Download Physics 535 lectures notes: 1 * Sep 4th 2007

Physics 249 Homework 5
Due Oct 19th
1) Consider two potential steps. One from V(x<0)=0 to V(x>0)=V0 and the other from
V(x<0)=0 to V(x>0)=-V0. Consider a particle incident on this step potential from
–x with total energy E=2V0. Note this is the total energy including the potential
energy. For each calculate.
a) The wave numbers in regions two (x>0) in terms of the wave number in region 1 (x<0)
and V0.
b) The reflection coefficients.
c) The transmission coefficients.
2) Consider potential barrier 0.5nm wide and 10eV high. Electrons accelerated through
5V approach the barrier. What percentage of electrons will tunnel through the
barrier?
3) Consider a infinite 3D box potential with L2=2L1 and L3=4L1. What are the quantum
numbers of the lowest degenerate energy levels? List, ordered by energy, the
quantum numbers and energies of all the levels up two the lowest energy set of
degenerate energy levels.
4) The radial probability distribution function for the hydrogen atom ground state can be
writer as 𝑃(𝑟) = 𝐶𝑟 2 𝑒 −2𝑟/𝑎0 , where C is a normalization constant.
a)Why is the factor of 𝑟 2 included?
b) Show that P(r) has it’s maximum value at r=a0.
c) If you considered equal infinitesimal volumes, what at what radius is the maximum
probability to find an electron?
d) Calculate the expectation value of r.