Statistical mechanics question -

[scigal89]

We're covering probability of the distance for free electrons with parallel spin (long-range oscillations should go to zero) and using that to get a correlation energy. My teacher wants us to elaborate the following 1D case.

$$\int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\Rightarrow Re\left (\frac{e^{ik(x-X)}}{i(x-X)} \right ) =\frac{sin[k(x-X)]}{x-X}$$

Taking X = 0 and evaluating for case 1 from 0 to k_0 and for case 2 from 0 to 2k_0 (where k_0 is at the Fermi level) my teacher wrote:

$$\psi_{1} \sim \frac{sin(k_{0}x)}{k_{0}x}$$

$$\psi_{2} \sim \frac{sin(2k_{0}x)}{2k_{0}x}$$

I think to get the k_0 in the denominator for the first one and 2k_0 in the denominator for the second one he multiplied both sides by k_0/k_0 and 2k_0/2k_0, respectively.

Here is my question:

My teacher says the first one is supposed to be the filled state, the second one the empty state - why? A lot of this, conceptually, is very unclear to me.

 

[_coyote_]

The answer to your question lies in the fact that the electrons with parallel spins behave differently than those with opposite spins. This is due to the Pauli Exclusion Principle, which states that no two electrons can occupy the same quantum state at the same time. As such, the first integral (from 0 to k_0) represents the states that are filled, while the second integral (from 0 to 2k_0) represents the empty states.