PHYS 541 Homework Set #3 1. The single particle density of states g is defined so that per unit volume there are g d states in the energy range to d . For a particle in a box in one, two, and three dimensions compute g . Ignore spin. 2. Compute the energy distribution for fermions of spin s, i. e., for a three dimensional system of fermions in volume V in equilibrium at temperature T, find N d , the number of particles with energy d . 3. At low temperatures, the internal energy of a system of free electrons may be written as an expansion, 2 4 T 3 5 2 T A ... . E N F 1 5 12 TF TF Obtain the value of the constant A, and indicate the order of magnitude of the terms that have been discarded. 4. A white dwarf star consists of helium nuclei and free electrons. The density of the star is on the order of 107 g/cm3. (a) Compute roughly the number of electrons per unit volume. (b) Compute the Fermi energy and Fermi temperature for the electron gas, assuming a classical spectrum p 2 2m . (c) We really have to treat the electrons relativistically with single particle energies p p 2 c 2 mc 2 2 . Show that the ground state energy per electron of the electron gas is of the form E0 N m 4 c 5 1 n f x F where 23 y xF p F mc and f y dxx2 1 x 2 . 0 5. Consider the collapse of the Sun into a white dwarf. (a) Calculate the Fermi energy of the Sun’s electrons. (b) What is the Fermi temperature? (c) What is the average speed of the electrons in the fermion gas? (d) What is the density of the electron gas?