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International Journal of Modern Physics B

GENERALIZED HANNAY ANGLE FOR THE MOST GENERAL TIME-DEPENDENT HARMONIC OSCILLATOR

Donald H. Kobe, Jiongming Zhu

Published Dec 30, 1993• 2 citations
Publisher

Abstract

The most general time-dependent Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with mass, spring “constant,” and friction (or antifriction) “constant,” all of which are time dependent, that is acted on by a time-dependent force. A generalized Hannay angle, which is gauge invariant, is defined by making a distinction between the Hamiltonian and the energy. The generalized Hannay angle is the classical counterpart of the generalized Berry phase in quantum theory. When friction is present the generalized Hannay angle is nonzero. If the Hamiltonian is (incorrectly) chosen to be the energy, the generalized Hannay angle is different. Nevertheless, in the adiabatic case the same total angle is obtained.

DOI
10.1142/s0217979293003851

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The most general time-dependent Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with mass, spring “constant,” and friction (or antifriction) “constant,” all of which are time dependent, that is acted on by a time-dependent force. A generalized Hannay angle, which is gauge invariant, is defined by making a distinction between the Hamiltonian and the energy. The generalized Hannay angle is the classical counterpart of the generalized Berry phase in quantum theory. When friction is present the generalized Hannay angle is nonzero. If the Hamiltonian is (incorrectly) chosen to be the energy, the generalized Hannay angle is different. Nevertheless, in the adiabatic case the same total angle is obtained.
The most general time-dependence Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with a time-dependent mass, a time-dependent friction (or antifriction) 'constant', and a time-dependent spring constant, acted upon by a time-dependent force. The energy, whose time derivative is the power, is in general different from the Hamiltonian. The generalized Berry phase for a given energy eigenstate has a state-dependent part, which vanishes if there is no damping, and an arbitrary state-independent part. If the Hamiltonian is identified as the energy, both the energy eigenvalues and the generalized Berry phase are different. In the adiabatic limit both approaches give the same total phase, which is the sum of the dynamical and Berry phases.
Canonical structure of a generalized time-periodic harmonic oscillator is studied by finding the exact action variable (invariant). Hannay's angle is defined if closed curves of constant action variables return to the same curves in phase space after a time evolution. The condition for the existence of Hannay's angle turns out to be identical to that for the existence of a complete set of (quasi)periodic wave functions. Hannay's angle is calculated, and it is shown that Berry's relation of semiclassical origin on geometric phase and Hannay's angle is exact for the cases considered.
A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.
In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.
The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.