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IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 NOTE BREVI

Self-consistent parameter for the Hill determinant:

quartic anharmonic oscillator

BISWANATHRATH(1) and K. PATNAIK(2)

(1) Physics Department, Khallikote College - Berhampur 760001, Orissa, India (2) Physics Department, Utkal University - Vanivihar, Bhubaneswar 751004, India (ricevuto il 9 Febbraio 1994; approvato il 29 Aprile 1997)

Summary. — We suggest a simple parameter calculated self-consistently for the eigenvalue calculation using the Hill-determinant approach. Energy levels for the anharmonic oscillator V(X) 4X2

1 lX4are calculated to a very high accuracy using a small-size determinant.

PACS 11.10 – Field theory.

The continued interests [1, 2] in the anharmonic oscillator devolve on their use as physical and mathematical models. The bulk of the quantum studies involves calculations of energy levels either using perturbation theory [4-12] or nonperturb-ative methods [13-20]. Among nonperturbnonperturb-ative approaches, the Hill-determinant method [13, 17, 18] produces the energy levels of high accuracy for any arbitrary values of the coupling constant. The most commonly studied anharmonic oscillator is

H 4P2

1 X21 lX4. (1)

In the energy level calculation using the Hill-determinant approach, one has to solve the eigenvalue problem

Hc 4Ec .

(2)

Biswas et al. [13] use the wave function of the form

c 4e2X2/2

!

CnX2 n.

(3)

Banerjee et al. [14] use the wave function of the form

cne2aX 2_/2

!

BISWANATH RATHandK.PATNAIK 924

(for details see ref. [14]) Killingbeck [15] has suggested a variational parameter in the wave function as

c(N , b) 4XN_e2bX2/2_.

(5)

However, the aim of this communication is to show that if a suitable parameter is introduced in the wave function then the size of the Hill determinant is reduced considerably. In addition to this, we suggest a method for calculating the parameter self-consistently using the knowledge of unperturbed Hamiltonian. The wave function to be used for eq. (2) can be calculated as follows:

Let

c 4

!

m 40

AmNmbW,

(6)

where the function NmbWsatisfies the equivalent unperturbed eigenvalue relations

HWNmbW4

y

z

and anNHNmbW4 0 for m c n .

The parameter W is determined by the condition

W2_{4 1 1 la}0 amNX 4 NmbW amNX2NmbW , (9) or W3 2 W 2 1.5 la04 0 ,

where a0is obtained from the condition [10]

y

z

The value of a0is found to be 2.

Now substituting the ansatz in eq. (6) in the eigenvalue equation (2) we find that

Am’s satisfy the following difference relation:

BmAm 241 CmAm 221 DmAm1 JmAm 121 KmAm 144 0 , (11) where Bm4 l 4 W2

k

k

SELF-CONSISTENT PARAMETER FOR THE HILL DETERMINANT:ETC. 925 with E04 W 2 1 1 2 W 1 3 l 4 W2 . (12f )

In the limit of large n, the zeros of the det U as a function of E give the energy eigenvalues, i.e.

det U 40 . (13)

Furher, due to the symmetric nature of the matrix elements anNHNmbW4 amNHNnbW

we can express U as U 4

N

N

N

N

N

N

In table I we compare first two energy levels with previous calculation.

In this approach we get very good result in a small-size determinant. Of course, this had been made possible due to our choice of W, which reduces the Hamiltonian to its effective value. In the language of second quantization, the coefficient of the quadratic term vanishes. In other words, the coefficient of the square of the creation operator as well as the annihilation operator vanishes automatically. One can check that with any arbitrary choice of W or W without a0 in eq. (9), one cannot achieve this condition.

Mathematically speaking, our choice of W in eq. (9) with a0 determined by eq. (10)

achieve the condition a 0 NHN2bW4 0 .

TABLEI. – Energy levels of the anharmonic oscillator V(X) 4X21 100 X4.

State n Present determinant size ( 15 315) Exact [13-20] 0 1 4.999 418 17.830 193 4.999 417 17.830 192

In this context we would like to state that the condition a 0 NHN2b 40 has been widely used in nuclear physics to find the most effective Hamiltonian [21]. Further scaled harmonic-oscillator wave function NmbW not only reduces the diagonal element

to its effective value but also reduces the strength of nondiagonal matrix element anNHNmbW considerably. For example, in the limit of large m, the nondiagonal matrix

elements reduced to m2

OW2. Hence the reduced nondiagonal matrix elements along with the effective Hamiltonian reduce the size of the Hill determinant. All these achievements had been made possible due to our choice of W which depends on the constant a0. In the opinion of the authors the constant a0 which contains the

contributions of state N0b and N2b plays an important role in the Hill-determinant approach of the anharmonic oscillator.

BISWANATH RATHandK.PATNAIK 926

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