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Principles of quantum statistical mechanics

The problem of quantum statistical mechanics is the quantum mechanical treatment of an N-particle system. Suppose the corresponding N-particle classical system has Cartesian coordinates

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and momenta

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and Hamiltonian

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Then, as we have seen, the quantum mechanical problem consists of determining the state vector tex2html_wrap_inline469 from the Schrödinger equation

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Denoting the corresponding operators, tex2html_wrap_inline471 and tex2html_wrap_inline473 , we note that these operators satisfy the commutation relations:

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and the many-particle coordinate eigenstate tex2html_wrap_inline475 is a tensor product of the individual eigenstate tex2html_wrap_inline477 :

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The Schrödinger equation can be cast as a partial differential equation by multiplying both sides by tex2html_wrap_inline479 :

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where the many-particle wave function is tex2html_wrap_inline481 . Similarly, the expectation value of an operator tex2html_wrap_inline483 is given by

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Mark Tuckerman
Tue May 9 19:40:24 EDT 2000