Stats

Question 1

What is the canonical partition function for a degenerate and undegenerate system?

Answer 1

Sum of g_j exp(beta E_j) for a degenerate system with g_j= 1 for and non

Question 2

Internal energy

Answer 2

Negative Partial differential of ln(Z) with respect to beta

Question 3

Hamiltonian of two state system

Answer 3

Energy multiplied by the occupation numbers. These are 1 or 0 dependent on which energy state the particle is in

Question 4

Microcanonical probability

Answer 4

One over number of micro states, number of way to distribute the wanted state among the number of particles

Question 5

Entropy

Answer 5

Boltzmann constant multiplied by the natural log number of microstates

Question 6

Heat capacity

Answer 6

The derivative of internal energy with respect to temperature

Question 7

Helmholtz Free energy, F

Answer 7

Begative Boltzmann constant multiplied by temperature time natural log of the canonical partition function. U-TS

Question 8

Pressure

Answer 8

The derivative of Gibbs free energy with respect to volume

Question 9

Equilibrium temperature

Answer 9

One over temperature is the derivative of entropy with respect to energy

Question 10

Grand canonical partition function

Answer 10

Q, sum over microstates the exponential of beta(muN- H) this is sum of exp( betamuN)Z

Question 11

Grand canonical probability

Answer 11

e^(betamuN)* Z/Q

Question 12

Grand potential

Answer 12

E-TS-muN=-kTlnQ

Question 13

Entropy of grand canonical

Answer 13

Negative of the differential of the grand potential with respect to temperature

Question 14

Number of particles in grand canonical

Answer 14

Negative of the derivative of the grand potential with respect to mu (chemical potential)

Question 15

Grand canonical pressure

Answer 15

Negative of the derivative of the grand potential with respect to volume

Question 16

Micro canonical

Answer 16

Try to start with multiplicity for probability . This is a system with no heat change or external work. N, V, E are fixed quantities

Question 17

Canonical

Answer 17

Macrostates allow heat input but no external work. The system is maintained at a constant temperature by using a reservoir. There is a Hamiltonian for both. Try to start with partition function for probability

Question 18

Canonical probability

Answer 18

The number of microstates of the reservoir divided by the microstates of the whole system. Becomes exponential of minus beta times Hamiltonian over Z

Question 19

Difference between fermions and bosons

Answer 19

Bosons can have many in one state whereas for fermions there can only be 1 or 0 particles in each state

Question 20

First law

Answer 20

dU=dQ-dW-mudN

Question 21

Second law

Answer 21

dS=dQ/T

Question 22

Gibbs free energy, G

Answer 22

G=U+PV-TS

Question 23

What does degeneracy do to the multiplicity?

Answer 23

Multiple the standard form by the degeneracy of the state to the power of how many particles exist in that state

Question 24

High temperature and low temperature limits quantum states

Answer 24

Low temp is discrete and high is continuous

Question 25

Landau free energy entropy and specific heat

Answer 25

Entropy is the negative of the derivative of F with respect to T. And specific heat is that differentiated with respect to T again and then multiplied by T

Question 26

Quantities fixed in the grand canonical

Answer 26

Volume and temperature

Question 27

Average number of particles

Answer 27

Sum of number of particles in a state multiplied by the probability of a particle being in that state