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445lec20

# 445lec20 - PHYS 445 Lecture 20 Quantum statistics Lecture...

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PHYS 445 Lecture 20 - Quantum statistics 20 - 1 © 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited. Lecture 20 - Quantum statistics What's Important: quantum statistics Text : Reif Quantum statistics The quantum approach to mechanics can be formulated in a statistical form using expectation values or means. For example, the mean energy of a classical gas (without momentum or position-dependent forces) can be cast as E = d 3 q 1 ... d 3 q N E exp( - ßE ) d 3 p 1 ... d 3 p N d 3 q 1 ... d 3 q N d 3 p 1 ... d 3 p N = E exp( - ßE ) d 3 p 1 ... d 3 p N d 3 p 1 ... d 3 p N (20.1) For a single particle in contact with a heat bath, this just becomes E = E exp( - ßE ) d 3 p d 3 p . (20.2) In other words, take the observable of interest ( E here), evaluate it using some weight function [ exp(- ßE ) ], then make sure the weight function is properly normalized (divide by d 3 p ). Quantum mechanics can be formulated in the same way. As you know from PHYS 285, Heisenberg's uncertainty principle prevents us from knowing the precise position and momentum of particles. Hence, we take averages or means: E = E d 3 p d 3 p (20.3) or x = x d 3 x d 3 x (20.4) The quantity is called the density matrix and is analogous to the Boltzmann weight. In terms of the quantum wavefunctions , is defined as * . (20.5) Now, all that really counts as far as observables ( e.g. < E > or < x >) are concerned is the density matrix, not the wavefunction. Nevertheless, quantum mechanics was

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