# Physics H133: Problem Set #7

Here are some hints, suggestions, and comments on the problem set.

## Two-Minute Problems

Remember to give a good explanation, no longer than two sentences.

1. 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.
2. 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.