Problem 2 (10 points):
a) [6 points] Consider a particle of mass m moving in a one-dimensional potential U(x). The energy
eigenvalues are nE? with n = 0, 1, 2, 3, etc. Seven of these identical non-interacting quantum-mechanical
particles are placed in this potential. When the particles are fermions of spin angular momentum ±?, the
ground-state energy of the system is E<sub>f</sub>. When the particles are bosons of zero spin angular momentum,
the ground-state energy is E<sub>b</sub>. Evaluate E<sub>f</sub> and E<sub>b</sub> in terms of given quantities.
b) [4 points] The possible wave functions of a single particle of mass m in a box (infinite well) of width
a in one dimension are
$\psi_n(x) = \sqrt{\frac{2}{a}} \sin(n\pi x/a)$,
with n a positive integer. What is the wave function $\Psi(x_1, x_2)$ for two identical, spinless, non-interacting
bosons in this box when the particles occupy the state with n = 1 and the state with n = 2?
