ,

dW dW dW dx - + --. (4.11)

d a a a d x d a

Now, differentiation of the conditions given in (4.2) with respect to a by taking x = -a and x = a, respectively, it follows that

Y a ( - a , y) - W x ( - a , y) = 0, W a ( a , y) + Wx(a, y) = 0, (4.12)

where the subscripts denote partial derivatives. Likewise for y = y ( a ) , from (4.9) and (4.3), we have

W a ( x , -01) - T y ( x , - a ) = 0 ,

Wa( x , a ) + WY( x , a ) = 0. (4.13)

Substituting (4.12) and (4.13) into (4.8) and using the symmetries of the wave function, we see that (4.8) can be written in the form

(4.14)

It is clear that d E / d a is strictly negative which implies that E( a ) decreases monotonically to its limit E(m) as a + 00. In other words, the larger region has smaller eigenvalues,

E(a) 2 E(m). (4.15)

This means that E(a)'s are upper bounds to the asymptotic eigenvalues; a property which is well known for the corresponding one-dimensional problems 19,121.

As a consequence of (4.14) and (4.151, the differ- ence IEJa,) - En[(a,)l = E , , for any quantum state ( n , I ) measures the inaccuracy of the results. In our calculations we estimate acr values in such a way that E , ! is less than 10-*", and therefore present eigenvalues to 20 significant figures.

198 VOL. 59, NO. 3


On the other hand, for any a, fixed, the relations

EII/(a, , N ) > E n l ( a , , N + 11, E J a , ) = lim Enl(a,, N ) (4.16)

are known from the variational principle. There- fore, stable digits of eigenvalues conforming between two consecutive values of N , imply the accuracy of En,(az). In the tables, the recorded truncation orders N are those for which

N-m

I E n [ ( & , , N + 1) - E n l ( ~ , , N)I < lop2’ (4.17)

showing that the Enl(a,)’s are correct up to 20 digits.

Set @ and set @ are used in evaluating the energy levels with the same parity, namely, both even or both odd. The eigenvalues, however, having different parity, one even and one odd, may be calculated by either set @ or set @ since they are doubly degenerate for all oscillators considered numerically in this work.

In Table I, we see that the exact eigenstates E n , = 2(n + 1 + 1) of the harmonic oscillator, V(x, y) = x2 + y2, are reproduced. This problem might have been treated in cylindrical polar coor- dinates by the separation of variables. In general, the separation process may cause a loss of some solutions. Indeed, solving the radial Schrodinger equation we could find the exact eigenvalues to be 2(2n + I + 1) [13]. Here, E2,, , is determined rather than En,, and hence the solutions corresponding to M odd are lost in the separation process. A similar situation occurs for the quartic and the sextic oscillators defined in (3.4) when a = 1 and b = 1, respectively, for which the systems have a special circular symmetry. For this reason, the eigenvalues of the circularly symmetric quartic oscillators characterized by En, in [3] and [13] should be com- pared with those of E2n,l in this study.

The structure of the present trigonometric basis sets provides a very natural and a simple way to order the eigenvalues according to the quantum numbers. If the energy levels E n , are characterized as groups denoted by the number rn = n + I , it is then easier to understand their certain ordering properties. In the case of the circularly symmetric oscillators we deduce, from Tables 11-VI for a = 1 and from Tables VII-X for b = 1, that if rn is even

- E 0 , m = E 1 , m - 1 < E 2 , m - z - E s , m - 3 < * * . < E m - 2 , 2

- - L 1 , l < E m , 0 (4.18)

while if m is odd

the numbers of degeneracies being 2,2, . . . , 2,l and 2,2, . . . , 2, respectively.

As another special case we have two uncoupled quartic anharmonic oscillators when a = 0. Simi- larly, for b = 0, the system becomes one of two independent sextic oscillators. For these cases, it is clear that

E n , = En + E, , M, I = 0,1,. . . (4.20)

where the Ek‘s are the kth eigenvalues of the corresponding one-dimensional problems. Thus the accuracy of the present results may be confirmed by means of (4.20) on recalling, especially, the numerical results of Banejee [14] and TaSeli and Demiralp [15] given in one dimension. The equation (4.20) implies also that En, = El , which can be seen immediately from our tables. Furthermore, the group characterized by rn has the energy levels in ascending order of magnitude for rn even, 2k say,

and for rn odd, 2k + 1 say,

- E k , k + l = E k + l , k < E k - l , k + 2 - E k + 2 , k - l < ’.’

- < E 1 , 2 k = E 2 k , / < E 0 , 2 k + l - E 2 k + 1 , 0 (4.22)

with the degeneracies equal, respectively, to 1,2,2,. . . ,2 and 2,2,. . . ,2.

The eigenvalues of the sextic oscillator in the group 2k + 1 remain doubly degenerate and un- split as b varies from - to 1. For the group 2k, however, the doubly degenerate levels at b = 0 and b = 1 split into two levels such that

E k , k < E k - l , k + l < E k + l , k - l < E k - 2 , k + 2 < ’ * *

< ‘ 1 , 2 k - - l < ‘ 2 k - 1 , l < ‘0,2k < ‘2k,0 (4.23)

when - i s b < 0 and 0 < b < 1. Although the numerical results are not quoted here, it seems that the eigenvalues of the quartic oscillators with -1 < a < 0 and 0 < a < 1 show the same trend.

As another important feature of energy levels crossings, we examine the ordering of eigenvalues


TA$ELI AND EID

belonging to two different groups. Let rn and rn, denote two groups such that rn = n + 1 and rn, =

n, + I,. If rn > rn, then we infer that

rn = 0,1, ..., M , (4.24) E n , > E n l , l l *

where M is a finite number representing the group M . It is noteworthy that the number M , up to which eigenvalues satisfy (4.241, depends mainly on the potential function being considered. Obvi- ously, M tends to infinity only for the exceptional case of harmonic oscillators. Hence in the near harmonic regime of eigenvalues, we observe that M is indeed a very large number. It is, however, considerably small in the boundary layer and in the pure anharmonic regime. More specifically, as can be seen from Table I11 for the quartic coupling with a = - 1 and c4 = 1, €,,, in group rn = 4 is less than Eo,3 in group rn, = 3 in spite of rn > rn,. Therefore, M = 3 for this potential. We notice, from Table VI for a = -1, that M becomes 2 as c4 + m since E,, , < E,,,. In general, the number M is more or less the same whenever the coupling constant c4 2 1. In the case of uncoupled quartic oscillator, a = 0, the eigenvalues may be ordered

according to (4.24) for rn = 0,1,. . . ,7 since E4,, < E , , which implies that M = 7. For the circularly symmetric oscillators with c4 2 1, by noting that inequality (4.24) is invalid for the first time at rn = 11 where E,,,, < E,,,, we find that M = 10.

Eigenvalues of the sextic oscillators behave very similarly. For c6 2 1, it is concluded that M is at least 3 and at most 7 as b varies from - 5 to 1. We believe that this interesting structure of the energy spectrum of a coupled system is discussed for the first time.

Numerical computations are performed using quadruple precision arithmetic on an IBM AIX computer system. Over a wide range of the coupling constants, we see that a truncation size of N about 20 is sufficient in determining eigenvalues to 20 significant digits, which consumes approxi- mately 4 CPU minutes. Only the cases of a = -1 and b = - 3 require to take higher truncation orders. This is related to the fact that the rate of convergence of our algorithm is relatively slower for the corresponding potentials.

The comments, given in [9], on the critical values of OL are completely representative in the two- dimensional case as well. Therefore, we do not

TABLE XI Eigenvalues of V ( x , y ) = - x 2 - y 2 + 0.1(x2 + y2I2.

En, -k 2.5 Basis set %T N n I

6.75

6.75

6.75

6.75

6.75

6.75

6.75

6.75

6.75 7.0

7.0

6.75

7.0

7.0

7.0

7.0

21

21

21

22

22

23

22

22

23 24

24

23

24

24

24

25

1.203 907 303 509 51 3 0486

1.61 0 523 081 475 256 2891

2.4024626758420828938

3.462 572 859 541 054 4225

3.5026729607879520038

4.635 103 020 741 069 8277

4.732 541 548 039 006 91 48

5.958 878 61 2 454 858 8725

6.1 76 997 91 9 702 665 3580 6.480 41 9 127 324 994 7378

7.449 056 91 1 248 673 8682

7.771 909 558 805 891 421 5

8.1974958604691072858

9.084 736 197 454 059 5765

9.973 273 992 548 445 0051

10.298 132 589 625 51 7 573

200 VOL. 59, NO. 3


duplicate here the fairly detailed discussion of [9] about aCr.

As an attempt of applying the method to an eigenvalue problem of a different nature we finally consider two-well quartic oscillators where

V ( X , y) = - x 2 - y2 + c4(x4 + 2ux2y2 + y4). (4.25)

Numerical results of a particular example, c4 =

0.1 and u = 1, are given in Table XI, which encour- age us to employ trigonometric basis sets in solving more complex systems having both symmetri- cal and unsymmetrical potentials with one or more than one minima.

References

1. T. Banks, C. M. Bender, and T. T. Wu, Phys. Rev. D 8,3346 (1973).

2. I. C. Percival and N. Pomphrey, Mol. Phys. 31, 97 (1976). 3. F. T. Hioe, D. MacMillen, and E. W. Montroll, Phys. Rep. 43,

4. R. A. Pullen and A. R. Edmonds, J. Phys. A: Math. Gen. 14,

5. N. Ari and M. Demiralp, J, Math. Phys. 26, 1179 (1985). 6. F. M. Fernandez, A. M. Meson, and E. A. Castro, Phys. Lett,

7. J. Killingbeck and M. N. Jones, J. Phys. A: Math. Gen. 19,

8. E. R. Vrscay and C. R. Handy, J. Phys. A: Math. Gen. 22,823

9. H. Taqeli, Int. J. Quantum Chem. 46, 319 (1993).

305 (1978).

L477 (1981).

112A, 107 (1985).

705 (1986).

(1989).

10. H. Taqeli, J. Comput. Phys. 101, 252 (1992). 11. H. W. Press, B. P. Flannery, S. A. Teukolsky, and W. T.

Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 19861, p. 335.

12. F. M. Fernandez and E. A. Castro, Int. J. Quantum Chem. 19, 533 (1981).

13. H. Taqeli, Int. J. Quantum Chem. 57, 63 (1996). 14. K. Banejee, Proc. R. Soc. A 364, 265 (1978). 15. H. Taqeli and M. Demiralp, J. Phys. A Math. Gen. 21, 3903

(1988).


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the Schrodinger Equation with Nonseparable Potentials