Density operator, microcanonical

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Miller et al. [2] showed more than 10 years ago that the CRP can be expressed exactly in terms of the microcanonical density operator,... 

With the microcanonical density operator given by Eq. (2.3), Seideman et al. [3] then showed that Eq. (2.1) for the CRP takes the simpler form... 

The difficult part of Eq. (38) to evaluate is the microcanonical density operator, 8(F = H), which is usually (19) expressed in terms of the outgoing wave Green s function (actually an operator)... 

With the microcanonical density operator given by Eq. (40) (with some choice for e), straightforward algebraic manipulations (also using Eq. (37)) lead to the following even simpler form for the cumulative reaction probability (4b) ... 

The next task is to eliminate the scattering eigenstate from equation (65). The easiest way to do this (at least formally) is to use the fact that the microcanonical density operator S(E — H) can be written in terms of the complete set of scattering eigenstates j/Eav as... 

S E - H) is the microcanonical density operator, pj is a projector onto states with total angular momentum J, and p+ (t) is a projector onto states which are or will be outgoing to the product side. 

The spectral density is most generally expressed as a trace of the microcanonical density operator in Hilbert space... 

As for classical systems, the microcanonical ensemble is used to characterize the macroscopic state of energetically isolated and closed quantum system of fixed volume. The density operator for a quantum microcanonical ensemble is given by... 

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