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ak416
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This is a question about thermal physics. There's this partition function Z = sum over all states s of the system ( exp(-E_s/T)). And its just used to calculate the probability of any state by taking the Boltzman factor exp(-E_s/T) of that state and dividing over the partition function. Theres one question that asks to show that the partition function for a combined system, Z(1and2) = Z(1)*Z(2). I understand the way its proved, you just take a double sum and say that E(1and2) =E(1)+E(2), so you can separate the sums. But by using a double sum arent you possibly overcounting some states? For example if E_s1 + E_s2 = 1 + 3 and E_s1 + E_s2 = 3 + 1, also 2+2... Shouldnt this just count as one state of the system, call it E_s = 4. Or would it be better to just keep it this way and then whenever you want to count the probability of observing an E = 4 of the double system, you would have to add all the possible boltzman factors corresponding to E = 4. I think i just answered my question..., but I am just wondering what's the right way to think about it, because there's another part in the book about ideal gases talking about how when you have a system with distinct particles you can overcount, but when you have a system with identical particles, you have to multiply the partition function by 1/N! They also say at the end that in our argument we have assumed that all N occupied orbitals (i guess they mean energy levels) are always different orbitals. How does this change anything?