Chapter 10: Quantum physics53For a macroscopic system in equilibrium holds[H,ρ] = 0. If all quantumstates with the same energy areequally probable:Pi=P(Ei), one can obtain the distribution:Pn(E) =ρnn=e-En/kTZwith the state sumZ=ne-En/kTThe thermodynamic quantities are related to these definitions as follows:F=-kTln(Z),U=H=∑npnEn=-∂∂kTln(Z),S=-k∑nPnln(Pn). For a mixed state ofMorthonormal quantum states withprobability1/Mfollows:S=kln(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 withStirling’s equation:ln(n!)≈nln(n)-n, and the conditions∑knk=Nand∑knkWk=W. For identical,indistinguishable particles which obey the Pauli exclusion principle the possible number of states is given by:P=kgk!
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