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Journal of Physics A Mathematical and Theoretical

Weyl-ordered series form for the angle variable of the time-dependent oscillator

Giulio Landolfi

Published Apr 18, 2008• 6 citations
Publisher

Abstract

The definition of the phase variable for a classical time-dependent oscillator as the natural variable canonically conjugated to the Ermakov invariant is revised. Some implications of the result at the quantum level are discussed and an exact formal expression in terms of Weyl-ordered operators is given for the associated phase operator.
DOI
10.1088/1751-8113/41/18/185302

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For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.
The Ermakov Lewis quantum invariant for the time dependent harmonic oscillator is expressed in terms of number and phase operators. The identification of these variables is made in accordance with the correspondence principle and the amplitude and phase representation of the classical orthogonal functions invariant. The relationship between the number and phase operators is established through this invariant as the system evolves from one frequency to another. In the specific case where the excitations represent the photon number, these relations are equivalent to the power density transport equations derived in nonlinear optical processes.
Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of qqq and the eigenstates of ppp. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.
We review some of the difficulties previously encountered in defining the phase operator for finite quantum systems. We then propose the Wigner–Weyl quantization of the angle function φ on phase space as the phase operator, and derive a closed expression for its matrix elements with respect to the Hermite functions. We also determine the quantization of e^{iφ} , which turns out to be a weighted shift operator, its spectrum and that of its adjoint. This is done in the framework of quantization of a certain symbol class of phase space distributions, specialized to those which depend on one variable only. After recalling some results for the position and momentum variables, we apply the scheme to functions of radius or angle. We give necessary and sufficient conditions for operators obtained by quantizing functions of the angle to be elements of \mathscr L^+ [\mathscr S(\mathbf R)] and \mathscr L^+ [\mathscr S (\mathbf R), L^2(\mathbf R)] , and a sufficient condition for boundedness. We then consider the associated questions of commutation relations and uncertainties for operators in \mathscr L^+ [\mathscr S(\mathbf R), L^2(\mathbf R)] , which we define as bilinear forms. As must be the case, the commutator between our phase operator and the number operator exhibits noncanonical terms. Not surprisingly, the Poisson bracket of their phase space symbols also exhibits a noncanonical term.