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 de±nitions as follows: F =-kT ln( Z ) , U = ± H ² = ∑ 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 = ² k
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.