Time-Dependent Coupled Harmonic Oscillators: Classical and Quantum Solutions

Abstract

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 invariant method. The exact wave functions are obtained by solving the respective Milne-Pinney equation for each system. We obtain the solutions for the system with m1 = m2 = m0e γt , ω1 = ω01e −γt/2 , ω2 = ω02e −γt/2 and k = k0.

Pre-Schmidt Mode Parameters in Quantum Entanglement

“…For simplification, the kinetic energy portion in Equation ( 1) is brought to rotationally invariant form (Han et al, 1999;McDermott & Redmount, 2004) so that:…”