# Physics 880.05: Assignment #1

Here are some hints, suggestions, and comments on the assignment.
1. One-dimensional delta function Fermi "fluid".
• Follow the same analysis as in class (check the online notes). The only real difference is that volume becomes simply L and the integrals are one-dimensional instead of three-dimensional. This makes a tremendous difference in the results, however!
• Don't forget that integrals run from -Infinity to +Infinity (and not 0 to Infinity).
• The integral you need to do for the exchange energy is a double integral over k and q. It is easy to get the answer by inspection if you consider it to be the area of a region in the k-q plane. (Or you can just shift one of the variables and get the answer trivially; it is more instructive to do it the "hard" way!)
• In 3-D, the first-order energy is proportional to (g-1). This should also be true in 1-D.
2. Collapse of the dilute Fermi gas.
• What do you know about variational calculations? This should apply at each density.
• You should draw different conclusions in three and one dimensions.
3. 2nd-order perturbation theory for dilute Fermi gas.
• There is no need to do any explicit calculations in this problem, because you just need to demonstrate a qualitative result.
• The state |0> is the same as |F>.
• Analyze <j|H1|F> as we analyzed <F|H1|F>, keeping in mind that |j> must be a different state than |F>.
• Which of the momentum variables p, k, q can get large? What happens when it does? Consider the calculation in the limit this variable is large (so throw away any subleading pieces).
4. (BONUS) Polarized dilute spin-1/2 Fermi gas.
• In this problem, the total fermion number N is held fixed. You are just examining how the energy changes at a given N as you flip some spins. So find E/N for fixed N as a function of zeta.
• Be sure to check that your answer for the energy per particle E/N reduces to our previous result when there are equal numbers of spin-up and spin-down (zeta=0).
• In part (b), the key word for the upper limit is "partially". For large enough values of (lambda N)/(Omega Tbar), the system is entirely magnetized (all spin-up or all spin-down).

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Physics 880.05: Assignment #1 hints.
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