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arXiv:quant-ph/0411084

Exact quantization of nonsolvable potentials: the role of the quantum phase beyond the semiclassical approximation

A. Matzkin

Published Nov 11, 2004• 0 citations
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Abstract

Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in $\hbar$, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.

DOI
10.48550/arxiv.quant-ph/0411084

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Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in ℏ\hbarℏ, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.
Semiclassical quantization is exact only for the so-called solvable potentials, such as the harmonic oscillator. In the nonsolvable case, the semiclassical phase, given by a series in $\ensuremath{\hbar}$, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows us to obtain the quantum wave function while locally approaching the best predivergent semiclassical expansion. An iterative procedure allowing us to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.
We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville–Green transformations of the Schrödinger equation. For anharmonic oscillators, first-order semiclassical quantization is seldom accurate and the higher order expansions eventually break down given the asymptotic nature of the series. A quantum phase that allows us in principle to retrieve the exact quantum mechanical quantization condition and wavefunctions is given along with an iterative scheme to compute it. The arbitrariness surrounding quantum phase functions is lifted by supplementing the phase with boundary conditions involving high order semiclassical expansions. This allows us to extend the definition of oscillation numbers, which determine the quantization of the harmonic oscillator, to the anharmonic case. Several illustrations involving homogeneous as well as coupling constant dependant anharmonic oscillators are given.
While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semiclassical Landauer–Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.
In the present investigation we assess the relevance of both classically forbidden phenomena and higher-order asymptotic contributions for the semiclassical energy quantization of a particle in the anharmonic oscillator potential V(x)=x 2 /2+ lambda x 4 . We propose an iterative method (Iq) for obtaining higher-order semiclassical corrections, which is similar to the WKB and phase-integral methods in the lowest order, but in higher orders the approximation differs significantly. A 'primitive' Bohr-Sommerfeld energy quantization is compared with a more complete semiclassical quantization, taking into account both complex trajectory contributions and higher-order (quantal) corrections.
Abstract An infinite potential well, truncated at finite height, provides a simple model for studying the effect of nonanalyticity on semiclassical approximations. An exact quantization condition for the bound states separates the effects associated with the untruncated well from those of the truncation. Because the truncation occurs beyond the classical turning points, it has no effect to any finite order in powers of Planck’s constant. The truncation contribution is exponentially small and depends on the potential in the classically forbidden region. The contribution associated with the well, when consistently approximated beyond all semiclassical orders, also leads to a small exponential, depending on the potential in the classically allowed region. Both exponentially small contributions can be extracted by asymptotic analysis, with explicit results in the simple case of a linear well. This combination of several different semiclassical techniques could be pedagogically useful as an exercise in teaching physical asymptotics at the postgraduate level.