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Communications in Theoretical Physics

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

Fan Hong‐Yi, Lu Hai-Liang

Published Oct 1, 2006• 4 citations
Publisher

Abstract

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1) algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis–Riesenfeld's invariant operator theory of treating time-dependent harmonic oscillators.

DOI
10.1088/0253-6102/46/4/006

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We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1) algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis–Riesenfeld's invariant operator theory of treating time-dependent harmonic oscillators.
A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.
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.
The unitary operator V ( t ) that diagonalizes the time-dependent quadratic Hamiltonian (TDQH) into a time-dependent harmonic oscillator (TDHO) is obtained using a Lie algebra. The method involves a factorization of the TDQH into a TDHO through a unitary Bogoliubov transformation in terms of creation and annihilation operators with time-dependent coefficients. It is shown that this operator can be easily achieved by means of the factorization, together with the commonly known Wei–Norman theorem. We discuss the conditions under which this unitary operator converges to the evolution operator U ( t ) of the Schrödinger equation for the TDQH, giving then a straightforward calculation of the evolution operator with respect to the procedures published in the literature.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.