# Physics H133: Problem Set #4

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. Q4T.4: Standard interference problem! Once you know the de Broglie wavelength, the problem goes through as with any two-slit interference problem involving waves of that wavelength.
2. Q4B.8: Is this ball relativistic or non-relativistic? Remember to make a comparison. Don't just say that the wavelength is small. Also, what determines the wavelength size (i.e., in an alternate universe, what could be different that would make it important to consider the wave nature of a pitched baseball?).

## Chapter Q4 Problems

• Q4S.2: How do you go from voltage difference to kinetic energy? (Hint: Express kinetic energy in eV.) The electrons are nonrelativistic if K << mc2; if they are nonrelativistic, you can use K = p2/2m. For a proton, repeat with a different value for mc2.
• Q4S.7: How does the de Broglie wavelength depend on v? You can estimate the mass of the buckeyball by noting it is 60 carbon atoms, each with 6 protons and 6 neutrons. (Don't try to calculate it too precisely, we're only interested in a few percent answer here.)
• Q4S.11: Do you think the de Broglie wavelength of the electrons should come out much smaller, smaller, about the same, or larger than the size of the nucleus? Why did Hofstadter get the Nobel prize for this work (look it up!)? He obviously saw something important. (c) What is the rest mass energy equivalent of an electron? Larger or smaller than 20 GeV?
• Q4R.2: The speeds are nonrelativistic, so the ordinary expression for the de Broglie wavelength (with the new h!) can be applied. You'll need to estimate the spacing of interference fringes (bright and dark spots) at the far end of the courtyard in order to judge if it is feasible to see the interference. Question: Does destructive interference of people hurt? :)
• Q4A.1: The relativistic formulas for energy E and momentum p are related by E/p = c2/v, which follows immediately from E = gamma*(mc2) and p = gamma*mv, with gamma = 1/(1-v2/c2)1/2. (These relations can be found in R9 and follow from a Lorentz transformation from the rest frame to a frame moving a velocity v.) The velocity of the crests of the waves is f*lambda. To evaluate d(omega)/dk, you'll need to substitute the relativistic expressions for E and p in terms of gamma and actually take the derivatives.