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International Journal of Modern Physics B

UNCERTAINTY RELATIONS FOR THE TIME-DEPENDENT SINGULAR OSCILLATOR

Jeong Ryeol Choi, In Hyun Nahm

Published Apr 19, 2006• 1 citation
Publisher

Abstract

Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216ℏ which is somewhat larger than that of the standard harmonic oscillator, ℏ/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)ℏ.

DOI
10.1142/s0217979206033802

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Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216ℏ which is somewhat larger than that of the standard harmonic oscillator, ℏ/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)ℏ.
We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.
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.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.