Physics 880.05: Assignment #4

1. Skyrme-type model for nuclear matter.
   1. This is a good opportunity to use Mathematica. You have two unknown parameters to determine, lambda and beta. You have two equations for them: i) E/A=-16. at rho=0.16 and ii) d(E/A)/d(rho) = 0 at rho=0.16. You can define these equations and then use Solve to find lambda and beta. You need to write the kinetic energy in terms of rho instead of k_F (which you can do explicitly or by defining a function kf[rho]). If you define EoverA[rho] and DerivEoverA[rho], with rho0=0.16 and EoverA0=-16., then
      Solve[{DerivEoverA[rho0]==0, EoverA[rho0]==EoverA0},{lambda,beta}]
      will give you your answers (note the double =)! One caution: be careful about units, since rho is in fm^-3 while energies are in MeV. I use the conversion factor hbarc = 197.33 MeV-fm. I.e., to have kf^2/2M come out in MeV, I use M=939 MeV for nucleons and multiply by hbarc².
   2. If you did the first part in Mathematica, plotting EoverA[rho] is immediate (after substituting the results of your Solve for lambda and beta). Review PS#1 problem 1 (my solutions are online) for calculating the presure and stability. You can plot the pressure and stability condition on the same graph. I defined the pressure as:
      Pressure[rho_] = rho^2 D[EoverA[rho],rho]
      (with an = rather than a :=).
   3. You can calulate the compression modulus in terms of derivatives with respect to rho rather than kf, or just substitute kf for rho before taking derivatives.
2. Lehmann representation for advanced and retarded functions.
   • The excerpt from Negele and Orland "Lehmann representation and quasiparticle pole" that is linked here (and on the 880 web page) may be useful (in fact, it may give the whole thing away; please try the problem before looking!).
3. Spin-dependent force.
   1. Try doing second quantization as we did at the very beginning last quarter to go from a potential to a form written in terms of field operators (e.g., pages 29 and 55 in the notes), but keep the spin indices.
   2. The Feynman rule should be similar to before except the spin sum has Pauli matrices rather than Kronecker deltas at the vertices.
   3. When you do the spin sums, you should get traces of the Pauli matrices. The Hartree term should vanish; why? You can check your result by doing first-order perturbation theory (like in PS#1).
   4. Just apply the Feynman rules again (more terms here!).
4. Fermi liquid theory in one spatial dimension.
   You might want to look at the exerpt from Negele and Orland, chapter 6, which is linked on the web page (under "Landau Theory of Fermi Liquids) or here. You will find explanations of the notation. Basically you need to repeat what is done there or in the class notes but in 1D rather than 3D. [Note: the sigma and sigma' here refer to the spin projection on a quantization axis and take values +1/2 or -1/2 (not +1 and -1). So if sigma=sigma', then we get f+phi, and if sigma=-sigma', we get f-phi.]
   1. IMPORTANT: There is a typo in the problem set, it should be F₁ and not F₀. Refer to 205-207 in the class notes and/or 300-302 in the Negele exerpt.
   2. Refer to 203-207 in the class notes.
   3. Refer to 208 in the class notes and/or 302-304 in the Negele exerpt.
   4. Skip this!