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phys documents (dragged) 59 - Chapter 10 Quantum physics 53...

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Chapter 10: Quantum physics 53 For a macroscopic system in equilibrium holds [ H, ρ ] = 0 . If all quantumstates with the same energy are equally probable: P i = P ( E i ) , one can obtain the distribution: P n ( E ) = ρ nn = e - E n /kT Z with the state sum Z = n e - E n /kT The thermodynamic quantities are related to these definitions as follows: F = - kT ln( Z ) , U = H = n p n E n = - kT ln( Z ) , S = - k n P n ln( P n ) . For a mixed state of M orthonormal quantum states with probability 1 /M follows: S = k ln( M ) . The distribution function for the internal states for a system in thermal equilibrium is the most probable func- tion. This function can be found by taking the maximum of the function which gives the number of states with Stirling’s equation: ln( n !) n ln( n ) - n , and the conditions k n k = N and k n k W k = W . For identical, indistinguishable particles which obey the Pauli exclusion principle the possible number of states is given by: P = k g k !
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