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Journal of Mathematical Physics

Renormalized perturbation series and the semiclassical limit of quantum mechanics

Francisco Marcelo Fernández

Published Aug 1, 1995• 8 citations
Publisher

Abstract

The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.

DOI
10.1063/1.530938

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The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
The detailed transition from classical mechanics to quantum mechanics, particularly in regard to the question of quantum corrections to the semiclassical quantization, can be completely understood for the x4{x}^{4}x4 perturbed harmonic oscillator. This is a consequence of five facts: (i) The quantum perturbation series for the energy can be solved order by order in closed form. (ii) The quantum series rearranges directly into the classical canonical perturbation series plus quantum corrections proportional to successively higher powers of ${\ensuremath{\Elzxh}}^{2}$. (iii) The classical series and the subseries for the quantum corrections converge (subseries by subseries) to the terms of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) expansion for the energy (to all orders of \ensuremath{\Elzxh}). (iv) The (all-order) JWKB expansion, conversely, yields the quantum perturbation series and consequently uniquely determines the exact eigenvalue. (v) The ratio of the quantum energy to the classical energy at action (n+1/2)\ensuremath{\Elzxh} differs from 1 asymptotically by 1/[9\ensuremath{\pi}(n+1/2)2{+1/2)}^{2}+1/2)2], independent of both \ensuremath{\Elzxh} and the perturbation coupling constant, as the harmonic-oscillator quantum number n approaches \ensuremath{\infty}. An interesting additional feature is the mechanism by which the convergence of the classical perturbation series and of the series for the quantum corrections gets turned into the divergence of the quantum perturbation series.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
The semiclassical approximation, particularly the Wentzel-Kramers-Brillouin (WKB) method, has been a cornerstone in quantum mechanics for understanding systems where quantum effects are present but allow for an approximate classical description. Despite its widespread success in providing intuitive insights into phenomena like tunneling and energy quantization, the standard WKB expansion often suffers from asymptotic divergence and limited accuracy, especially for lower quantum numbers or near turning points. This paper introduces a novel approach to enhance the precision and convergence of the WKB method by integrating variational principles with advanced resummation techniques. We propose a framework where the WKB expansion for key physical quantities is treated as an asymptotic series, and its coefficients are optimized through a variational procedure derived from the underlying quantum dynamics. Subsequently, these optimized series are subjected to resummation methods, such as generalized Borel transforms or Padé approximants, to extract highly accurate and convergent results. Our methodology aims to transcend the limitations of conventional WKB by systematically incorporating non-perturbative information and improving the analytic properties of the semiclassical series. We present the theoretical foundations of this "Resummed Variational WKB" approach, outlining its potential to yield unprecedented precision in calculating energy eigenvalues, tunneling rates, and other quantum observables, thereby bridging the gap between purely classical and fully quantum mechanical descriptions with enhanced quantitative fidelity.
Epstein’s [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman–Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein’s rederivation of the Rayleigh–Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones–Brillouin–Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.