Tunneling with Tamm-Dancoff method

TL;DR

This paper compares semi-classical and Tamm-Dancoff methods to estimate quantum tunneling energy splitting in one-dimensional systems, validating semi-classical results and exploring various potentials.

Contribution

It introduces the use of Tamm-Dancoff method for precise numerical tunneling calculations and compares it with semi-classical approximation across different potentials.

Findings

Tamm-Dancoff provides accurate numerical results for finite coupling.

Semi-classical approximation is validated in weak coupling regimes.

Different potentials exhibit consistent tunneling behavior.

Abstract

The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approximation the splitting is calculated in the weak coupling limit. The other approach, Tamm-Dancoff method gives very precise numerical results for finite coupling. Comparing these two approaches we are able to confirm results of the semi-classical approximation and determine range of its applicability. Several potentials are considered: anharmonic double well, cosine potential and cosine potential in periodic space with finite number of minima. We also address a non-generic problem of anharmonic triple well potential.