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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2018

Problem Set 7

Due Tuesday, April 24, by noon in the homework locker

  • Problem 1 [20 points]

    Consider a degenerate Fermi gas of non-interacting, non-relativisitic, particles in two dimensions (this might be a model for electrons in a thin metallic film).

    a) Find the density of states 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 = g(ε)
    eβ(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<<TF. Does µ(T) have a power series expansion in T at low T, like we found in three dimensions with the Sommerfeld expansion?

  • Problem 2 [35 points]

    Consider a degenerate two-dimensional gas of N indistinguishable non-interacting spin 1/2 fermions of mass m in an external isotropic harmonic potential V(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(nx, ny) = ℏ ω0 (nx + n2 + 1), where nx and ny = 0, 1, 2, . . . are integers. Assume the thermal energy is much greater than the spacing between the energy levels, i.e. kBT >> 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 (nx, ny) 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). The condition kBT >> ℏω0 should be helpful here.

    b) What is the Fermi energy εF 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 << TF.