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American Journal of Physics

Path integral for the quantum harmonic oscillator using elementary methods

Scott M. Cohen

Published Jun 1, 1998• 23 citations
Publisher

Abstract

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.

DOI
10.1119/1.18896

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Theoretical Physics•Mar 22, 2017•0 citations
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.