Physics 880.05: Assignment #1
Here are some hints, suggestions, and comments on the assignment.
- 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.
- 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
- 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).
- (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).
Physics 880.05: Assignment #1 hints.
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