Practice Final
1) An electron in a hydrogen atom sees a potential that goes as -Eh/r where r is the
distance from the proton to the electron and Eh is a constant. You are to determine the
probability that hydrogen is in its first excited state as a function of temperature. Provide
a graph. Begin by carefully explaining how you will approach the problem and then go
ahead and solve it. Things you will need to know: The constant Eh is called a Hartree
and is equal to 27.2 eV. The quantized energy levels for the electron in this potential are
given by En = -Eh/(2n2) where n is a quantum number equal to 1, 2, 3, … and each level
is n2 degenerate. Explain why we do not need to consider the electronic contributions to
the heat capacity for atoms and molecules at common temperatures.
2) An Einstein solid is a model of a crystal developed, obviously, by Albert Einstein. It
assumes a crystal of N atoms each with 3 degrees of vibrational freedom (vibrations in
the x, y, and z directions). Derive an expression for the solid’s internal energy, entropy
and heat capacity as a function of temperature. What is the internal energy at 0 K?
Explain why this is the case.
3) Consider a system of N non-interacting spins in a magnetic field of strength B (often
measured in units of Tesla). The spins can be either parallel to the magnetic field or antiparallel (i.e. up or down). In the former case the energy of the spin is given as +µ Β and
in the later case as -µ Β, where µ is the Bohr magneton (check the inside cover of your
book for it’s value in Joules/Tesla). Derive an expression giving the internal magnetic
energy, heat capacity and entropy for this system as a function of field strength and
temperature. Plot these expressions.