Physics 880.05: Assignment #2

1. Equilibrium and Stability.
   1. See the lecture notes, where these are discussed (and derived!).
   2. Go back to the thermodynamic definition of mu (which follows from the one here in the limit N --> infinity) and convert from E to epsilon. Be careful about what is held fixed and what you need to differentiate.
   3. Nothing tricky here --- just identify where the saturation point is.
   4. Directly compare the sum of the energies in the two perturbed half volues, 1/2[E(rho+delta)+E(rho-delta)] to the energy in the unperturbed full volume E[rho]. Since the volume is held fixed, it is convenient to work with the energy density:
      ==> energy/volume = (energy/particle)*(particles/volume) = epsilon * rho
      Expand in delta!
2. Nuclei as a Fermi gas.
   1. There are spins and isospins. The latter arises when the neutrons and potons are treated as different isospin states (like spin up and spin down) of identical particles. "Each level" refers to the momentum value. For a Fermi gas of electrons, the degeneracy factor per level is two (spin up plus spin down).
   2. For a noninteracting Fermi gas, k_F is related to the number density and e_F is related to k_F. How does the density vary with A, according to the formula for R?
   3. Back to thermodynamics!
   4. Neglecting the Coulomb energy (which arises because the protons are charged), the binding energy per nucleon is about 16 MeV. What is the average kinetic energy per nucleon?
   5. Compare the Fermi and Bose occupation numbers expressions. What is the condition that they give the same result?
3. Symmetry factors.
   1. This is just a review of the rules for diagrams (e.g., how is the number of vertices related to the power of lambda?).
   2. Follow the rules from the lecture notes. Remember when calculating the vertex permutation factor (the third factor) that the "external lines" for <x²> are nailed down. There are 10 distinct diagrams.
   3. Just plug in the numbers and add the fractions.
4. Field operators.
   1. Write out the commutator of H and N explicitly and then use the (anti-)commutation relations for the field operators repeatedly on one of the two terms to bring it to the form of the other.
   2. What does the matrix element of a three-body potential look like? (Generalize from a two-body potential.)
5. Path integral --> Schrodinger equation.
   1. Go back to the definition of U. Break up the interval from t_i to t_f into intervals from t_i to t_m and t_m to t_f. Can you insert a complete set of |y> states at t_m?
   2. It is implied that you work in units where hbar=1 and m=1 (otherwise comparing |x-y| to the square root of epsilon wouldn't make sense). What happens to the first term in the exponential if |x-y|² is greater than epsilon as you integrate over y? This term is the key. When you then expand in powers of epsilon and eta, you CANNOT expand this term. To zeroth order in epsilon and eta, you should still have an integral to do and the constant 1/A, so A is determined.
   3. Nothing fancy here, just remember the limits of the integral.
   4. Remember that order epsilon means either epsilon or eta² terms are kept. Remember the title of the problem as a hint to what you should find!