Semiclassical calculation of the energy dispersion relation in the valence band of the quantum pendulum

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Abstract
The sine-Gordon soliton is used to calculate the energy dispersion relation of the lowest-lying energy band of the quantum pendulum. The kernel for propagation between two arbitrarily separated minima is approximated, in the limit of large imaginary time, by the sum of the contributions of multisoliton and antisoliton trajectories. The dispersion relation which results has the general form of expressions derived by the tight-binding approximation. In the limit of low probability for barrier penetration, the asymptotic expression for the width of the band turns out to be in perfect agreement with known mathematical results and better by a factor of 0.93 than standard WKB results.