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Physical Review A

Group-theoretical approach to the classical and quantum oscillator with time-dependent mass and frequency

Gabriella Profilo, Giulio Soliana

Published Aug 1, 1991• 41 citations

Abstract

We investigate the classical and quantum oscillator with time-dependent mass and frequency, in the framework of the Lie group theory. Our analysis shows that both systems possess five time-dependent Noether invariants, where only two of them are functionally independent. In the classical case, three of these invariants form the su(1,1) Lie algebra under the Poisson bracket operation, while in the quantum case the corresponding invariants form the su(1,1) Lie algebra under the commutator operation. Since this algebra underlies the noncompact group SU(1,1), we are thus able to define unambiguously the dynamical group related to the generalized quantum oscillator. Furthermore, the Hamiltonian of this system can be expressed in terms of the Noether invariant operators satisfying the su(1,1) Lie algebra. This form is useful in the evaluation of the energy spectrum and in the study of coherence and squeezing. Some particular cases are briefly illustrated. The other two Noether invariants can be interpreted as the quadrature-phase amplitudes involved in the quantum nondemolition measurements and in the problem of the generation of coherent and squeezed states.

DOI: 10.1103/physreva.44.2057

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We investigate the classical and quantum oscillator with time-dependent mass and frequency, in the framework of the Lie group theory. Our analysis shows that both systems possess five time-dependent Noether invariants, where only two of them are functionally independent. In the classical case, three of these invariants form the su(1,1) Lie algebra under the Poisson bracket operation, while in the quantum case the corresponding invariants form the su(1,1) Lie algebra under the commutator operation. Since this algebra underlies the noncompact group SU(1,1), we are thus able to define unambiguously the dynamical group related to the generalized quantum oscillator. Furthermore, the Hamiltonian of this system can be expressed in terms of the Noether invariant operators satisfying the su(1,1) Lie algebra. This form is useful in the evaluation of the energy spectrum and in the study of coherence and squeezing. Some particular cases are briefly illustrated. The other two Noether invariants can be interpreted as the quadrature-phase amplitudes involved in the quantum nondemolition measurements and in the problem of the generation of coherent and squeezed states.

For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of

SU(2)

and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of

SU(1,1)

. We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of

SU(1,1)

for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.

Taking the generalized time-dependent harmonic oscillator (GTHO) and singular time-dependent harmonic oscillator (STHO) as illustrative examples, we discuss the connection between the quantum basic invariants and classical Noether's theorem and, for the system, find: (1) The classical basic invariants and general form of classical invariants. (2) The current algebra and corresponding operator algebra. (3) A kind of converse of the Noether's theorem. (4) The connection between the so-called symmetry algebra ({ G k } algebra) and the classical current algebra through the operator algebra.

The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.

Using the transformations from paper I, we show that the Schrödinger equations for (1) systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators all have isomorphic Lie space–time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is demonstrated.

We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2ν}\partial_{xx} +h^{(2)}(t)e^{2ν}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P^2 and X^2 terms. We give the specific transformations to a different quantum Schrödinger(TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries and (x,t) representations for number states, coherent states, and squeezed states. These general results include earlier work as special cases.

In this paper, we study the generalized harmonic oscillator with arbitrary time-dependent mass and frequency subjected to a linear velocity-dependent frictional force from classical and quantum points of view. We obtain the solution of the classical equation of motion of this system for some particular cases and derive an equation of motion that describes three different systems. Furthermore, with the help of the quantum invariant method and using quadratic invariants we solve analytically and exactly the time-dependent Schrödinger equation for this system. Afterwards, we construct coherent states for the quantized system and employ them to investigate some of the system's quantum properties such as quantum fluctuations of the coordinate and the momentum as well as the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary system. Finally, we evaluate the dynamical and Berry phases for three special cases and surprisingly find identical expressions for the dynamical phase and the same formulae for the Berry's phase.