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Homework set #14
Phy 3031 Modern physics, Spring 2007
To be handed in on Wednesday, April 18, 2007. Note that there was no homework set #13.
For numerical answers give two significant digits. The first should be correct, the second can
be approximate.
1. A free particle of mass m, with wave number k1 is traveling to the right. At x = 0 the
potential jumps from zero to V0 = h̄2 k12 /4m and remains at V0 for all x > 0.
(a) If the total energy is E = h̄2 k12 /2m = 2V0 , then what is the wave number k2 in
the region x > 0? Give your answer in terms of k1 .
(b) What is the reflection coefficient at the step in the potential at x = 0?
(c) What is the transmission coefficient?
(d) If one million particles are incident on this step, with these conditions, then how
many particles are reflected by the potential step?
(e) How many of the one million particles would be reflected if classical Newtonian
physics were correct?
2. A free particle of mass m, with wave number k1 is traveling to the right. At x = 0
the potential jumps from zero down to −V0 = −h̄2 k12 /4m and remains at −V0 for all
x > 0. To be clear: We are assuming that V0 is positive, so that the potential V = −V0
is negative for x > 0.
(a) If the total energy is E = h̄2 k12 /2m = 2V0 , then what is the wave number k2 in
the region x > 0?
(b) What is the reflection coefficient at the step in the potential at x = 0?
(c) What is the transmission coefficient T ?
(d) If one million particles are incident on this step, with these conditions, then how
many particles are reflected by the potential step?
(e) How many of the one million particles would be reflected if classical Newtonian
physics were correct?
3. A beam of electron, each with kinetic energy 40 eV, approaches a step potential with
a height V = V0 = 30 eV.
(a) What fraction of the electrons are reflected?
(b) What fraction would have been reflected if the particles had been protons, rather
than electrons?
4. Five identical, noninteracting particles are placed in a one-dimensional infinitely high
square well with length L = 1.0 nm.
(a) Compute the lowest total energy for the system if the particles have the same
mass as the electron, and spin s = 1/2.
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(b) Compute the lowest total energy for the system if the particles have the same
mass as the electron, and spin s = 0.
5. Consider the wave functions for the energy levels of an electron in the hydrogen atom.
If n = 3
(a) What are the possible values of l?
(b) For each of these values of l, what are the allowed values of m? (You should have
a separate list of m values, for each of the l’s in the l list.)
(c) Recall that the spin of the electron allows the possible electron states to occur in
pairs. What is the total number of electron states with n = 3.
6. Write the electron configuration of
(a) Carbon
(b) Oxygen
(c) Germanium