Quadratic Hamiltonians in phase-space quantum mechanics
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Fecha
1989-07
Tipo
artículo original
Autores
Gadella, Manuel
Gracia Bondía, José M.
Nieto, Luis M.
Várilly Boyle, Joseph C.
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Resumen
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. We 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 appendix.
Descripción
Palabras clave
Moyal propagator, Quantum mechanics in phase space, Quadratic Hamiltonians