# Spin density in an (ideal) Fermi gas

• I
Gold Member

## Summary:

Notation used:

With ##n_m( \mathbf x) = \hat \psi_m^{†}( \mathbf x) \hat \psi_m( \mathbf x) ## I mean the density of the particle with spin ##m## and ##\psi_m^{†}## ##\hat \psi_m## are the field operator

##|F>## = (ground) state of a ideal fermi gas (no interaction)

## Main Question or Discussion Point

Hi,

some time ago our professor told us (en passant) to evaluate this quantity:
$$<F|n_m( \mathbf x) n_{m'}(\mathbf x) |F> - <F|n_m( \mathbf x)|F><F|n_{m'}(\mathbf x) |F>$$
And then he said: "you'll find that this quantity may not be zero. In particular when the electron are correlated it will not be zero." Then he said something about the spin of the electrons being "parallel". I'm really sorry if I'm being vague but I really don't remember because it was some time ago. Anyway... I calculated the first part using Wick's theorem and I got:
$$<F|n_m( \mathbf x) n_{m'}(\mathbf x) |F> = <F|\hat \psi_m^{†}( \mathbf x) \hat \psi_m( \mathbf x) \hat \psi_{m'}^{†}( \mathbf x) \hat \psi_{m'}( \mathbf x) |F>=$$
$$=<F|\hat \psi_m^{†}( \mathbf x) \hat \psi_m( \mathbf x)|F><F|\hat \psi_{m'}^{†}( \mathbf x) \hat \psi_{m'}( \mathbf x) |F> + <F|\psi_m^{†}( \mathbf x) \hat \psi_{m'}( \mathbf x)|F><F|\psi_m( \mathbf x) \hat \psi_{m'}^{†}( \mathbf x)|F> =$$
$$= <F|n_m( \mathbf x)|F><F|n_{m'}(\mathbf x) |F> + <F|\psi_m^{†}( \mathbf x) \hat \psi_{m'}( \mathbf x)|F><F|\psi_m( \mathbf x) \hat \psi_{m'}^{†}( \mathbf x)|F>$$
So the interesting quantity is:
$$<F|\psi_m^{†}( \mathbf x) \hat \psi_{m'}( \mathbf x)|F><F|\psi_m( \mathbf x) \hat \psi_{m'}^{†}( \mathbf x)|F>$$
which I can re-write in momentum base as (I didn't use vector notation here sorry):
$$<F| \sum_{kk'} e^{(k-k')x} c_{k,m}^{†} c_{k',m'}|F><F|\sum_{qq'} e^{(q-q')x} c_{q,m} \hat c_{q',m'}^{†}|F>$$

Is it okay ?

Because from here I don't really see what kind of conclusion I could draw... The only thing I could think of is the following:
in order for that term not to vanish I should have (correct me if I'm wrong) a state with the energy levels filled like this:

I'm again very sorry if it all seems vague but I can't remember and I hope that maybe some of you can recognize it as a well know example or something...

Thanks
Ric