Physics 880.05: Assignment #1

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).