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I want to calculate the number of ways for putting N_1,N_2,N_3,...,N_i,... bosons in energy levels with degeneracies g_1,g_2,g_3,...,g_i,....
The particles are indistinguishable and there can be any number of particles in a state.
The level with degeneracy g_i has N_i particles in it.The first of this N_i particles has g_i states to choose.The second,again has g_i choices and the same for all of them.So there are g_i^{N_i} ways for putting N_i particles in g_i sates.But the particles are indistinguishable so their order is not important and so g_i^{N_i} reduces to \frac{g_i^{N_i}}{N_i!}. So the number of ways for putting N_1,N_2,N_3,...,N_i,... bosons in energy levels with degeneracies g_1,g_2,g_3,...,g_i,... is:
<br /> \prod_i \frac{g_i^{N_i}}{N_i!}<br />
But the above result will give us sth like Boltzmann distribution,not Bose-Einstein's and we know that the answer should be like below:
<br /> \prod_i \frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}<br />
But what is wrong?
In deriving my formula,I assumed only that the particles are indistinguishable and don't follow Pauli's principle,the same assumptions made for bosons.So what was different?
Thanks
The particles are indistinguishable and there can be any number of particles in a state.
The level with degeneracy g_i has N_i particles in it.The first of this N_i particles has g_i states to choose.The second,again has g_i choices and the same for all of them.So there are g_i^{N_i} ways for putting N_i particles in g_i sates.But the particles are indistinguishable so their order is not important and so g_i^{N_i} reduces to \frac{g_i^{N_i}}{N_i!}. So the number of ways for putting N_1,N_2,N_3,...,N_i,... bosons in energy levels with degeneracies g_1,g_2,g_3,...,g_i,... is:
<br /> \prod_i \frac{g_i^{N_i}}{N_i!}<br />
But the above result will give us sth like Boltzmann distribution,not Bose-Einstein's and we know that the answer should be like below:
<br /> \prod_i \frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}<br />
But what is wrong?
In deriving my formula,I assumed only that the particles are indistinguishable and don't follow Pauli's principle,the same assumptions made for bosons.So what was different?
Thanks
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