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Physics of Particles and Nuclei

Bohr–Sommerfeld Quantization Rule in the Case of a Two-Dimensional Motion of a Quantum Particle in the Field of Decreasing Power-Like Potential

Vasily Veniaminovitch Pupyshev

Published Jul 1, 2020• 1 citation
Publisher

Abstract

We study the finite two-dimensional motion of a quantum particle in the field of the potential $$V(r) = - {{V}_{0}}{{r}^{{ - \beta }}}$$ with parameters $${{V}_{0}} > 0$$ and $$\beta \in (0,2)$$ . An analysis of an asymptotic equation equivalent to the Bohr–Sommerfeld quantization rule is given. As a result, a simple and explicit approximation for the energies of weakly-bound states of this particle has been obtained.

DOI
10.1134/s1063779620040590

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We study the finite two-dimensional motion of a quantum particle in the field of the potential V(r)=−V0r−βV(r) = - {{V}_{0}}{{r}^{{ - \beta }}}V(r)=−V0​r−β with parameters V0>0{{V}_{0}} > 0V0​>0 and β∈(0,2)\beta \in (0,2)β∈(0,2) . An analysis of an asymptotic equation equivalent to the Bohr–Sommerfeld quantization rule is given. As a result, a simple and explicit approximation for the energies of weakly-bound states of this particle has been obtained.
An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
The generalized Bohr-Sommerfeld quantization condition up to the ninth-order approximation is given explicitly, in terms of complete elliptic integrals, for two kinds of quartic oscillators V(z)=v2{\mathit{v}}_{2}v2​z2{\mathit{z}}^{2}z2+v4{\mathit{v}}_{4}v4​z4{\mathit{z}}^{4}z4: (i) for the quartic double barrier, where v2{\mathit{v}}_{2}v2​>0 and ${\mathit{v}}_{4}$0, and the energy is below the potential maximum; and (ii) for the quartic double well, where ${\mathit{v}}_{2}$0 and v4{\mathit{v}}_{4}v4​>0. Resonance energies for some quartic double barriers and energy levels for some quartic double wells were evaluated by solving the quantization condition numerically. In some cases the results are very good.