1. May 12, 2010

### 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.

Last edited: May 13, 2010