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Physica Scripta

Conserved quantities for the density matrix and the degree of statistical mixing

PAUL M. RADMORE, Stephen Mark Barnett, Arthur Marshall Stoneham

Published Aug 1, 1990• 2 citations
Publisher

Abstract

We investigate the conserved quantities for a closed, N -state quantum system. A simple derivation from the equation of motion for the density matrix shows that there is an infinite number of such conserved quantities. However, only the first N of these constants of the motion are independent. These constants uniquely determine the degree of mixing in the system, which is itself conserved. We discuss the relationship between these results, the properties of Hamiltonians under unitary transformation and van Vleck's principle of spectroscopic stability.

DOI
10.1088/0031-8949/42/2/001

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We consider an NNN fermion system at low temperature TTT in which we encounter special particle number values NmN_mNm​ exhibiting special traits. These values arise in focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling constant of the Hamiltonian, the DMs stay constant for all NNN-values, but experience sudden jumps at the NmN_mNm​. For a quantum state described by the matrix ρ\rhoρ, its purity is expressed by Trρ2Tr \rho^2Trρ2 and then the degree of mixture is given by 1−Trρ21 - Tr \rho^21−Trρ2, a quantity that coincides with the entropy SqS_qSq​ for q=2q=2q=2. Thus, Tsallis entropy of index two faithfully represents the degree of mixing of a state, that is, it measures the extent to which the state departs from maximal purity . Macroscopic manifestations of the degree of mixing can be observed through various physical quantities. Our present study is closely related to properties of many-fermion systemsn that are usually manipulated at zero temperature. Here we wish to study the subject at finite temperature. Gibbs' ensemble is appealed to. Some interesting insights are thereby gained.