Dynamical symmetries of the time-dependent Lewis-Riesenfeld harmonic oscillator

Neil J. Günther

Elaboration is given to a previous discussion concerning dynamical symmetries of the Lewis–Riesenfeld time-dependent harmonic oscillator. A boson operator formalism is used to define a generator of impicitly time-dependent unitary transformations. Bilinear combinations of the transformed boson operators are shown to span the symplectic algebra Sp(2n,IR) regarded as a larger noninvariance group for the Lewis–Riesenfeld Hamiltonian. The multiplet structure of the embeddings, Sp(2n) ↓SU(n), and SU (n+1) ↓SU(n), is determined for the case n=3, using established branching rules. Boson operator realizations are presented for the multiplet structures in each SU(n) declension of these dynamical groups.

Ettore Salusti B, Francesco Zirilli

M. Harrat, N. Mebarki, A. Redouane Salah

Kyoungho Cho, S P Kim

For a time-dependent harmonic oscillator we show a connection between the integrals of classical equations of motion and the Lewis-Riesenfeld (LR) invariant, and express the LR invariant in terms of the integrals of classical equations of motion. As applications we find the LR invariant for the Guth-Pi Hamiltonian and the asymptotic form of the LR invariant for a general time-dependent harmonic oscillator.

B. Bassetti, E. Montaldi, Mario Raciti

Stanley Santiago Coelho, Lucas Queiroz, Danilo T. Alves

Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.