the schrodinger equation with nonseparable potentials
TRANSCRIPT
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 oscil- lator 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 oscilla- tor, 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, nega- tive values of v2, v2 < 0, enable us to consider two-well quartic oscillators (Fig. 1). Numerical re- sults 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 dimen- sions 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 nonnega- tive 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 condi- tion 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 discus- sion 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 de- fined 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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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 nat- ural 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 im- plies 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 con- sider 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 prob- lem 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 eigenval- ues 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 investiga- tion of the potentials considered here requires too many numerical tables since the number of cou- pling 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 oscilla- tor 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 sim- ple 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 parame- ters. 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 corre- sponding 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 oscilla- tors 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 criti- cal 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 symme- tries with the accompanying Dirichlet boundary conditions
188 VOL.59, NO. 3
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 89
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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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
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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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 193
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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 95
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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 197
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 explic- itly 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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
On the other hand, for any a, fixed, the rela- tions
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 be- tween 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, hav- ing 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 oscil- lators defined in (3.4) when a = 1 and b = 1, re- spectively, for which the systems have a special circular symmetry. For this reason, the eigenvalues of the circularly symmetric quartic oscillators char- acterized 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 equa- tion (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
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 99
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 cou- pling 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 val- ues 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
NONSEPARABLE POTENTIALS IN TWO DIMENSIONS
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 solv- ing more complex systems having both symmetri- cal and unsymmetrical potentials with one or more than one minima.
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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 201