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Physical Review A

Wentzel-Kramers-Brillouin method in the Bargmann representation

André Voros

Published Dec 1, 1989• 154 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.6814

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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.
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 apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.
A long-standing problem and major shortcoming of semiclassical (WKB) bound-state wave functions is their divergence at the classical turning points. Presently available regularization schemes are accurate but rather complicated. It is shown how finite wave functions can be simply obtained by reorganizing the WKB perturbation expansion in powers of ℏ and retaining appropriate higher-order terms. The wave functions obtained by this method are compared with exactly known ones for the harmonic-oscillator and Morse potentials.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
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 WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.