Lect9_ThermalEquilibrium

# Lect9_ThermalEquilibrium - Review of Density Matrix A more...

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Review of Density Matrix A more general description of a quantum system is via a density operator, ! . The density operator description incorporates ignorance into the theory Answers the question: What equation do we use if we don’t know the exact state of our system? The general form of a density operator is: Here p j is the (classical) probability to be in state | " j # . The normalization condition is: Tr { }=1 For a pure state we have a single state with probability 1, so that: When the state is not pure, the form of the density operator depends on the situation The expectation value of any operator is given by: j j j j p # = = { } m A m A Tr A m " = = Quantum Description of a System in Thermodynamic Equilibrium Having the most information about a system means knowing the exact state vector | j # . Having no knowledge of a state means that it is equally likely to be in any state in the N dimensional Hilbert space: It should be clear that in this sense ignorance is equivalent to microscopic ‘disorder’ in classical statistical mechanics The measure of ignorance is called the Entropy The entropy is a measure of the size of the available state-space, subject to the constraint of fixed mean energy N I n n N N n = = ! = 1 1

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Von-Neumann Entropy The entropy should be maximum for the state ! =I/N and minimum for the pure state There are many such functions of . The one which gives a result which agrees with experiment is the ‘Von-Neumann Entropy’: Method to evaluate the matrix logarithm ln( A ) 1. First find the eigenvectors and eigenvalues of \$ . Using these as a basis gives:
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## This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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Lect9_ThermalEquilibrium - Review of Density Matrix A more...

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