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Quantum statistics: a formal approach Reminder to axioms of quantum mechanics state vector of the Hilbert space describes quantum mechanical system Observables are represented by hermitian operator  expectation value System in state with and Eigenstates of  Measurement of A yields a n with probability with First consider a truly isolated system
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We stated: Matrix representations: Closer inspection of the density operator of a pure state with Let’s show that in fact
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Now let‘s consider a quantum system interacting with the external world Instead of coherent superposition for isolated system Coupling with the external worlddecoherence State of a system in equilibrium is an incoherent superposition of eigenstates We can recall the idea of a Gibbs ensemble, an infinite collection of systems each in an N-particle eigenstate Simple example for the difference between coherent superposition & statistical mix interference term
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coherent superposition & statistical mix with Next we see that in the canonical ensemble Equivalent considerations can be made for the micro- and grandcanonical ensamble and with is our thermal average in the canonical ensemble
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The notation gets meaning via So far we wrote: Let’s reconsider the Boltzmann factor Eigenstates of N-particle Schroedingder eq. From basis independent
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Let’s consider internal energy U In general
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