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Acta Physica Polonica A

Explicit Computation of the Quantum Phase of the Time Perturbed Harmonic Oscillator

Henryk Gzyl

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.143.19

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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.
A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrodinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
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.
For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.
A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.
A generalization of a driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization, which gives a general quadratic Hamiltonian system, does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous solutions and one particular solution. The kernels are then used to find wave functions that satisfy the Schr\"odinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization, which gives a rather complicated quadratic Hamiltonian, is simply interpreted as applying unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.