Energy eigenvalues of d-dimensional quartic anharmonic oscillator

Abstract

On the basis of a radial generalization of the JWKB quantization rule, which incorporates higher orders of the approximation, an explicit analytical formula is derived for the energy levels of the three-dimensional quartic anharmonic oscillator. The formula exhibits the scaling property of the exact eigenvalues, and is readily generalized to any dimension. Together with the Hellmann–Feynman theorem, it yields the values of the diagonal moments of r2k. The predicted energies and moments are in excellent agreement with known numerical results.

Phase-integral approach to quantal two- and three-dimensional isotropic anharmonic oscillators

“…Denoting by f(u2, A) the exact energy eigenvalues of the differential equation (2.4), when B(r) is given by (2.5), V(r) is given by (2.1), and m and L are kept fixed, one easily finds that [14] of the phase-integral approximation. For evaluating the complete elliptic integrals appearing in the right-hand members of (5.10a) -(5.10c) we used very rapid standard library routines.…”

“…. , aqua and expressions (5.9a) -(5.91) for J2~2~2", we finally obtain I Pk'~+ 3(1 P)~It @+(P2 k2) 1 5 1 2~360AsP~2 ps q(5) g7 2P'(14k" -66k" + 375k'+ 250k'+ 375k' -66k'+ 14) +3P ( -168k +807k -1550k + 15k + 15k -1550k +807k -168) +6P (224k -1086k + 2100k -2005k + 2430k -2005k + 2100k -1086k + 224) + 7( -128k + 624k -1215k + 1182k -591k -591k + 1182k -1215k + 624k -128) (5.10a) (5.10b) Ps(28k'2 -lllk -315k -250k -435k + 234k + 56) 360A' »qs . +3P ( -168k + 681k -1055k + 2160k -2175k + 2605k -1488k + 336) +3P (448k -1836k + 2865k -2015k -2430k + 6025k -7065k +4008k -896) + 7( -128k + 528k -831k + 603k -198k +789k -1785k + 2046k -1152k + 256) The derivation in the present paper is based on the assumption that n ) 0, but one can find reasons for believing that the resulting final formulas, expressed in terms of complete elliptic integrals, remain valid also when o.…”

Harmonic oscillator potential with a quartic anharmonicity in the prolate γ-rigid collective geometrical model

“…Here one presents an approximate analytical formula for the energy spectrum of a prolate γ-rigid collective Hamiltonian with a harmonic oscillator potential corrected by a quartic term [18], which is based on the forth order approximation made on the JWKB quantization rule [19]. Due to the scaling property of the problem, the corresponding energy spectrum depends up to an overall factor on a single free parameter, which when vanishing leads to a parameter free X(3)-β 4 model.…”

“…For this, one adapts the method from Ref. [19] for finding the eigenvalue W , to the present case of the modified centrifugal term. Following the procedure of Ref.…”

“…Following the procedure of Ref. [19] one can write the eigenvalue W as function of λ , n -the β vibration quantum number and the intrinsic angular momentum I, as follows:…”