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Journal of Mathematical Physics

Quantum systems with time-dependent harmonic part and the Morse index

Jorge Rezende

Published Nov 1, 1984• 22 citations
Publisher

Abstract

A simple relation between two quantum systems with a time-dependent, respectively, time-independent, harmonic part is established. Using this we give a computation, valid for all times, of the Green’s functions of the time-dependent harmonic oscillator with and without a perturbation of the type g/x2. The asymptotic expansion of the wave function in powers of Planck’s constant is discussed using a new representation of the Morse index.

DOI
10.1063/1.526073

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A simple relation between two quantum systems with a time-dependent, respectively, time-independent, harmonic part is established. Using this we give a computation, valid for all times, of the Green’s functions of the time-dependent harmonic oscillator with and without a perturbation of the type g/x2. The asymptotic expansion of the wave function in powers of Planck’s constant is discussed using a new representation of the Morse index.
The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (SHO) system ( ) can be obtained by solving the equation directly. It has a compact expression, which gives correct eigenvalues and eigenfunctions easily. The Green's function G with an additional delta-function potential can be obtained readily. The same technics of solving the Green's function G 0 can be used to solve the eigenvalue problem of the SHO with an generic delta-function potential at an arbitrary site, i.e. . The Wronskians play an important and interesting role in the above studies. Furthermore, the approach can be easily generalized to solve the quantum system of a SHO with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration.
The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.
We consider a simple harmonic oscillator subject to a time-dependent, spatially homogeneous perturbation and show that it is canonically equivalent to a non-perturbed oscillator. The canonical transformation that relates both systems can be implemented as a unitary transformation mapping the perturbed system and the non-perturbed systems onto each other. This unitary transformation allows us to explicitly compute the time dependence of the states of the perturbed system, and in particular to compute the phase that affects all states. The phase turns out to be a classical action along the classical trajectory of the origin (in phase space), along its motion under the action of perturbation. In this simple system, the transition probabilities due to the perturbation can also be computed explicitly. Our approach is independent of the magnitude of the perturbation, and does not require an adiabaticity assumption.
The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.