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arXiv:quant-ph/9701002

Quantum theory of motion of a time-dependent harmonic oscillator in the pilot-wave theory

Jeong‐Young Ji, Kwang‐Sup Soh

Published Jan 3, 1997• 1 citation
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Abstract

The de Broglie-Bohm quantum trajectories are found in analytically closed forms for the eigenstates and the coherent state of the Lewis-Riesenfeld (LR) invariant of a time-dependent harmonic oscillator. It is also shown that an eigenstate (a coherent state) of an invariant can be interpreted as squeezed states obtained by squeezing an eigenstate (a coherent state) of another invariant. This provides ways for a whole description of squeezed states.

DOI
10.48550/arxiv.quant-ph/9701002

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The de Broglie-Bohm quantum trajectories are found in analytically closed forms for the eigenstates and the coherent state of the Lewis-Riesenfeld (LR) invariant of a time-dependent harmonic oscillator. It is also shown that an eigenstate (a coherent state) of an invariant can be interpreted as squeezed states obtained by squeezing an eigenstate (a coherent state) of another invariant. This provides ways for a whole description of squeezed states.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
A closed solution of the Schrdinger equation for a timedependent two coupled harmonic oscillators including a driving part and a twomode coupled part is derived by adopting the LewisRiesenfeld invariant theory and properly chosen Hermitian invariant operators. The evolution operator of the system and the condition for generating twomode squeezed states are obtained. It is shown that the quantum fluctuation of squeezed states of the system is independent of the driving part, but depends on the initial states of the system.
We show that the nonlinear auxiliary equation for the invariant introduced by Lewis and Riesenfeld for a time-dependent harmonic oscillator is satisfied by the amplitude of a complex soltuion to the classical equation of motion. One-parameter squeezed Gaussian states are found whose parameter denotes the mixing of positive and negative frequency solutions. The minimization of the energy expectation value is shown to provide a criterion to select the vacuum state. PACS number(s): 42.50.D; 03.65.G; 03.65.-w Typeset using REVTEX 1 Harmonic oscillators have played many important roles in quantum physics, partly because they are exactly solvable quantum mechanically and partly because any system around an equilibrium can be approximated as a harmonic oscillator system. As a non-stationary system, a time-dependent quantum harmonic oscillator can also be exactly solved. One can find typical time-dependent harmonic oscillators in a system of harmonic oscillators interacting