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

Abstract

A two-or three-dimensional quantal isotropic anharmonic oscillator is treated by means of the phase-integral method of Froman and Froman. The generalized Bohr-Sommerfeld quantization condition for the radial wave function is expressed in terms of complete elliptic integrals up to the Sfth order of the phase-integral approximation. The quantization condition is solved numerically, and energy levels are obtained for various quantum numbers. Comparison with numerically exact results is also made.PACS number(s): 03.65.Sq

Anharmonic oscillators energies via artificial perturbation method

“…Also, the effect of the [4,5] Padé approximant on the leading terml 2 E (−2) is reported as E [4,5]. In table 1 we list our results along with the exact numerical ones and the (best estimated) eigenvalues obtained from the fifth-order phase-integral method (PIM) reported by Lakshmanen et al [5]. Obviously, our results compare excellently with the exact numerical ones and surpass those from PIM.…”

“…Obviously, our results compare excellently with the exact numerical ones and surpass those from PIM. Whilst the [4,5] Padé approximant had no dramatic effect on the energy eigenvalues for l = 0, it had no effect on the energy eigenvalues for l ≥ 1. A common feature between PSLET and PIM is well pronounced here; the precession of both methods increases as l increases.…”

“…Such studies already lie beyond the scope of our present methodical proposal. Table 1: Eigenvalues from the fifth-order phase-integral method E P IM [5], the pseudoperturbative shifted-l expansion technique E P SLET , the effect of the [4,5] Padé approximant on our leading energy term E [4,5], and from the exact numerical calculations [5] for the three-dimensional anharmonic oscillator V (r) = 1 2 r 2 + 1 2 r 4 , with n r = 0. Three-dimensional ground state energies or equivalently onedimensional first excited state energies for V (q) = −aq 2 /2 + q 4 /2.…”

“…Interest in this model Hamiltonian arises in quantum field theory and molecular physics [1][2][3][4][5][6].…”

“…For the sake of comparison, we use results from exact numerical methods reported in [2,5], the best estimation of the phase-integral method (PIM) [5], an open perturbation technique [2], and a perturbative-variational method (PVM) [6]. Section 4 is reserved for concluding remarks.…”