ELSEVIER Physica A 216 (1995) 452-458

PHYSICA

Energy eigenvalues for anharmonic and double-well oscillators with even power polynomial potential

Mamta, Vishwamittar*

Department of Physics, Panjab University, Chandigarh-160014, India

Received 19 January 1995

Abstract

Energies for different states of anharmonic and double-well oscillators described by V(x) = ~ = l ( a 2 j / 2 j ) x 2j have been determined using the renormalized hypervirial-Pad6 scheme. A comparison of the results with available exact values shows that the method is quite successful for anharmonic oscillators but fails in the case of double-well oscillators. Also included is the discussion of the dependence of the energy on various parameters.

1. Introduction

The investigation of the characteristics of one-dimensional anharmonic and double-well oscillators (AHOs and DWOs) described by even power polynomial potentials

V ( x ) = m a2j

j= 1 ~J X2j (1)

has drawn commendable attention because of their usefulness in developing models for various phenomena in different branches of physical and biological sciences. One of the frequently studied aspects of these oscillators is the determination of energy eigenvalues and this aim is achieved either by employing some formalism that yields exact analytical results for a few wavefunctions for specific potentials satisfying certain constraints on the values of the coefficients a2j arising from solvability of the equations involved, or by using some perturbation technique that generally suffers from the limitations associated with convergence of the series employed. The main thrust of these studies has been on the potentials for quartic AHOs and DWOs (s -- 2),

* Corresponding author.

0378-4371/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 0 3 2 - 1

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though occasional efforts have been directed to the case of quartic-sextic AHOs and

DWOs (s = 3) (see, e.g. Refs. [1,2] and references therein). The energy eigenvalues for V(x ) with s > 3 have been determined only by few workers [3-7], who have con- sidered the potential with s = 5 in view of its relevance to different models of charmonium system [4]. While Magyari [3] and Flessas [4] obtained relations between the coefficients a2j without carrying out numerical calculations, Adhikari et al. [5], who used factorization method based on the ideas of supersymmetry, also evaluated energies for the potentials describing anharmonic, double-well and triple- well oscillators conforming to the conditions for a2j's derived by them. Furthermore, Chaudhuri and Mondal [6] and Agrawal and Varma [7] employed the modified Hill determinant method to compute energies for oscillators with three-, four- and five- wells. Although Witwit [8] has mentioned that the potential considered was of the form of Eq. (1) with s = 10, he has actually performed finite-difference computations of energy values for AHOs with j = l and only one value of j > 2 at a time, rather than a multiterm polynomial.

No doubt perturbational methods are handicapped due to the difficulty of conver- gence, they do offer an opportunity to study the dependence of energy values on various coefficients in a systematic manner and thus facilitate the development of a suitable model. In view of these facts it was decided to use renormalized hypervirial- Pad6 technique [9-12] to study the AHOs and DWOs described by Eq. (1) and this communication is an outcome of such an effort. First, we develop the formalism for general potential (1), using terms with j ~> 2 as perturbation to the a2( -- to 2) term, and then compute energies for different arbitrary combinations of values of a2~ for anharmonic as well as double-well oscillators with s -- 5 to compare the findings with those of Adhikari et al. [5], and to discuss the dependence of energy on various coefficients. The extent of applicability of the technique to such potentials has also been analyzed. It has been found that the method is completely unsuitable for double-well potentials studied by Adhikari et al. and as such we have not used it for the potentials with more than two wells reported by different authors [5-7].

2. Essentials of the formalism

In this derivation we replace a 2 by _+ co 2 and treat other terms as perturbations; the positive sign of O) 2 leads to an AHO and the negative sign with similar sign for a4 to a DWO. Thus, a2j with j /> 2 become perturbation parameters and the presence of s - 1 independent anharmonicity parameters is bound to create problems in handling the formalism. To overcome such a difficulty and to keep the first-order contribution from the term (aEJ2S)x 2s of the same order as the (s - 1)th-order effect o f ( a 4 / 4 ) x 4, we introduce a perturbation parameter 2 and substitute

a2j = 2 j - 1 o~j 2 (2)

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for j /> 2, with So = 1. Accordingly, the Hamiltonian corresponding to V(x) in Eq. (1) becomes

h 2 d 2 ~-~ ~ J - l o ~ j - 2 H - 2m dx 2 +- ½mt°ZxZ + m z., xZJ. (3)

j = 2 2 j

With a view to implement the renormalized hypervirial-Pad~ scheme, we add and 1 2 subtract ~m2Kx to Eq. (3) and get:

h 2 d 2 H - 2mdx2+½mco'ZxZ+m2(~xg-½KxZ)+m ~ 2J-'°tJ-2 j = 3 2j X 2j, (4)

where

(D '2 = -[- 0) 2 -1 t- 2 K . (5)

Here, K is a variational parameter and in the case of DWOs it has to be such that co '2 is positive.

Using the hypervirial theorems and the required commutation relations for Eq. (4), writing

E, = ~ E~k)2 k, (6) k=O

(x u ) = ~ C~2 t, (7) / = 0

and comparing the terms of the same powers of 2 on the two sides of the equation so obtained, we get:

Here,

and

- - ~ n U p k + - - ' ~ C(pU+2' (N + 5)mco 'z (N + 1) ~(k,,-(m mKC(pN+12) k=O

i N + j + 1 f , ( N + 2 j ) h2 } - m j=2 2j o~j_2L.,p_j+ 1 +-~m(N- 1)N(N + 1)C(p N-2) (8)

E (°) = (n + ½)h~o' (9)

c(O) = fro. (10) P

N The remaining E(~ ) are expressed in terms of C v with the help of Hellmann- Feynman theorem and are given by

E(k, ~ ( 1 ~(2, ~ ( J - - 1 ) c t j _ 2 . ~ ( z j ) ) (11) n = - - g K L k - 1 "+ L - ' k - j+ l •

j = 2 2 j

This equation together with Eq. (6) gives E. as a function of K and C(p m and the latter are found from the recurrence relation (8) in a sequential manner. It turns out that

c ( 2 M + 1) = 0 (12) p

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for p, M = 0, 1, 2, -... Furthermore, the variational parameter K is determined in such

a way that for a particular n-value, the maximum number of digits in the final expression for E, is stable. The L, values corresponding to the K so obtained are employed to determine the Pad6 approximants to energy, E,(P), using the algorithm given in Ref. [11].

3. Calculation of energy values and their discussion

We have carried out numerical calculations for the potential with s = 5 for AHOs and DWOs and have taken h = m = 1. The computations have been executed with double precision and by performing summation in equation (6) up to k = 50. How- ever, the results presented here pertain to the situation for which convergence was the best and the maximum number of digits was stable.

3.1. AHOs

The E,,(P) values for various combinations of air for the AHO with tenth-degree even power potential are listed in Table 1, which also includes the available exact values reported by Adhikari et al. [5] and kindly supplied by an anonymous referee; the last two entries based on Ref. [5] have been recalculated to get more significant figures. The values chosen for a2~ have been guided by the values used in [5], which, in

turn, were obtained by them on the basis of constraints found on a2fs from their analysis.

A study of the contents of Table 1 leads us to the conclusions enumerated below. (1) The energy eigenvalues for different n that can be compared with the exact

values are in excellent agreement with them; this is all-the-more heartening because the present results are highly accurate while the findings of the shifted 1/N method reported in [5] and given in parenthesis under the exact values of [5] in Table 1 show some departure. Furthermore, the preciseness of these four values and the degree of convergence observed in all other cases encourage us to believe that these results too must be correct.

(2) The E,(P) values differ significantly from the corresponding values for harmonic oscillators implying that the anharmonicities are quite large. E,(P) increase with increase in the a2j-value for a particular ~o 2. Similarly, if ~o z is increased or decreased keeping other parameters fixed, the E,(P) increase or decrease; the enhancement is prominent when e~ 2 and all other azfs are doubled.

(3) For a specific potential, K generally increases with increase in the quantum number n. However, the effect on K of change in a2j ( j ~> 2) values keeping ~o 2 the same is marginal. If to 2 and a4 are both halved without changing other parameters in l/(x), the K values depict an increase which becomes more marked with increase in n. On the other hand, when ~o 2 and all a2/s are doubled the K values get somewhat reduced.

456 Mamta, Vishwamittar / Physica A 216 (1995) 452-458

Table 1 Energy eigenvalues of the AHOs governed by tenth-degree even power potential for the ground and some excited states

~2 a4 a6 a8 alo n K E.(P) E.(ex)

12.5625 6.5 3.0 0.4 0.05 0 0.0 1.875000000000 1.875000 a (1.8748)

1 0.5 5.826524340175058 5.82652 b 2 1.5 10.17947433609 10.17947 b 3 2.5 14.92535070965 14.92535 b

13.0 3.0 0.4 0.05 0 0.5 1.945561971132390 1 0.5 6.132983478655940 2 1.5 10.86436984047071 3 2.0 16.06301633872712

6.0 0.8 0.1 0 0.0 1.9573225234229 1 0.5 6.199104494269944 2 1.0 11.0556190013413 3 1.5 16.459246850608

6.5 6.0 0.8 0.1 0 0.5 1.889309371079 1 0.5 5.909747374 2 1.5 10.42454666 3 2.0 15.4358394

6.28125 6.5 3.0 0.4 0.05 0 1.0 1.4346240936866893 1 2.0 4.612887369647 2 2.5 8.3644718499 3 4.0 12.615325889

25.125 13.0 6.0 0.8 0.1 0 0.0 2.607856157211825 1 0.5 8.05269518524889 2 1.0 13.84738059241272 3 1.5 20.07219093333619

11.5625 6.1 3.0 0.4 0.05 1 1.0 5.625000000000 5.625000 a (5.6245)

10.5101 5.7 3.0 0.4 0.05 2 2.5 9.509415590120 9.50941722 a (9.5182)

9.4108 5.3 3.0 0.4 0.05 3 4.5 13.53315418467 13.53313824 a (13.5425)

a Ref. [5]. b An anonymous referee.

(4) The convergence of E, and hence the number of stable digits in the E,(P) values are also influenced by the relative magnitudes of 092 and other parameters. An

increase in the a4 values for the same set of other parameters improves convergence of E, for all n and a decrease in a4 with increase in other a2fs renders the convergence poorer. In fact the convergence was worst for ~o2= 12.5625, a4 = 6.5, a6 = 6.0, a8 = 0.8 and alo = 0.1. The convergence was not affected by halving co 2 but keeping all other four parameters unchanged and was definitely improved when t~ 2 and all a2fs were doubled. These trends are also observed in the E,(P) values as is clear from

Mamta, Vishwamittar / Physica A 216 (1995) 452 458

T a b l e 2

C a l c u l a t e d e n e r g y v a l u e s for s o m e D W O s

457

- co 2 a4 a6 as a l o Vmi. n K E , E, (ex)

1.3594 - 9.25 3.0 4.0 5.0 -- 1.50996 0 - 13.5 0.1653 0.1875"

(0.1593)

1 - 25.0 1.328 1.55222 b

1.0 - 1.0 1.0 1.0 1.0 - 0 .39365 0 - 33.0 0.364

3.0 4.0 5.0 - 0 .20326 0 - 100.0 0 .6230

6.0 8.0 10.0 - 0 .14295 0 - 40.0 0.71

-- 5.0 3.0 4.0 5.0 - 0 .59108 0 - 27.5 0 .45753

- 10.0 3.0 4.0 5.0 - 1.50930 0 11.0 0 .18237

2.0 - 1.0 3.0 4.0 5.0 - 0 . 4 7 8 3 2 0 - 100.0 0 .4910

- 2.0 3.0 4.0 5.0 - 0 .57670 0 -- 65.0 0 .4515

- 2.0 6.0 8.0 10.0 -- 0.40651 0 -- 60.0 0.6203

a Ref. [5] .

b A n a n o n y m o u s referee.

the contents of Table 1. Of course, even when the E, values showed very poor

convergence for the case referred to above (we got Eo = 1.889, E~ = 5.92, E2 = 10.4,

E3 = 15.5), their Pad6 approximants led to a good number of stable digits as seen

from the table.

These considerations indicate that the renormalized hypervirial-Pad6 scheme can

safely be used when co 2 and a4 are larger than a2j ( j /> 3) and with caution when a6 is

comparable to a4.

3.2. DWOs

The potential in Eq. (1) with s = 5 produces a double well if a 2 and a 4 are taken to

be negative. Parameters for some such potentials together with their energy eigen-

values for n = 0 or 1 are projected in Table 2. Also included in this table are the

potentials at the minima (Vmin) and the available exact results. It may be mentioned

that the E. values found from Eq. (6) could not be improved by using Padb approxi-

mants and, therefore, we have listed E, values rather than E,(P) in this table.

Furthermore, the scheme did not work well in all the cases reported in [5]. In fact, it

could be used only when V m i n > - - 1.51 while the cases considered by Adhikari et al.

have Vmin down to -- 10.034. The renormalized hypervirial method failed to yield a convergent value of E. even for - m2 = 4, a4 = - 1, a , = as = alo = 1 for which

Vmln = -- 1.964. These observations are similar to our earlier findings that in the case

of cubic-quartic double-well oscillator this approach gives reasonably correct values

only when Vmin >~ -- 1.10 [12]. A perusal through the contents of Table 2 leads us to the following conclusions.

(1) The accuracy of the energy values found for the DWOs is poorer than that for AHOs. In the case for which comparison could be made with the exact values the

458 Mamta, Vishwamittar / Physica A 216 (1995) 452 458

er ror is a b o u t 12% for n = 0 and 14% for n = 1. N o doub t , the agreement is qui te bad,

it is s l ightly bet ter than tha t ob ta ined with the shifted 1 /N expans ion technique whose

result for n = 0 [5] too is inc luded in the table.

(2) A negat ive value of a4 requires tha t K should be negat ive so that ¢o '2 is posit ive.

F o r specific values of - 0) 2 and a4, if o ther pa rame te r s are increased, then more

negat ive values of K are needed and Eo values have increased. However , doub l ing

- ~ o z for the same value of the a z f s does not a l ter K though Eo has increased.

F u r t h e r m o r e , if the magn i tude of a4 is increased keeping - o ) 2, a6, a 8 and

a l o unchanged , then IKI, as well as Eo decrease.

Acknowledgements

The au thors are grateful to an a n o n y m o u s referee of Physics Let ters A for supply ing

and pe rmi t t ing us to use some exact results l isted in the paper .

References

[1] S. Srivastava and Vishwamittar, Mol. Phys. 72 (1991) 1285. [2] M. Bansal, S. Srivastava and Vishwamittar, Phys. Rev. A 44 (1991) 8012. [3] E. Magyari, Phys. Lett. A 81 (1981) 116. [4] G.P. Flessas, J. Phys. A 14 (1981) L209. [-5] R. Adhikari, R. Dutt and Y.P. Varshni, Phys. Lett. A 141 (1989) 1. [-6] R.N. Chaudhuri and M. Mondal, Phys. Rev. A 43 (1991) 3241. [7] R.K. Agrawal and V.S. Varma, Phys. Rev. A 49 (1994) 5089. [8] M.R.M. Witwit, J. Phys. A 25 (1992) 503. [9] J. Killingbeck, J. Phys. A 14 (1981) 1005.

[-10] J. Killingbeck, J. Phys. A 20 (1987) 601. [-11] J. Killingbeck, Phys. Lett. A 132 (1988) 223. [12] M. Bansal, S. Srivastava, Mamta and Vishwamittar, Chem. Phys. Lett. 195 (1992) 505.