## Midterm II reflection

Posted by peeterjoot on March 10, 2013

Here’s some reflection about this Thursday’s midterm, redoing the problems without the mad scramble. I don’t think my results are too different from what I did in the midterm, even doing them casually now, but I’ll have to see after grading if these solutions are good.

## Question: Magnetic field spin level splitting (2013 midterm II p1)

A particle with spin has states . When exposed to a magnetic field, state splitting results in energy . Calculate the partition function, and use this to find the temperature specific magnetization. A “sum the geometric series” hint was given.

## Answer

Our partition function is

Writing

that is

Substitution of gives us

To calculate the magnetization , I used

As [1] defines magnetization for a spin system. It was pointed out to me after the test that magnetization was defined differently in class as

These are, up to a sign, identical, at least in this case, since we have and travelling together in the partition function. In terms of the average energy

Compare this to the in-class definition of magnetization

For this derivative we have

This gives us

After some simplification (done offline in \nbref{midtermTwoQ1FinalSimplificationMu.nb}) we get

I got something like this on the midterm, but recall doing it somehow much differently.

## Question: Pertubation of classical harmonic oscillator (2013 midterm II p2)

Consider a single particle perturbation of a classical simple harmonic oscillator Hamiltonian

Calculate the canonical partition function, mean energy and specific heat of this system.

There were some instructions about the form to put the integrals in.

## Answer

The canonical partition function is

With

the momentum integrals are

Writing

we have

The mean energy is

The specific heat follows by differentiating once more

Differentiating the integral terms we have, for example,

so that the specific heat is

That’s as far as I took this problem. There was a discussion after the midterm with Eric about Taylor expansion of these integrals. That’s not something that I tried.

# References

[1] C. Kittel and H. Kroemer. *Thermal physics*. WH Freeman, 1980.

## Midterm II reflection, take II, with approximate anharmonic oscillator solution « Peeter Joot's Blog. said

[…] Midterm II reflection […]