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American Journal of Physics

Semiclassical treatment of the double well

Barry R. Holstein

Published Apr 1, 1988• 32 citations
Publisher

Abstract

The double well potential V(x)= 1/4 λ(x2−α2)2 is addressed using both semiclassical path integral and instanton techniques. The basic physics of the two-state system is shown to arise, and energy levels calculated via the two methods are compared.
DOI
10.1119/1.15634

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In the semiclassical limit, the functional-integral description of a quantum-mechanical system is dominated by certain classical paths of motion. We identify the relevant paths for a double-well potential and study how they are affected by coupling to a dissipative environment. When the coupling exceeds a critical value, the system becomes localized in one or the other of the two wells. The classical-path method reproduces previous computations of the critical coupling. This description is applicable for all values of the dissipative coupling and for any low-lying state of the system.