Solution of the Schr dinger equation for time-dependent 1D harmonic oscillators using the orthogonal functions invariant

Abstract

An extension of the classical orthogonal functions invariant to the quantum domain is presented. This invariant is expressed in terms of the Hamiltonian. Unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent parameter. The solution thus obtained is in agreement with the results derived using other methods which invoke the Lewis invariant in their procedures.

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

“…In fact, this classical generator precisely corresponds to the exponential operator found in [16] for a quantized version of the time-dependent harmonic oscillator. It is worth noting that, regardless of the choice of coordinates, G takes the same form in either q, p or Q, P, that is it holds that G q(Q, P), p(Q, P) = G Q, P .…”

“…Further note that the inverse square of the time scaling function ξ(t) precisely corresponds to the Lagrange multiplier λ(s) = dt/ds that is involved in the extended classical Hamiltonian (2.13). Given this result we can now give an explicit solution of the Schrödinger equation as was already shown in [16]:…”

Solution to the Time-Dependent Coupled Harmonic Oscillators Hamiltonian with Arbitrary Interactions

“…In the case of the two oscillators, we have shown how to solve the Hamiltonian for arbitrary functions of time, as formerly it had been solved only when the functions were related in specific ways [23]. We did it by using the orthogonal functions invariant introduced in [36] which allowed us to split the Hamiltonian in such a way that it was left to solve a single time-dependent harmonic oscillator, which is a well-known problem [23,29].…”

“…We note that the Hamiltonian in Equation (31) has been separated in two parts: one of them is a time-dependent harmonic oscillator that depends only onŷ andp y (and powers of them) and therefore there are Ermakov-Lewis methods to solve it; the other depends only onp x (and its powers) and therefore is integrable [36]. In addition, within the framework of shortcuts to adiabaticity [6], Equation (32) provides quick routes to obtain final results of slow adiabatic processes.…”

“…In the following sections, through a series of unitary transformations, some of them time-dependent, we manage to take the Hamiltonian for the two coupled harmonic oscillators to an integrable form. We do so by generalizing some operator methods for the single time-dependent quantum harmonic oscillator introduced in reference [36].…”