Problem Set 9

Problem Set 9

Due Monday, April 22, uploaded to Blackboard by 11pm

Problem 1 [10 points]
The Stefan-Boltzmann law states that a black body at temperature T radiates power per unit surface area equal to σT⁴ where σ is a universal constant independent of the material properties of the body. Assuming that the sun and the earth are black bodies, and that the earth is in thermal equilibrium with the sun [i.e. energy absorbed = energy emitted] calclate the temperature of the earth in terms of the temperature of the sun. Look up the parameters you need in order to compute a number for this estimate of the earth's temperature.
Problem 2 [15 points]
Consider a degenerate gas of indistinguishable, non-interacting, non-relativisitic, spin 1/2 fermions of mass m in two dimensions (this might be a model for electrons in a thin metallic film).
a) Find the density of states per unit volume g(ε).
b) Find the Fermi energy and the T=0 energy density.
c) Using the the fact that the particle density n is given by

n = ∫ dε g(ε)
e^β(ε-µ)+1

find the chemical potential as a function of temperature, µ(T), for fixed density n, by doing this integral exactly. You may have to look up an integral in an integral handbook! Using the exact expression for µ(T), find a simpler approximation that holds at low T<<T_F. Does µ(T) have a power series expansion in T at low T, like we found in three dimensions with the Sommerfeld expansion?
Problem 3 [25 points]
Consider a degenerate two-dimensional gas of N indistinguishable, non-interacting, non-relativistic, spin 1/2 fermions of mass m in an external isotropic harmonic potential U(r) = (1/2)mω₀² |r|², where r = (x, y). This might be taken as a model for electrons in a magnetic trap. The quantized single particle energy levels are given by ε(n_x, n_y) = ℏ ω₀ (n_x + n_y + 1), where n_x and n_y = 0, 1, 2, . . . are integers. Assume the thermal energy is much greater than the spacing between the energy levels, i.e. k_BT >> ℏω₀.
a) If the number of single particle states between energies ε and ε+dε is g(ε)dε, then find the density of states g(ε).
Hint: Consider the shape of the contour of constant energy ε in the (n_x, n_y) plane, and find the number of states G(ε) whose energy is less than or equal to ε. Using G(ε) you can then find g(ε).
b) What is the Fermi energy ε_F as a function of the number of particles N?
c) What is the total energy E of the gas, as a function of N, at T=0?
d) Give an estimate of the spatial extent R of the Fermi gas about the origin of the harmonic potential.
e) Give a rough estimate of the specific heat of the gas C at the low temperatures T << T_F. The condition k_BT >> ℏω₀ should be helpful here.