Physics – Condensed Matter – Mesoscale and Nanoscale Physics
Scientific paper
1998-08-12
Ann. Phys. (N.Y.), vol.272, pp196-242 (1999).
Physics
Condensed Matter
Mesoscale and Nanoscale Physics
Submitted to Annals of Physics. This work has been submitted to Academic Press for possible publication
Scientific paper
10.1006/aphy.1998.5887
It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.
Creagh Stephen C.
Whelan Niall D.
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