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European Journal of Physics

A simple evaluation of a determinant in a path integral calculation of a harmonic oscillator propagator

Kiyoto Hira

Published Sep 10, 2013• 1 citation
Publisher

Abstract

The calculation of a one-dimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical significance. We focus on the evaluation of a determinant resulting from the action integral in the discretized form of the path integral. As a consequence we can make the commonly accepted calculations in the literature simpler and clearer.

DOI
10.1088/0143-0807/34/6/1373

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The calculation of a one-dimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical significance. We focus on the evaluation of a determinant resulting from the action integral in the discretized form of the path integral. As a consequence we can make the commonly accepted calculations in the literature simpler and clearer.
We present a purely analytical method to calculate the propagator for the quantum harmonic oscillator using Feynman’s path integral. Though the details of the calculation are involved, the general approach uses only matrix diagonalization and well-known integrals, techniques which an advanced undergraduate should understand. The full propagator, including both the prefactor and the classical action, is obtained from a single calculation which involves the exact diagonalization of the discretized action for the system.
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.