Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension

TL;DR

This paper analyzes the behavior of quantum wave functions tunneling through a potential barrier in one dimension, showing that the leading tunneling component is exponentially small and Gaussian for small , with results valid over large and moderate times.

Contribution

It provides a rigorous analysis of the leading order tunneling wave function in the semiclassical limit, establishing Gaussian form and exponential smallness.

Findings

Leading tunneling component is exponentially small in 1/

Tunneling wave function is Gaussian for small 

Results hold for large and moderately large times

Abstract

We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in $1/\hbar$. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small $\hbar$. We prove this for both the large time asymptotics and for moderately large values of the time variable.