Physics H133: Problem Set #7
Here are some hints, suggestions, and comments on the problem set.
Remember to give a good explanation, no longer than
- Q7T.5: Since de Broglie relates lambda to p,
you should first express the energy in terms of
momentum, and then use the de Broglie relation.
Take a look at the equations on page 130 to help
you do this.
- Q8T.6: What are the possible projections of a spin vector
of length given by s=5/2 along the given axis?
Chapter Q7 Problems
- Q7S.3: The pendulum's amplitude is the maximum angle by which
it deviates from the vertical position. How is it related
to the total energy of the pendulum bob? What is
its value if this total energy is the minimal value allowed
by quantum mechanics for a harmonic oscillator? To see
whether this is easy to measure, try to convert the angle
amplitude into a length that can be measured.
- Q7S.4: Review the discussion of the Bohr atom on p.130 and replace
everywhere the Coulomb attraction from the proton by its
gravitational attraction. Can you find a dimensionless ratio
which allows you to translate all relevant results for the
Bohr atom into corresponding versions for the gravitationally
bound hydrogen atom?
- Q7S.9: This just requires algebraic manipulations, remembering
that eix = cos x + i sin x.
- Q7R.1: This is a 3-dimensional problem, and treating it as a
1-dimensional box is only good for obtaining a rough
order-of-magnitude estimate of the energies and length scales
involved. (This is a little more quantitative than simple
dimensional analysis.) -- You are supposed to show that the
electron's (positive) kinetic energy will necessarily be much
larger than its (negative) electrostatic potential energy; so
you should try to arrive at a lower limit for
the former and an upper limit for the latter and
compare these two. -- How can you calculate the electron's
kinetic energy in an energy eigenstate of the
box potential? In which state will it be smallest? To estimate
its potential energy in the electric field of the nucleus,
the box model is no good (why?). To obtain an upper limit,
you can assume that the nuclear charge Ze=50e is concentrated
in a point. What should you take as a reasonable average
distance for the electron's distance from this point charge?
You can use the knowledge from chapter Q9 that 3-dimensional
wave functions have zero probability at r=0 (i.e. at the position
of the point charge).
- Q7A.1 Follow the hints and instructions in each part.
Your comments and
suggestions are appreciated.
Physics H133: Hints for Problem Set 7.
Last modified: 05:19 pm, April 15, 2012.