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Physical Review Letters

Wentzel-Kramers-Brillouin Quantization of Pseudospin Hamiltonians

Abraham Klein, Ching‐Teh Li

Published Apr 6, 1981• 22 citations
Publisher

Abstract

A new method is proposed for studying the semiclassical limit for model Hamiltonians which are polynomials in the generators of the Lie algebra $\mathrm{SU}(n)$. The symmetric representation, characterized by a single independent Casimir invariant is mapped onto a subspace of a set of canonical coordinates, and a systematic expansion in the reciprocal of the Casimir invariant is carried out. WKB calculations for a specific example yield improved accuracy compared to previous work.

DOI
10.1103/physrevlett.46.895

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A new method is proposed for studying the semiclassical limit for model Hamiltonians which are polynomials in the generators of the Lie algebra SU(n)\mathrm{SU}(n)SU(n). The symmetric representation, characterized by a single independent Casimir invariant is mapped onto a subspace of a set of canonical coordinates, and a systematic expansion in the reciprocal of the Casimir invariant is carried out. WKB calculations for a specific example yield improved accuracy compared to previous work.
We examine the semiclassical limit of the quantum energy spectrum in many dimensions: by means of a WKB-like ansatz leading to Einstein-Brillouin-Keller (EBK) quantization, by means of a path integral, hence associating a bound state with a particular classical periodic trajectory, and by the Birkhoff-Gustavson (BG) transformation to action-angle variables. We extend the EBK method to many-fermion systems using coherent states; and apply both EBK using surfaces of section, and the BG transformation to an SU(3) schematic nuclear shell model. We describe a new algorithm for finding periodic trajectories of a Lagrangian system with polynomial potential. It is applied to the Henon-Heiles system with good results, and these trajectories are used to quantize the system. The EBK and BG methods have some success, while periodic trajectory quantization fails. We discuss possible reasons for this failure and future approaches to these problems.
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.
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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.