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Physica A Statistical Mechanics and its Applications

Integrals of motion of pure and mixed quantum systems

V. V. Dodonov, Vladimir I. Man’ko

Published Dec 1, 1978• 28 citations
Publisher
DOI
10.1016/0378-4371(78)90075-4

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We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. In the third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, we find the state liftings and observable liftings corresponding to state vectors and observables respectively in the different pictures of motion. The equations of motion for these quantities are derived. Using the results obtained, problems concerning the integrals of motion are considered from the bundle viewpoint. Necessary and sufficient invariant bundle conditions for a dynamical variable to be an integral of motion are found.
The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decoherence are constructed, the importance of zero-modes in the absolute value [Formula: see text] of V for the generation of pure states from arbitrary mixed states is illustrated. The universal rôle of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.