self-interacting one-dimensional oscillators

PRAMANA __ journal of

physics

~,~g: Printed in India Vol. 46, No. 3, March 1996 pp. 203-211

Self-interacting one-dimensional oscillators

MAMTA and VISHWAMITTAR* Department of Physics, Panjab University, Chandigarh 160 014, India *Author for correspondence

MS received 26 October 1995

Abstract. Energy eigenvalues and (x2), for the oscillators having potential energy V(x) = (to2x2/2) + 2 < x 2' > x 2' have been determined for various values of 2, r, s and n using renormalized hypervirial-Pad6 scheme. In general, the results show an improvement over the findings of earlier workers. Variation of the evaluated quantities and of the renormalization parameter with 2, r, s and n has been discussed. In addition, this potential has been employed as an illustrative example of the applicability of alternative formalism of perturbation theory developed by Kim and Sukhatme (J. Phys. A25 647 (1992)).

Keywords. Self-interacting oscillators; energy eigenvalues; perturbation theory.

PACS No. 03.65

1. Introduction

There are a number of important and interesting situations in physical and biological studies where the interaction between the system and its surroundings influence not only the former but also the latter. This results in the modification of environmental field so that the interaction potential acting on the system also depends upon its own state ~,. and is, therefore, written as V(~.). Accordingly, the time-independent Schr6dinger equation for the system becomes non-linear and reads [1-3];

H(~.)~,, = [H 0 + V(q,.)]~. = E,~,. 0 )

with H o as the Hamiltonian for the isolated system. A typical example of such a self- dependent system is one-dimensional oscillator described by the Hamiltonian (h = m = 1)

H = - (1 /2 ) (dZ /dx 2) + (to2 x2/2) + 2 (x 2 " )x 2s, (2)

where both r and s are integers. Obviously, when r = s = 1, (2) becomes Hamiltonian of a self-interacting harmonic oscillator whose force constant is linearly dependent on the mean square of displacement of vibration, and r >/1, s >/2 correspond to the self- interacting anharmonic oscillators whose anharmonicity depends on (x2"). Also for to = 0, r ~> 1, s t> 2 we have the self-dependent oscillators with extremely large magnitudes of anharmonicity - the so-called infinite-field limit of the oscillators. Further- more, for r = 0 and s = 2 or 3 we get the special cases of well-studied quartic or sextic anharmonic (when to 4: 0) and pure quartic or sextic (to = 0) oscillators.

Since (1) cannot be solved exactly, some iteration and approximation methods have been used to determine the energy eigenvalues. An adaptation of Rayleigh-Schr6dinger

203

Mamta and Vishwamittar

perturbation theory (RSPT) has been put forward for this purpose and applied to the problem of a molecule in a polarizable medium [1,3,4]. Surj/m [3] also pointed out limitations of iterative method, configuration interaction approach and variational technique for solving the time-independent non-linear Schr6dinger equation. How- ever, since perturbation theory suffers from the uncertainty about its convergence, particularly for large perturbations, Cioslowski [5] employed an extension of connec- ted moments expansion to obtain the solution for (1) and illustrated it by finding the values of energy and mean square displacements of vibration <x2 > for the ground state (n = 0) of self-interacting harmonic oscillator for different magnitudes of 2. Vrscay [2] analyzed the perturbation expansion for (2) using hypervirial and Hellmann-Feynman theorems and performed numerical calculations to determine lower and upper bounds to energy for different values of r and s -- 2 corresponding to to = 0 as well as co 4:0 employing a renormalized RSPT wherein summation was carried out by the Pad~ approximants method.

Killingbeck [6-8] presented a variational parameter-based renormalized hypervirial Pad6 scheme (RHPS) that yields very accurate energy eigenvalues (and also (x 2 ) values) in a simple manner for anharmonic oscillators. We examined the extent to which this technique is successful in finding the energies and the expectation values of x 2 for self-interacting harmonic and anharmonic oscillators and this communication is an outcome of the effort for a wide range of values of r, s, ;t and n. We shall discuss in the sequel the dependence of energy, (x 2) and the variational parameter on various quantities.

It is pertinent to mention that Kim and Sukhatme [9] have developed an alternative formalism for Rayleigh-Schr6dinger perturbation theory (ARSPT) for a one-dimensional problem described by linear Schr/Sdinger equation. Being an expansion in the powers of perturbation parameter, this leads to the same results as RSPT but without requiring cumbrous sums over intermediate eigenstates and also reduces to the logarithmic perturbation theory of Au and Aharonov [10]. We have also obtained expression for energy of the self-interacting oscillators in the framework of this approach and compared the results with the findings of RHPS.

2. Renormalized hyperviriaI-Pad~ calculations

2.1 Theoretical f ramework

Following Killingbeck's [6-8] prescription, the Hamiltonian (2) is renormalized by adding and subtracting (2KxZ/2) so that denoting (to 2 + ~,K) 1/2 by to', we get

H = ( - 1/2)(d2/dx 2) + (to'2x2/2) + A ( x Z ' ) x 2s - (2Kx2/2) • (3)

Using the hypervirial theorem together with the relevant commutation relations, expanding E n and (xN)n as

E .= ~ -~lT~J~'~-~, (4) j=O

<xN>= ~, C~N~2k (5) k=O

204 Pramana - J. Phys., Vol. 46, No. 3, March 1996


and equating coefficients of the same powers of 2 on the two sides of the resulting equation, we obtain the recurrence relation

P C~N + 2~ = [(2 (N + 1)) /((N + 2)f0'2) "] ~ E(i~n C(N)'p-j _'4- [,,/tt.nllc'/'J2~]l...p_f'(N +12)

j=O

p - I - [ ( ( 2 ( N + s + l ) ) / ( ( N + 2 ) w ' 2 ) ] ~ C '2") C~N+2~) p-k-I

k=O

Here

C~ °1 = ~op and

,2 (N-2) + [ ( N - 1 ) N ( N + I)/4(N + 2)o~ ]C,, . (6)

(7)

°) = (n + ¢8)

The E~ ) forj /> 1 are related with C~ N) by using the Hellmann-Feynman theorem and this gives us

_,E ti~ = (l/j) ( - K / 2 ) C ~ 1 + ~ (k + l~C(2r)c k i - k - t " (9) k=0

Equations (8) and (9) when substituted in (4) give E, in terms of K and C~ N~ and the latter are determined from (6) in a hierarchial manner. Similarly, substitution of(6) in (5) with N = 2 leads to the formula for evaluation of (x 2). . It may be pointed out that the product terms C~2_'~ k_l C~ff ÷ 2s)in (6) a n d C(k2r)c~2_S)k_l in (9) allow the calculation to treat the term 2 (x 2') x 2s in (3) directly, without any need for iteration of any kind.

2.2 Numerical results and discussion

In order to execute the computations the parameter K is determined in such a way that maximum number of stable digits in the value is obtained from the final expression for E, for a particular state. Also it is found that

C(e2M+ 11= 0 (10)

for p, M = 0, 1, 2, 3,..., which is a consequence of even power terms in the potential. The computations for energy as well as (x 2 ) , have been executed with double precision and by terminating the series summation in (4) at j = 32; though in some cases, particularly those for which ). is large, summation had to be extended up to j = 52 (or even 80) to obtain proper convergence. However, in all cases the values being reported correspond to a situation for which the maximum number of digits was stable. The energy values so determined (E~) and those improved upon by finding Pad6 approximants, E,(P), to the series expansion with chosen K parameters are listed in tables 1-3 for different values of r, s, n, co and 2. Also projected in these tables are the relevant (x 2) . obtained by performing sum in (5) by the Pad6 approximants. With a view to compare our results with those of other workers, their values have also been included in these tables. It may be mentioned that the entries in the tables have been kept up to such an unrealistic number of significant figures just to emphasize the degree to which results of the present calculations can be trusted.

Pramana - J. Phys., Vol. 46, No. 3, March 1996 205

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Se!f-interacting one-dimensional oscillators

Perusing through the computational data displayed in the three tables, we arrive at the following conclusions:

(1) In self-interacting oscillators and the infinite-field limit oscillators (tables 1 and 3) wherever comparison with the findings of other workers is possible, our results for E,(P) as well as (x z). are more accurate. This prompts us to infer that other values found by RHPS are also quite reliable for these categories of oscillators. As far as self-interacting anharmonic oscillators with s = 2 (table 2) are concerned, Eo(P ) are an improvement over the results of Vrscay for r = 1"0 for 2 up to 10.0 and for r = 2"0, 3"0 for 2 up to 1-0, in addition to being so for r = 0 for all 2. However, in all other cases, the convergence of Eo(P ) is worse than the two bounds of energy reported by Vrscay. Furthermore, such a comparison is not possible for (x z )o as these values have not been determined by Vrscay. Nonetheless, the extent to which these sums are convergent, they too must be correct. (2) For self-interacting oscillators, the renormalization parameter K is zero for 2 up to 0.1 and 0"5 for 2 values ranging from 0"5 to 10,000. Thus, for very low 2 values even ordinary hypervirial-Pad~ or simple hypervirial method is successful. In the case of anharmonic oscillators whether to is 0 or not, K does not vary significantly or monotonically with 2, r or n. For a particular set of values of s and r, it decreases with increase in n; and for specific values of r and n, K increases if s is increased. (3) If we consider a particular type of oscillator with to 4 0 then the ground state energy increases while (x 2)o decreases with increase in the contribution of the perturbation term, i.e., in the value of 2. For a particular 2, the values of E o and ( x : ) o for r = 0 are, respectively, higher and lower than the corresponding values for r :~ 0 cases though for r = 1, 2, 3 the variations of E o as well as (x 2)0 with r do not exhibit a well defined trend. In the case ofo9 = 0, 2 = 1 oscillators there is no definite trend in the variation of E, and (x 2) . .

3. ARSPT

In the framework of perturbation theory, the interaction

V(~k.) = 2(xZ') x 2~ (11)

is treated as a perturbation and since it depends upon ~O.(x), similar to ¢. and E. this too is expressed as a power series in the perturbation parameter 2. Thus, following Surj/m and ,~ngy/m [1], Surj/m [3] and Kim and Sukhatme [9], we write

~,(x) = O~°~(x)[1 + 2 f ~ ' + U 7 ~ + ...], (12)

E, = E;o' + 2e~" + 22Ep' + ..., (13) and

V(O.) = 2 V~.'~(x) + 2 2 Vp~(x) + . . . . (14)

Here, the wavefunctions are taken to be real because we are concerned with one- dimensional bound state situation with local confining potential [11, 12]. From (11), (12) and (14) we have

V:"(x)= xZ~ f ~., O~.°'"(x')(x')2"dx' (15)

Pramana J. Phys., Vol. 46, No. 3, March 1996 209

Mamta and Vishwamittar

Table 4. A comparison of the ground state energy for the Hamiltonian (2) with r = 1 determined by ARSPT (Eo) and RHPS Eo(P). All entries are in the units corresponding to h = m = o ~ = 1.

s = l s = 2

2 0-1 1-0 0'1 l-0

E o 0.5225 0'5000 0-5361 - 0.5547 Eo(P)* 0'5233 0"6624 0'5297 0"6491

*Eo(P) values have been rounded off to four significant figures.

and

S V~2)(x) = 2x 2~ (x')2'O~°)2(x')f~l)(x')dx'. (16) - - G O

Following the cus tomary procedure, we get

E~ 1' f oo ~1) (o,~ = V, (x)~k. (x)dx, (17)

d - o0

and

E~ 2, f ~ (~) (~) [E. f . (x) V(.2)(x) ") ") (°~ . . . . V. ( x ) f . ( x ) ]¢ . (x)dx (18)

with

f~t)(x) = - 2 [dx'/O~°)2(x')] [E~ l ~ - Vn~t)(X,,)]0,~O~ . . . . . (X)dx . 09) - - o 0

With a view to compare the results obta ined by using A R S P T with those of R H P S we have determined ground state energy for the self-interacting oscillators with r = 1, s = 1,2 keeping co = 1. We have for r = 1, s = 1

E o = 0"5 [ 1 + (2/2) - ( ~ . 2 / 2 ) ] (20)

and for r = 1, s = 2

E o = 0.51-1 + (32/4) - (18322/64)]. (21)

The values found for 2 = 0"1 and 1.0 by two methods are compared in table 4. As expected the E o values differ significantly from Eo(P) and the deviation is higher for higher s and ,~.

References

[1] P R Surj/tn and J G Angyfin, Phys. Rev. A28, 45 (1983) [2] E R Vrscay, J. Math. Phys. 29, 901 (1988) [3] P R Surjfin, J. Math. Chem. 8, 151 (1991) 1-4] J G Angy/m a n d P R Surjfin, Phys. Rev. A44, 2188 (1991) [5-1 J Cioslowski, Phys. Rev. A36, 374 (1987)

210 Pramana - J. Phys., VoL 46, No. 3, March 1996


[6] J Killingbeck, J. Phys. AI4, 1005 (1981) [7] J Killingbeck, J. Phys. A20, 601 (1987) [8] J Killingbeck, Phys. Lett. A132, 223 (1988) [9] I M Kim and U P Sukhatme, J. Phys. A25, 647 (1992)

[10] C K Au and Y Aharonov, Phys. Rev. A20, 2245 (1979) [11] L D Landau and E M Lifshitz, Quantum mechanics-non-relativistic theory (Oxford,

Pergamon, 1965) [12] M Razavy and A Pimple, Phys. Rep. 168, 307 (1988)

Pramana - J. Phys., Vol. 46, No. 3, March 1996 21 !

self-interacting one-dimensional oscillators

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