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Physical Review A

WKB wave functions without matching

M. Casas, Angel Ricardo Plastino, Antonio Puente, N. Canosa, R. Rossignoli

Published May 1, 1993• 13 citations
Publisher

Abstract

A method is presented for the construction of WKB wave functions that avoids explicit consideration of any matching between results pertaining to classically allowed and classically forbidden regions. The formalism is based on the maximization of a suitably defined quantal pseudoentropy, subject to the constraints posed by the expectation values of a reduced number of operators, which are evaluated according to special WKB techniques that consider expansions up to orders \ensuremath{\Elzxh} and ${\mathrm{\ensuremath{\Elzxh}}}^{2}$. The algorithm is tested with reference to the harmonic and quartic anharmonic oscillators.

DOI
10.1103/physreva.47.3530

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A method is presented for the construction of WKB wave functions that avoids explicit consideration of any matching between results pertaining to classically allowed and classically forbidden regions. The formalism is based on the maximization of a suitably defined quantal pseudoentropy, subject to the constraints posed by the expectation values of a reduced number of operators, which are evaluated according to special WKB techniques that consider expansions up to orders \ensuremath{\Elzxh} and \ensuremath\Elzxh2{\mathrm{\ensuremath{\Elzxh}}}^{2}\ensuremath\Elzxh2. The algorithm is tested with reference to the harmonic and quartic anharmonic oscillators.
A method is presented for the construction of Wentzel-Kramers-Brillouin (WKB) wave functions that avoids explicit consideration of any matching between results pertaining to classically allowed and forbidden regions. The formalism is based on the maximization of a suitable defined quantal pseudoentropy, subject to the constraints posed by the expectation values of a reduced number of operators, which are evaluated according to special WKB techniques. A self-consistent procedure is developed that simultaneously yields the nodes of the wave function and the information theory parameters.
Using the Frieden and Soffer principle of extreme physical information we present a method for the construction of WKB wave functions that avoids the matching problem between the classically allowed and forbidden regions. When this formalism is applied in conjunction with the semiclassical methods for N-fermion systems, it provides one with Thomas-Fermi and Wigner-Kirkwood densities, valid beyond the turning points that also include shell effects. The method is illustrated with reference to both the sextic bistable and the Morse potential.
The extended WKB method recently presented is applied to the construction of a high-lying semiclassical wave function, corresponding to the quantum numbers (30,30), for the Barbanis Hamiltonian, a well-known two-dimensional nonseparable system; the results are compared with the semiclassical wave function obtained, for the same state, by Davis and Heller, by means of the coherent Gaussians method. The extended WKB method shows that the semiclassical wave function is the sum of four wave functions, differing on parts of the classically allowed region, each of these separately satisfying the Einstein–Brillouin–Keller quantization rules. The results presented here show that our method, which is the direct extension to integrable nonseparable Hamiltonians of the usual WKB scheme, is suitable for the semiclassical quantization of high-lying states also, and enables a closer investigation of the fine structures of the semiclassical wave functions.
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 develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.