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Acta Physica Sinica

Exact wave function of dual-coupled two-dimensional harmonic oscillators with time-dependent and anisotropic mass and frequency

Ling Rui-liang, Feng Jin-fu, Yun Hu, 常熟理工学院物理与电子工程系，常熟 215500

Published Jan 1, 2010• 2 citations
Publisher

Abstract

Using a special rotational transformation of coordinates and momentum，we solve successfully the diagonalization of Hamiltonian. Furthermore，we obtain the exact wave function of dual-coupled two-dimensional harmonic oscillators with time-dependent and anisotropic mass and frequency by means of trial function methed. The wave function is also verified.

DOI
10.7498/aps.59.759

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Using a special rotational transformation of coordinates and momentum，we solve successfully the diagonalization of Hamiltonian. Furthermore，we obtain the exact wave function of dual-coupled two-dimensional harmonic oscillators with time-dependent and anisotropic mass and frequency by means of trial function methed. The wave function is also verified.
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.
The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion.Mathematically,the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator.Based on the invariants obtained here,the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate.
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.