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American Journal of Physics

Propagator for the simple harmonic oscillator

Nora S. Thornber, Edwin F. Taylor

Published Nov 1, 1998• 30 citations
Publisher

Abstract

Through direct simulation with a computer program, we have verified that the published formula for the propagator of the one-dimensional simple harmonic oscillator (1DSHO), when applied in the straightforward manner in which physicists might interpret it, predicts the wrong development of the wave function for half the future times. This note presents a new and simple derivation of the (known) explicit phase factors of the normalization of the 1DSHO propagator that yield the correct development of any initial wave function for all times.

DOI
10.1119/1.19003

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Through direct simulation with a computer program, we have verified that the published formula for the propagator of the one-dimensional simple harmonic oscillator (1DSHO), when applied in the straightforward manner in which physicists might interpret it, predicts the wrong development of the wave function for half the future times. This note presents a new and simple derivation of the (known) explicit phase factors of the normalization of the 1DSHO propagator that yield the correct development of any initial wave function for all times.
The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.
A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.
Two methods for obtaining the ladder spectrum characteristic of the simple harmonic oscillator are widely used in elementary quantum mechanics. In the first method, that of series solution, quantization is shown to follow from the requirement that the wave function be normalizable. In the second, it is common to derive quantisation in an operator formalism based on factorization of the wave equation. In this latter method the role of normalizability is not customarily discussed. We show explicitly that the same boundary condition implicit in the operator manipulations leads to quantization in both methods.