00:01
So in this question we're told we have two non -interacting spin -zero particles, and the energy of the n -thuygens state is n times epsilon, and we also have the degeneracy of the n -thigens state for each particle is 2n plus 1.
00:26
So let's say that one of them is in state n, and one of them is in state m, then the combined energy, e -n -m, is going to be n plus m times epsilon.
00:38
And the combined degeneracy, g and m, is just going to be the product of the two.
00:43
2m plus 1, 2n plus 1.
00:49
So let's say that their total energy has fixed value, e equals n -epsilon.
00:57
Then the partition function is going to be the sum over possible states of e to the minus beta times the energy of that state.
01:14
So the possible states are going to be indexed by n and m.
01:17
But what we know is that if enm is n plus m epsilon is equal to n -epsalon, then we know that m is going to be n minus n.
01:38
So actually z is going to be the sum over from n equals zero to n of, sorry, this should have a factor of the degeneracy.
01:53
Well, no, because we're summing over all possible states, that includes the degeneracy.
02:00
But if we're summing over n equals zero to n, then what we're going to have is we need a factor of the degeneracy, g n, n minus n, times e to the minus beta, times the energy.
02:15
But the energy is just n epsilon.
02:17
So that's the same every time.
02:19
So what we actually have here is e to the minus beta n epsilon, times the sum from n equals 0 to big n of this particular degeneracy, which is 2n plus 1, times 2n minus 2 little n plus 1.
02:40
So let's break this sum down.
02:44
It's the sum from n equals 0 to big n of minus 4n squared 4 n, but then we're going to get a minus 2n plus 2n, so that doesn't give us anything...