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Physical Review

Wave Functions of a Harmonic Oscillator

Jerzy F Plebanski

Published Mar 15, 1956• 52 citations
Publisher

Abstract

By means of some special unitary operators, theorems are obtained extending classical properties of of Kennard's packets (classical motion of the center, and periodic time-depencence of $\ensuremath{\Delta}x$) over all solutions of the oscillator wave equation. These results are also generalized for the quantum field theory. Using a special "functional representation," nonspreading wave packets with classical properties are given for a general free field.

DOI
10.1103/physrev.101.1825

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By means of some special unitary operators, theorems are obtained extending classical properties of of Kennard's packets (classical motion of the center, and periodic time-depencence of \ensuremathΔx\ensuremath{\Delta}x\ensuremathΔx) over all solutions of the oscillator wave equation. These results are also generalized for the quantum field theory. Using a special "functional representation," nonspreading wave packets with classical properties are given for a general free field.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.
The dynamical properties of the wave and particle aspects of the harmonic oscillator can be studied with the help of the time-dependent Schroedinger equation (SE). Especially the time-dependence of maximum and width of Gaussian wave packet solutions allow to show the evolution and connections of those two complementary aspects. The investigation of the relations between the equations describing wave and particle aspects leads to an alternative description of the considered systems. This can be achieved by means of a Newtonian equation for a complex variable in connection with a conservation law for a nonclassical angular momentum-type quantity. With the help of this complex variable, it is also possible to develop a Hamiltonian formalism for the wave aspect contained in the SE, which allows to describe the dynamics of the position and momentum uncertainties. In this case the Hamiltonian function is equivalent to the difference between the mean value of the Hamiltonian operator and the classical Hamiltonian function.
By using a test-function method, we construct nnn exact solutions of a quantum harmonic oscillator with a time-dependent "spring constant". Any nnn-th solution describes a wave-packet train consisting of n+1n+1n+1 packets. Its center oscillates like the classically harmonic oscillator with variable frequency, and width and highness of each packet change simultaneously. When the deformation is small, it behaves like a soliton train, and the large deformation is identified with collapse and revival of the wave-packet train.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.