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 (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 m 1 = m 2 = m 0 e γt , ω 1 = ω 01 e −γt/2 , ω 2 = ω 02 e −γt/2 and k = k 0 .

Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators

“…Notice that Equation ( 13) contain coupling coordinates which are easier to handle with the aid of coordinate transformation [17], that is, by decoupling the coordinates one at a time. Considering first the decoupling of coordinates 𝑥 1 and 𝑥 2 with the matrix given by Equation (14).…”

“…Several mathematical explorations concerning coupled oscillations, including the linear chains and ring geometry, have been addressed [8][9][10][11][12][13][14][15][16][17][18][19][20]. Researches conducted by Hong-Yi [9] and Butanas [19] focus on analyzing the dynamics of three coupled oscillators by solving the wave function and quantum propagator.…”

An investigation of algebraic quantum dynamics for mesoscopic coupled electric circuits with mutual inductance

“…For classical and quantum collective processes, requires that time behavior of coupled harmonic oscillators be considered. Recently the time-dependent coupled harmonic oscillators are studied in [5,6].…”

Time-dependent coupled harmonic oscillators: Comment

“…with constant frequency equal to one. Following Macedo and Guedes [1,5] we could do x = M(t)X and p = 1 M(t) P, such that we would obtain a Hamiltonian that looks time independent…”

“…In fact, eliminating the time dependent mass is not a an simple task, as we show here for the case of two coupled time dependent oscillators Hamiltonian. Given the Hamiltonian [1,5]…”