PHYS 403: HOMEWORK ASSIGNMENT No. 3: QUANTUM GASES and SUPERFLUIDS 2022
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PHYS 403: HOMEWORK ASSIGNMENT No. 3:
QUANTUM GASES and SUPERFLUIDS
2022
QUESTION (1) EARLY UNIVERSE: At the ‘recombination time’ τR (roughly 400,00 yrs after the Big Bang), the main constituents of the universe were photons, H atoms, protons, and electrons. Let’s ignore the photons here, and assume that the 3 remaining species have chemical potentials µH , µp , and µe , and number densities nH , np , and ne , respectively. Assume a hydrogen ionization energy Eo , and that that there are 2 relevant states for the proton and electron (they are spin-1/2), and hence 4 states for the H atom.
1(a) Suppose we can treat this system as low density. Then what are nH , np , and ne in terms of µH , µp , and µe ? 1(b) What defines thermal equilibrium for this system, and at equilibrium, what are nH , np , and ne ?
1(c) Using values for Eo and for the mass me of an electron that you can get from the literature, find the density ne when nH = np = ne (ie., half the H atoms are ionized), which gives the density at the time τR .
QUESTION (2) BOSE GASES:
2(a): Draw two graphs as a function of energy E which shows (i) the 1-particle density of states, and (ii) the Bose distribution function, for a 3-dimensional Bose system of massive particles, for the cases T > Tc and T < Tc . Here Tc is the BEC condensation temperature. Then draw two graphs showing the product of these 2 functions as a function of energy, again for these 2 cases.
2(b) A criterion for BEC to occur in a 3-d gas of bosons is that the chemical potential µ = 0. Explain this criterion with reference to the relevant mathematical expressions.
2(c) Rederive the criterion for 2-d and 1-d systems. What do the results tell you about BEC in these cases?
2(d) Consider now the photon gas. Why is µ = 0 always for photons? Now, derive an expression for the energy density u(T) for a photon gas in n dimensions, where n is a positive integer; and show that u(T) x Tn+1 .
QUESTION (3) SUPERFLUIDS:
3(a) Suppose a mass M is moving through a fluid with constant viscosity coefficient η. Find the equation of motion of the particle, assuming there is an external force f (t) acting on it. If the initial velocity at t = 0 is v(t = 0) = vo , then show the solution to this equation of motion is
v(t) = vo e —γt + t dt′ f (t′ ) e —γ(t —t尸 )
where γ = η/M . Then show that after a long time the particle will reach a constant velocity vf , and find vf .
3(b) In a superfluid the friction depends on the velocity. Suppose that η(v) = ηo (v - vc ) θ(v - vc ), where θ(x) = 0 for x < 0, and θ(x) = 1 for x > 0. Find the new terminal velocity vf , without solving the new equation of motion.
3(c) Superfluids have quantized vortex ring excitations. For a circular ring of radius R, the energy E ~ ρκ2 R ln[R/ao ], and the momentum p ~ πρκR2 , where ρ is the superfluid density, κ the circulation quantum, and ao ~ 0.1 nm is a vortex core radius. If the critical velocity for formation of a vortex ring is vc ~ min(E/p), then show that in an infinite system, vc → 0; and also find vc if the superfluid is moving through a cylindrical tube of radius Ro . Finally; since the vortex ring velocity is v = dE/dp, find v(R) as a function of R, and sketch a graph of it.
2022-03-23