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arXiv:quant-ph/9610027 · Journal of Physics A Mathematical and General

WKB to all orders and the accuracy of the semiclassical quantization

Marko Robnik, Luca Salasnich

Published Mar 7, 1997• 30 citations
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Abstract

We perform a systematic WKB expansion to all orders for a one-dimensional system with potential . We are able to sum the series to the exact energy spectrum. Then we show that at any finite order the error of the WKB approximation measured in the natural units of the mean energy level spacing does not go to zero when the quantum number goes to infinity. Therefore we make the general conclusion that the semiclassical approximations fail to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.

DOI
10.1088/0305-4470/30/5/031

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We perform a systematic WKB expansion to all orders for a one-dimensional system with potential . We are able to sum the series to the exact energy spectrum. Then we show that at any finite order the error of the WKB approximation measured in the natural units of the mean energy level spacing does not go to zero when the quantum number goes to infinity. Therefore we make the general conclusion that the semiclassical approximations fail to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.
We perform a systematic WKB expansion to all orders for a one--dimensional system with potential V(x)=U0/cos⁡2(αx)V(x)=U_0/\cos^2{(αx)}V(x)=U0​/cos2(αx). We are able to sum the series to the exact energy spectrum. Then we show that any finite order WKB approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.
We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.
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.
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.