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arXiv:1201.6607 · Journal of Mathematical Physics

Wave functions of log-periodic oscillators

Vagner Henrique Loiola Bessa, I. Guedes

Published Jun 1, 2011• 4 citations
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Abstract

We use the Lewis and Riesenfeld invariant method [J. Math. Phys. 10, 1458 (1969)]10.1063/1.1664991 and a unitary transformation to obtain the exact Schrödinger wave functions for time-dependent harmonic oscillators exhibiting log-periodic-type behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with m = m0t/t0 and ω = ω0t0/t, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with m and ω constant.

DOI
10.1063/1.3601739

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We use the Lewis and Riesenfeld invariant method [J. Math. Phys. 10, 1458 (1969)]10.1063/1.1664991 and a unitary transformation to obtain the exact Schrödinger wave functions for time-dependent harmonic oscillators exhibiting log-periodic-type behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with m = m0t/t0 and ω = ω0t0/t, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with m and ω constant.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.