Quantum/Classical Phase Space Analogies

Charles Jaffé, Sheldon Kanfer, Paul W. Brumer

We introduce an analog, in classical phase space, of the fundamental building block of time dependence in quantum mechanics, i.e., the superposition state. Applications to two-level systems and quantum beats illustrate this conceptually pleasing phase-space picture of the phase interference between energy eigenstates which comprises pure-state quantum dynamics.

Wesley E. Brittin

Garnet N. Ord

John R. Klauder

The quantum analog of the classical phase space is a complex Hilbert space, and the quantum analog of a phase-space point is a vector lying in this Hilbert space. A complex Hilbert space has many properties in common with a complex, _nite-dimensional, Euclidean space. In particular, such a space is a linear vector space with an inner product.

Wei-Min Zhang, D.H. Feng, Jian-Min Yuan

Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (\ensuremath{\subset}scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$. The phase-space representation of the system is realized on scrG/scrH, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of ``quantum chaos'' via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.

Daniela Dragoman, Mircea Dragoman