Single Particle Partition Function

Question 1

What is the single partition function?

Answer 1

z = sum over i of exp(β*Ei)

Question 2

What is the equation for the internal energy u?

Answer 2

u = -dlnz/dβ

Question 3

What is the equation for the entropy S?

Answer 3

S = k(B)sum over i of Pi(βEi+lnz) = k(B)*(βu + lnz)

Question 4

What is the equation for the Helmholtz free energy?

Answer 4

F = u-TS = k(B)T*ln(z)

Question 5

What is another equation for the entropy?

Answer 5

S = (u-F)/T = k(B)ln(z) + k(B)T(dln(z)/dT)at const V

Question 6

What is the equation for heat capacity Cv?

Answer 6

C = T*(dS/dT) at const V = (du/dT) at const V

Question 7

For a two level system from -Δ/2 to Δ/2 (so energy change is Δ), what does z equal?

Answer 7

z = sum over i of exp(-βEi) = exp(βΔ/2)+exp(-βΔ/2) = 2cosh(βΔ/2)

Question 8

What can we find from z for the two level system? What does this tell us?

Answer 8

We can find the internal energy u, which tells us that at high T, βΔ/2«1 o u->0, and as T->0, βΔ/2 large and u-> -Δ/2

Question 9

What else can we find from z? What does this tell us?

Answer 9

Can find the helmholtz free energy and therefore the entropy, which tells us that at hight T, βΔ/2«1 so s-> k(B)*ln(2), and at low T, βΔ/2 is large so s-> 0

Question 10

What is the difference between s at high temperatures and low temperatures?

Answer 10

ΔS = k(B)*ln(2)

Question 11

What does En equal for the Harmonic oscillator?

Answer 11

En = (n+1/2)*ћω

Question 12

What does z equal for the harmonic oscillator?

Answer 12

z = sum over n of exp(-β(n+1/2)ћω) -> can take out the constant exp term and then remaining sum is a geometric progression

Question 13

What is the equation for the geometric progression?

Answer 13

a*sum over n to inf of r^n = a/1-r -> can use this for z for the harmonic oscillator.

Question 14

Which values do we need to find in SM from the single partition function?

Answer 14

u, S, F, Cv