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## Homework Statement:

- I don't know if I am correct or not....

## Relevant Equations:

- Probabilities

b)

Consider P_j(n) as a macrostate of the system,

Bosons: P_1(1) = P_2(1) = 1/2*1/2=1/4 ,P_1(2)=P_2(2)=1/2*1/2=1/4

Fermions: P_1(1)=P_2(1)=1 (Pauli exclusion principle), P_1(2)=P_2(2)=0

Different species: P_1(1)=P_2(1) = 2*1/2*1/2=1/2 (because there are two microstates with corresponding to one atom in each well, atom A in well 1, atom B in well 2 and atom B in well A, atom A in well B).

c)

Bosons: Suppose well 2 has an energy of E. 3 microstates as above

Z = 1 + e^(-2bE) + e^(-bE),

P_1(1)=P_2(1) = e^(-bE)/Z, P_1(2)=1/Z, P_2(2)=e^(-2bE)/Z

Fermions: P_1(1)=P_2(1)=1 (Pauli exclusion principle), P_1(2)=P_2(2)=0

Distinct species: Z = 1 + e^(-2bE) + 2e^(-bE)

P_1(1)=P_2(1) = 2e^(-bE)/Z, P_1(2)=1/Z, P_2(2)=e^(-2bE)/Z

d)

Bosons:

Z = e^(-bU) + e^(-2b(E+U)) + e^(-bE)

P_1(1)=P_2(1) = e^(-bE)/Z, P_1(2)=e^(-bU)/Z, P_2(2)=e^(-2b(E+U))/Z

Fermions: P_1(1)=P_2(1)=1 (Pauli exclusion principle), P_1(2)=P_2(2)=0

Distinct species: Z = e^(-bU) + e^(-2b(E+U)) + 2e^(-bE)

P_1(1)=P_2(1) = 2e^(-bE)/Z, P_1(2)=e^(-bU)/Z, P_2(2)=e^(-2b(E+U))/Z

For U->infinity, Z->e^(-bE) for bosons, 2e^(-bE) for distinct atoms,

P_1(1)=P_2(1) -> 1 for bosons and distinct atoms and P_2(2)=P_1(2)->0 as U->infinity.

I am not sure that whether this method is correct for distinct atoms and is there any more general methods that work for more than 2 wells? For example, I don't know what to do if the question ask: What is the probability of p_j(n) if there are N atoms and J wells....