Statistical mechanics question - please help

  • Thread starter scigal89
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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.

[tex]


\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}


[/tex]

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:

[tex]

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

[/tex]

[tex]

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

[/tex]

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.
 
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