Statistical Physics Pt.3

Question 1

How can we relate the wavefunctions for indistinguishable particles?
(Hint: Phase factor, K)

What does it mean for a phase factor of -1 or 1?

Answer 1

K = exp(-ia)
Probability denisities must stay the same for indistinguishable particles.
-1 for anti-symmetric, +1 for symmetric wavefunctions.

Question 2

What are particles with symmetric wavefunctions called? Give examples
What are particles with antisymmetric wavefunctions called? Give examples

State the degeneracy of quantum states for a particle with total spin quantum number l.

Answer 2

Question 3

Question on image: FOR BOSON!

Answer 3

Where i and j represent any eigenfunction.

Question 4

For non-interacting particles, what is teh energy of this state?

Answer 4

Question 5

What is the form of the wavefunction for two fermions?
What does i=j show?

Answer 5

i=j shows that fermions can not exist in the same state, PAULI EXCLUSION PRINCIPLE.

Question 6

If i=j, Psi = 0, Pauli exclusion principle.

Answer 6

Question 7

Question 8

For a large reservoir, how do we treat the chemical potential and Temperature if a small number of particles is added/removed?

What is the temperature of the reservoir by definition?

Answer 8

Question 9

By integrating the chemical potential and/or the temperature of a reservoir, find the number of microstates in the reservoir.

Now consider a system connected to a reservoir, forming a grand canonical ensemble, re-write in terms of the total number of particles and energy.
Hint: Consdider the fact that energy and number of particles is conserved.

DON’T LOOK AT IMAGE AS IT GIVES WORKED INTEGRATION

Answer 9

Question 10

Suppose at some instant System A is in a particular quantum state, state β,
comprising identical particles, distributed amongst the system’s singleparticle energy levels so that their total energy [of the form in Eq. (11.2)] =Eβ.

What is teh number of accesible quantum states/microstates for the TOTAL system, W_b?
(eq 12.6 in ans)

Answer 10

Question 11

Working on from the image, what is the OVERALL number of states for the TOTAL system, corresponding to all possible different states (j) of system A.
*Total system = System A + Reservoir

State the GRAND partition function (12.6 + 12.7)
Next, find the probability of being in a state Beta.

Answer 11

Question 12

Find the probability that a state is occupied.

Answer 12