Approximation of the Propagator for the Harmonic Oscillator

Logan A. Thomas

University of Massachusetts Dartmouth Mathematics Department

Professor Dana Fine

University of Massachusetts Dartmouth Mathematics Department

Abstract

Equations

Results

The propagator, which is a function of an initial position, a final position and a time, completely determines the quantum mechanics of a point particle in a given potential. The approximate propagators are explicitly calculated for the case of the harmonic oscillator potential. This project presents numerical investigation of the accuracy of certain approximate propagators for various values of N and t. This is feasible for this case, as the exact propagator is known.

One of the equations used in this project is the exact propagator for the harmonic oscillator, the other equation used is an estimation of the first equation, a first-order propagator of the harmonic potential.

Simplification

Because these two equations are hard to put into a computing software to analyze the percent error between the actual propagator and the estimate. In order to simplify we will have to apply certain conditions that are specified in the derivation of the estimate. The conditions applied are:

Harmonic Oscillator

Conclusions

The propagator for the harmonic oscillator gives a probability amplitude of a point particle moving to a new place at a given time. The reason we can compare experimental approximations of the propagator because the actual Propagator for a 1 dimensional particle is known. The experimental first order Propagator was derived by N. Makri and W. Miller

Though this approximation to the harmonic oscillator is quite accurate as N increases, it could use some improvement, It is a future goal of this project to offer improvements to this approximation.

Through some substitution and manipulation of these two equations, the simplified equations are: