Two level systems, Quantum Harmonic Oscillators

Question 1

What is the total energy of a two level micro-canonical system? What does n represent in this equation?
IMPORTANT FOR CARD 3

Answer 1

Hint: n represents the number of exited states.

Question 2

What is the number of microstates of a two level system? Or paramagnet?
IMPORTANT TO REMEMBER

Answer 2

Question 3

Using the equation total energy of a two-level microcanonical system and number of microstates, find the number, n of excited states and hence the total energy in terms of this derived n.

Hints
n is the total number of exited states in a 2 level system = number of quanta in a 2 level system.
1. W= ?, E = nepsilon
2. Find the Entropy, use Stirling approximation
3. Temperature
4. What is the relationship between temperature, energy and entropy?
If really stuck:
3. S= k lnW, then apply ln(a!) = aln(a) - a
4. partial(dS/dE) = 1/T
5. partial: dE = epsilon dn
6. Plug into 4, re-arrange for n
7. E=nepsilon, plug in n.

Answer 3

Question 4

Important geometric series, also useful for partition function for harmonic operators.
State the result.
Sum(e^ [- kxi] )

Answer 4

Sum(e^ [- kxi] ) = 1/(1-e^[-kx])

Question 5

What is the number of microstates for a Harmonic oscillator?

Answer 5

C(M+N-1, N-1) = (M+N-1)!/M!(N-1)!

Question 6

What is the total energy of a Quantum Harmonic Oscillator?
Need to know for card 7!!!
What is a good way of thinking how to distribute M=5 quanta to N=3 Oscillators?

Answer 6

Question 7

Using the total energy of a quantum harmonic oscillator, and the number of microstates for a quantum harmonic oscillator, find the number of quanta M, and hence the total energy of all the oscillators.

Hint:
1. W= ?, E = Mhw
2. Find the Entropy, use Stirling approximation
3. Temperature
4. What is the relationship between temperature, energy and entropy?
If really stuck:
3. S= kb lnW, then apply ln(a!) = aln(a) - a
4. partial(dS/dE) = 1/T
5. partial: dE = epsilon dn
6. Plug into 4, re-arrange for M, approximate (N - 1 ~ to N)
7. E= Mhw, plug in M.

Answer 7

Question 8

Answer 8

Question 9

Use the boltzmann distribution and partition function to find the probability of being in an excited or non-excited state for a two-level CANONICAL system (not micro-canonical).

Answer 9

Non-excited: E= 0
Excited: E = epsilon

Question 10

You can do this question using the earlier derived M, or can use the equations from slide 11.

Answer 10

Solution uses equations from slide 11.

Question 11

What is the energy per oscillator in a Quantum Harmonic Oscillator?
How would you get the total energy if there are N oscillators?

Answer 11

To get total energy, multiply by N oscillators.

Question 12

Derive the energy of a system per oscillator/excited state from the partition function (used for Quantum Harmonic Oscillator).
Use: dF = -p dV -S dT, and E = TS + F
Hint: Consider partial derivatives from the first equation listed above.

Answer 12

F = -kTln(z)

Question 13

Derive the total energy of a two level system (can be used for both canonical and micro-canonical) using the equation for the energy per oscillator (or excited state in this case).

Answer 13

This method is nice as it does not require probabilities of each each level (like with the normal canonical calculation) or the number of states (like with the micro-canonical which is long and a little more tedious).

Question 14

State the number of microstates for a random walk of N steps.

Question 15

Using the number of microstates for a random walk, find the entropy of the system, and hence the free energy. F = E - TS

Answer 15