Physics 880.05: Assignment #2
Here are some hints, suggestions, and comments on the assignment.
- Equilibrium and Stability.
- See the lecture notes, where these are discussed (and derived!).
- 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
- Nothing tricky here --- just identify where the saturation point
- 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!
- Nuclei as a Fermi gas.
- 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
- For a noninteracting Fermi gas, kF is related to the
number density and eF is related to kF.
How does the density vary with A, according to the formula for R?
- Back to thermodynamics!
- 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?
- Compare the Fermi and Bose occupation numbers expressions.
What is the condition that they give the same result?
- Symmetry factors.
- This is just a review of the rules for diagrams (e.g., how is
the number of vertices related to the power of lambda?).
- 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.
- Just plug in the numbers and add the fractions.
- Field operators.
- 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.
- What does the matrix element of a three-body potential look
like? (Generalize from a two-body potential.)
- Path integral --> Schrodinger equation.
- Go back to the definition of U. Break up the interval from
ti to tf into intervals from
ti to tm and tm to tf.
Can you insert a complete set of |y> states at tm?
- 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
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
- Nothing fancy here, just remember the limits of the integral.
- 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!
Physics 880.05: Assignment #2 hints.
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