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Physical Review A

Equation for the WKB approximation and Wigner-Kirkwood expansions for systems with magnetic field

D. Bollé, D. Roekaerts

Published Oct 1, 1984• 13 citations
Publisher

Abstract

We discuss semiclassical expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent, magnetic field. In the process, we derive a new equation for the WKB approximation for such systems. In particular, we provide recursion relations and a diagrammatic method for calculating explicitly the coefficients in these expansions. As an illustration we present the off-diagonal coefficients up to order ${\ensuremath{\hbar}}^{3}$, and the on-diagonal coefficients up to ${\ensuremath{\hbar}}^{6}$ for a uniform magnetic field. We also give the on-diagonal coefficients up to order ${\ensuremath{\hbar}}^{4}$ for a nonuniform field.

DOI
10.1103/physreva.30.2024

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We discuss semiclassical expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent, magnetic field. In the process, we derive a new equation for the WKB approximation for such systems. In particular, we provide recursion relations and a diagrammatic method for calculating explicitly the coefficients in these expansions. As an illustration we present the off-diagonal coefficients up to order \ensuremathℏ3{\ensuremath{\hbar}}^{3}\ensuremathℏ3, and the on-diagonal coefficients up to \ensuremathℏ6{\ensuremath{\hbar}}^{6}\ensuremathℏ6 for a uniform magnetic field. We also give the on-diagonal coefficients up to order \ensuremathℏ4{\ensuremath{\hbar}}^{4}\ensuremathℏ4 for a nonuniform field.
We study the expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent magnetic field in physically relevant parameters. In particular, the high-temperature expansion and a new heavy-mass expansion is discussed. We provide recursion relations and a diagrammatic method to calculate the coefficients in these expansions. We compare these expansions in detail with the semiclassical expansion derived previously [Phys. Rev. A 30, 2024 (1984)]. As an illustration we present the first few coefficients explicitly.
The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.
This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function FFF. For an NNN-body quantum system with Hamiltonian HHH and inverse temperature $\ensuremath{\beta}$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}H}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}{H}_{0}}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}$. Expansion of lnF\mathrm{ln}FlnF in Planck's constant hhh leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\ensuremath{\beta}$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.