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Springer eBooks

Tunneling and the semiclassical propagator: A new perspective

Neepa T. Maitra, Eric J. Heller

Published Apr 7, 2008• 1 citation
Publisher
DOI
10.1007/bfb0105971

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•Jan 1, 2001•0 citations
We discuss how the semiclassical approximation arises within the (classically improved) Langevin quantization. We derive a new representation of the semiclassical propagator (at imaginary time) in the form of a white noise average, which is valid over an arbitrary (imaginary) time interval. Although our result is ultimately equivalent to the standard representation of the semiclassical propagator, it nevertheless turns out to be more advantageous in dealing with certain problems that involve tunnelling and metastability.
A continuous path integral is used to obtain the semiclassical expressions of quantum propagators by means of coherent states in the holomorphic representation together with the appropriate boundary conditions. It allows us to express the total actions including boundary terms, classical trajectories, and extra phases in the stationary phase approximation. In particular, in the semiclassical expression for the quantum spin propagator, we pay attention to the extra quantum phase discovered by Solari and Kochetov. It is shown that this extra phase is related to an anomaly in the fluctuation determinant. It is verified that once this extra phase is included, the semiclassical propagator leads to the correct short time behaviour to O(T²), and shows its consistency under dissection of the path. In the application to the instanton methods, the extra quantum phase yields the correct formula for the tunnel splitting including both the exponential factor and the prefactor. The quantum phase leads to the correct spectrum with a shift of the spin quantum number by 1/2. We apply the semiclassical propagators and the tunnel splitting to a few models in magnetic molecular systems. It is shown that the formula gives the correct exponential factor and prefactors for quantum tunneling of a large spin if the extra quantum phase is taken into account.