Semiclassical approximation with zero velocity trajectories

TL;DR

This paper introduces a local semiclassical method using zero velocity trajectories to approximate quantum wavefunctions, enabling efficient calculation of tunneling probabilities and eigenvalues with high accuracy.

Contribution

The paper presents a novel semiclassical approach based on zero velocity trajectories that is local, easily extendable to imaginary time, and effective for specific quantum calculations.

Findings

Accurate tunneling probability calculations with a single trajectory.

Precise low energy eigenvalues matching exact quantum results.

Method is local and does not require trajectory communication.

Abstract

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.