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Journal of Physics A Mathematical and General

On the equivalence of the classical-limit scattering matrix to the Wentzel-Kramers-Brillouin formalism

K.K. Loh, K. S. Low

Published Apr 21, 1991• 7 citations
Publisher

Abstract

The classical-limit scattering matrix (CLSM) formalism was developed by Miller as a simple semiclassical procedure for the analyses of complex scattering and reaction processes. It invokes simple classical dynamics to define trajectories and superposition procedures to obtain interference effects. In one-dimensional systems, Miller (1974) has shown that the CLSM is equivalent to the standard WKB approximation. In this paper, it is shown that the equivalence between the CLSM and WKB extends to general systems. This equivalence between the WKB and CLSM formalisms is attributed to the general equivalence of the h(cross)-expansion procedure in the WKB with the stationary phase approximation used in the CLSM.

DOI
10.1088/0305-4470/24/8/018

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The classical-limit scattering matrix (CLSM) formalism was developed by Miller as a simple semiclassical procedure for the analyses of complex scattering and reaction processes. It invokes simple classical dynamics to define trajectories and superposition procedures to obtain interference effects. In one-dimensional systems, Miller (1974) has shown that the CLSM is equivalent to the standard WKB approximation. In this paper, it is shown that the equivalence between the CLSM and WKB extends to general systems. This equivalence between the WKB and CLSM formalisms is attributed to the general equivalence of the h(cross)-expansion procedure in the WKB with the stationary phase approximation used in the CLSM.
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.
The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrödinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner-Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.
We have obtained a set of coupled differential equations from the continuous limit of the transfer matrix method. Decoupling such a set of equations yields an extension to the Wentzel-Kramers-Brillouin (WKB) approximation for the Schrödinger equation with the position-dependent effective mass (PDEM). In the classically allowed region, the decoupling is to ignore the reflection resulting from the variations of both the potential and effective mass. By considering an infinite-well example with the PDEM, it is shown that the extended WKB approximation can provide not only an estimation to eigenenergies, but also an analytic form to approximate wavefunctions.
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.