More Problems

Problem 1:

Consider a free particle of mass M in two dimensions.  Since continuum states are not traditionally normalizable, we assume that the particle is confined to a cubical 2D box with periodic boundary conditions. Its eigenfunctions are
ψnq(x,y) = (1/L)exp(ikxx)exp(ikyy),
with kx = 2πn/L,  ky = 2πq/L, n, q = 1, 2, ... .
Derive and expression for the density of states dN/dE = ρ(E).

Solution:

Problem 2:

In a one-dimensional structure a particle of mass m is in the ground state of a potential energy function U(x) = ½kx2.  A phase transition occurs, and the effective spring constant suddenly becomes k' = k/2.  What is the probability that the particle will be found in an exited state after the phase transition?  Give a numerical answer.

Solution:

Problem 3:

An electron in a hydrogen atom is in one of the degenerate |n,l,m>  = |4,2,m> states.
For all possible |4,2,m> states, which radiative transitions to a lower energy state are allowed by the dipole selection rules?
Neglect spin.

Solution: