Combining Partition Functions

Question 1

Supposing the energy of a system depends on two independent, distinct contributions, Eij = E1i + E2j, what does z equal?

Answer 1

z = sum over i of sum over j of exp(-β(E1i+E2j)) = sum over i of exp(-βE1i) * sum over j of exp(-βE2j) = z1*z2

Question 2

What is the general equation for multiple partition functions?

Answer 2

z(N) = z1z2…

Question 3

From the general equation for z, what can we say about other quantities such as u, F etc?

Answer 3

Any functions which depend on ln(z), contributions add: u(N) = u1+u2+…, F(N) = F1+F2+…

Question 4

For a 3D SHO, what does the energy equal?

Answer 4

E(3D) = (n(x)+1.2)ћω + (n(y)+1/2)ћω + (n(z)+1/2)*ћω

Question 5

What does z(3D) equal for the 3D SHO?

Answer 5

z(3D) = z(x)z(y)z(z), but all these are the same (found before), so z(3D) = (z(SHO)^3), and u(3D) = 3u(SHO)

Question 6

What is the equation for the magnetic moment of a spin-1/2 paramagnet?

Answer 6

μ(B) = e*ћ/2m

Question 7

What is the equation for z1 for the spin-1/2 paramagnet?

Answer 7

z1 = exp(βμ(B)B) + exp(-βμ(B)B) = 2cosh(βμ(B)B)

Question 8

What is a paramagnet in terms of spins?

Answer 8

A paramagnet is an array of N non-interacting spins.

Question 9

What is the general equation for z(N) for a spin-1/2 paramagnet? What is the equation for du?

Answer 9

z(N) = (z1)^N, du = TdS - mdB, where m is the magnetic moment.

Question 10

What are the two possible configurations of the paramagnet?

Answer 10

All moments up, so m = N*μ(B), or half up half down, so m = 0

Question 11

What does dF equal for the spin-1/2 paramagnet?

Answer 11

dF = -S dT - m dB, where m = -(dF/dB) at const T, thus F = -Nk(B)Tln(2cosh(βμ(B)*B))

Question 12

What does m equal for the spin-1/2 paramagnet?

Answer 12

m = Nμ(B)tanh(βμ(B)B)