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Physica Scripta

Thermal State of General Time-Dependent Harmonic Oscillator with a Singular Perturbation

Jeong-Ryeol Choi

Published Jan 1, 2003• 5 citations
Publisher

Abstract

Taking advantage of the dynamical invariant method and the unitary transformation method together, we obtained quantum-mechanical solutions of the general time-dependent harmonic oscillator with a singular perturbation. The wave functions of the system are represented in terms of the associated Laguerre polynomial. We introduced a density operator satisfying the Leouville–von Neumann equation and used its diagonal elements to evaluate various expectation values in thermal state. We confirmed that the time-dependency of variables such as and 2 in thermal state are exactly the same as that in number state.

DOI
10.1238/physica.regular.068a00036

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Taking advantage of the dynamical invariant method and the unitary transformation method together, we obtained quantum-mechanical solutions of the general time-dependent harmonic oscillator with a singular perturbation. The wave functions of the system are represented in terms of the associated Laguerre polynomial. We introduced a density operator satisfying the Leouville–von Neumann equation and used its diagonal elements to evaluate various expectation values in thermal state. We confirmed that the time-dependency of variables such as and 2 in thermal state are exactly the same as that in number state.
A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to arbitrariness of time variations of parameters. The parametric conditions required for formulating this invariant are definitely specified. By using the unitary transformation method, the invariant operator is transformed to the one that corresponds to three independent simple harmonic oscillators. Inverse transformation of the well-known quantum solutions associated with such a simplified invariant enables us to identify quantum solutions of the coupled original systems. These solutions are exact since we do not use approximations not only in formulating the invariant operator but in the unitary transformation as well. The invariant operator and its eigenfunctions provided here can be used to characterize quantum properties of the systems with various choices of the types of time-dependent parameters.
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.
We investigated the coherent states of nonconservative harmonic oscillator with a singular perturbation. The invariant operator represented in terms of lowering and raising operators. We confirmed that if the difference between two eigenvalues, α and β, of coherent states is much larger than unity, the states |α> and |β> are approximately orthogonal to each another. We calculated the expectation values of various quantities such as invariant operator, Hamiltonian and mechanical energy in coherent state. The mechanical energy of the system described by the Kanai–Caldirola Hamiltonian decreased exponentially depending on γ as time goes by in coherent state.