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arXiv:cond-mat/9907197 · Journal of Mathematical Physics

Application of the discrete Wentzel–Kramers–Brillouin method to spin tunneling

Anupam Garg

Published Oct 1, 1998• 22 citations
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Abstract

A discrete version of the Wentzel–Kramers–Brillouin (WKB) method is developed and applied to calculate the tunnel splittings between classically degenerate states of spin Hamiltonians. The results for particular model problems are in complete accord with those previously found using instanton methods. The discrete WKB method is more elementary and also yields wave functions.

DOI
10.1063/1.532563

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A discrete version of the Wentzel–Kramers–Brillouin (WKB) method is developed and applied to calculate the tunnel splittings between classically degenerate states of spin Hamiltonians. The results for particular model problems are in complete accord with those previously found using instanton methods. The discrete WKB method is more elementary and also yields wave functions.
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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.
We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.