Partition function for electrons/holes

[Welshy]

Homework Statement

By shining and intense laser beam on to a semiconductor, one can create a collection of electrons (charge -e, and effective mass m_e) and holes (charge +e, and effective mass m_h) in the bulk. The oppositely charged particle may pair up (as in a hydrogen atom) to form a gas of excitons, or they may dissociate into an electron hole plasma.

a) Write down the single particle partition functions Z_e(1) and Z_h(1) at temperature T in a volume V for electrons and holes respectively. (The thermal wavelength $$\lambda$$ for a particle is $$\lambda$$ = $$\frac{h}{\sqrt{2 \pi mkT}}$$

Homework Equations

The Attempt at a Solution

I know that the partition function is exp[-E/kT] summed over all energies or integrated with a density of states over all energies. But how do I go about it in this case?

 

[aurora14421]

In our notes somewhere (section 11/12/13), you can see that he uses the equation:

Z(1) = (V/λ^3), where V is the volume.

I'm not sure where it comes from, but I think we just need to learn it.

 

[Count Iblis]

You can obtain that formula by summing over all the energy states by approximating the summation by an integral using the fact that the number of quantum states within a volume V_p of momentum space is

V V_p/h^3

So, if you integrate over all momenta, the energy is E = p^2/(2m) and you can write:

Z1 = Integral of V d^3p/h^3 exp[-beta p^2/(2m)] =

V/h^3 Integral from |p| = 0 to infinity of 4 pi p^2
exp[-beta p^2/(2m)] d|p|

For electrons you must multiply this by 2 because there are two spin states for each energy eigenstate.