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Journal of Mathematical Physics

Unitary declension of dynamical symmetries for the time-dependent harmonic oscillator

Neil J. Günther

Published Jul 1, 1979• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.524214

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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.
Dynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time-dependent oscillator: H (t) = 1/2 𝒥3i[pipi+w2(t) qiqi]. This is achieved by finding a simple expression for the time-dependent Hamiltonian in the interaction picture: H=Ho+gH1. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time-dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.
By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters.
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.
We construct Lewis–Riesenfeld invariants from two-dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic sp(4) Lie algebra. This allows for an alternative systematic approach to find Lewis–Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time-dependent Dyson maps from their respective time-independent counterparts in the reference system.
We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.