Exclusion principle and the distance between two particles

[dRic2]

Hi, I'm reading the book "Quantum theory of many-particle system" by Fetter and Walecka. I can not understand the following quote from the book:

The exclusion principle prevents two particles of the same spin from occupying the same single-particle state. As a result, the two particles density correlation function for parallel spins vanishes thought a region comparable with the interparticle spacing. If the range of the potential (they're talking about the interaction potential between two particles) is less then the interparticle spacing then this exclusion hole is crucial in determining the ground-state energy.

Ok so the first thing I don't quite get is the part about the correlation function. I've tried to Google it a bit but didn't find anything which I am able to relate to this. To my knowledge the definition of the density correlation function for two particles is the following
$$<gs| \delta n_m(x) \delta n_{m'}(x')|gs>$$

Where $\delta n_m(x) = n_m(x) - <gs|n_m(x)|gs>$. "gs" is the ground-state and $n_m(x)$ is the density operator for spin m particles.

How do I verify that this function does actually vanish in the space between two particles with the same spin?

My Secondo question is a bit more vague, hope it makes sense. When they say

this exclusion hole is crucial in determining the ground-state energy.

I get the feeling that there may be a link with the exchange Energy, am I right ?

 

[DrClaude]

https://en.wikipedia.org/wiki/Fermi_heap_and_Fermi_hole

 

[dRic2]

Thank you. I'm very slow when it comes to learn new stuff and I'm super busy in these days. I will be back maybe next week. Sorry.

 

[dRic2]

Ok, let's see if I got it right. I have a ground state of fermions, i.e. something like $\sum_p (-1)^p |u_1, u_2, ..., u_N>$ where the sum is intended over all possible permutations. If I use spin-position $<m_i, \mathbf x_i|$ basis and I set $m_i = m_j$ and $\mathbf x_i = \mathbf x_j$ then the terms containing those permutation vanishes. The remaining ones are orthogonal and when I take the expectation value they give a null contribution. In conclusion the expectation value of $n(x)$ is zero.

For the second part, regarding the long/short range of the potential I still need some time. I have an idea which goes something like this.

In second quantization the potential operator is (I have in mind a Coulomb-like potential)
$$ \hat V = \frac 1 2 \sum_{m, m'} \int d^3 \mathbf x d^3 \mathbf x' ψ^†_m(x)ψ^†_{m'}(x')ψ_{m'}(x')ψ_m(x)v(x, x')$$

when I take the expectation value, if V is a long range potential it can be considered constant in the region $x \approx x'$ and taken outside the integral. Then the creation and destruction filed operator can be re-arranged to give the density operator and we saw that its expectation value vanishes in that region so for $x \approx x'$ I should get no contribution from the potential. If on the other hand in the region $x \approx x'$ V is rapidly varying (assuming very high values) I may get a finite contribution. I still don't quite get how to correlate this (assuming its correctness) to the exchange energy though.

What do you think, am I getting it right?