Modern Physics, Spring '10

Homework 8 (covers Townsend Chapter 5)

1. Problem 5.2 (Show that the operator...)

2. Problem 5.5 (For a particle in a harmonic oscillator potential...)

3. Problem 5.8 (Let the operator...)

4. In section 5.4 Townsend develops the famous Heisenberg Uncertainty Principle for position and momentum (equation 5.50). He says that this is just one of "many other important uncertainty relations" -- one of which is then developed in the next section. Let's think about some other possibilities. For example, consider a free particle ("free" meaning that V(x) = 0 everywhere). Is there some minimum value to the product of the uncertainties in its momentum and its energy? How about for a particle in a harmonic oscillator potential? (Note, you don't have to figure out what the HUP type equation would look like exactly... the question is only whether the right hand side is zero or nonzero in the two cases.)

5. Well, I can't resist asking you some things about Section 5.6. On page 171, Townsend summarizes the EPR argument and notes that "For Einstein, this 'spooky action at a distance' was unacceptable, something that 'no reasonable definition of reality' should permit." Is Townsend here disagreeing with Einstein, i.e., saying that 'spooky action at a distance' is acceptable? More generally, what do you take him to be saying here?

6. Again with Section 5.6. Near the top of page 170, Townsend talks about the collapse postulate and says: "How this collapse happens is a mystery. It is referred to as the measurement problem." Then later, at the very end of the section, he suggests instead that "the crux of the measurement problem" is the fact that, according to Schroedinger's equation, the final state (of e.g. a cat in a box) should be a superposition if the initial state is. What is the relationship between these two statements about what the measurement problem is? That is, do they amount to the same thing, or are they totally different and unrelated, or what? Stepping back, do you see any serious "problem" associated with "measurement" for this theory?