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Journal of Mathematical Physics

Symbols of operators and quantum evolution

I Borsari, S. Graffi

Published Sep 1, 1994• 1 citation
Publisher

Abstract

An algebraic procedure to compute the higher order symbols of the Heisenberg quantum observables is explicitly implemented. As an application, it is proven in a very simple way that a time evolution under which the symbol of any Heisenberg quantum observable is exactly given by the evolution along the classical flow is necessarily generated by a quadratic Hamiltonian.

DOI
10.1063/1.530863

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An algebraic procedure to compute the higher order symbols of the Heisenberg quantum observables is explicitly implemented. As an application, it is proven in a very simple way that a time evolution under which the symbol of any Heisenberg quantum observable is exactly given by the evolution along the classical flow is necessarily generated by a quadratic Hamiltonian.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.
We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.