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College Physics

Exact wave function of the time-dependent harmonic oscillator in the invariant representation

Feng Jin

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The two quantum invariant operators are constructed from the time dependent Harmiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schrdinger equation for the system is derived by using these invariant operators.

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The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schr\"odinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the invariant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of unitary transformed operators, i.e., squeezing operators.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.
The time-dependent Schrdinger equation for the harmonic oscillator with time-dependent frequency ω=ω(t) and mass m=m(0)ω(0)ω(t)exp[2c∫t0ω(t)dt] can be solved by means of simple transformation and separation of variables.The wave functions are obtained for c21,c2=1 and c21,respectively.When |x|→∞,the wave function for c21 tends to zero but it is not so for c2≥1.Finally,some special cases are given for the time-dependent harmonic oscillator and one of them is similar to a damped harmonic oscillator.
An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely transforming the mathematical representations of quantum quantities belonging to the simple harmonic oscillators. As a case in point, the eigenfunctions of the invariant operator in the original systems are found through inverse transformation of the well-known eigenfunctions associated with the simple harmonic oscillators.
By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.