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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Phase space representation of quantum mechanics in terms of coherent states

Michele Cini, Maurizio Serva

Published Jul 1, 1992• 1 citation
Publisher
DOI
10.1007/bf02728568

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The phase-space picture of quantum mechanics and some examples illustrating it are presented. Since the position and momentum are c numbers in this picture, it is possible to introduce the concept of phase space in quantum mechanics. The uncertainty relation is stated in terms of an area element in phase space, whose minimum size is Planck’s constant. Area-preserving canonical transformations in phase space are therefore uncertainty-preserving transformations. The wave-packet spread, coherent-state representation, and squeezed states of light are discussed as illustrative examples.
An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.