END; include ("common_short.php"); physics_header("",""); print <<Physics H133: Problem Set #\$ps_num 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. Q6T.5: What determines the probability of finding the quanton at a given position?
2. Q6T.9: For a free particle with well-defined momentum (or wave length), what can we say about the probability to find the particle at a given point? Are there any positions more likely than others? So if we measure a definite position (i.e. the quanton's state collapses to a position eigenstate), can the momentum be still well-defined?

## Chapter Q6 Problems

• Q6S.5: (a) What is the expression for the probability of collapse of either |psi> or |psi'> to |psi_f>? Now simply work it out. (b) You can use Eq. (Q5.7).
• Q6S.9: (a) Just use the normalization condition with the result for the integral in (Q6.31). (b) You need to do an integral over the region near the origin. If x is always small in this interval, you can do a Taylor expansion of the integrand (how many terms should you keep), leaving an integral you can do in your head.
• Q6S.11: (a) Is there experimental evidence suggesting that the die has settled down before we look? (b) Is there experimental evidence which is in contradiction with the assumption that the quanton has a definite position before we measure it (e.g. in the two-slit experiment)?
• Q6R.2: Write down a general normalized 2-component complex state |psi> and translate the statements given in the problem into conditions for these complex components. Which rule provides these conditions? After you have written them down, solve these conditions. There is more than one solution, but you need to find only one.
• Q6A.1: Simply follow the logic given in the problem statement. Write down general expressions for the desired pairs of 2-component vectors which satisfy the additional special properties stated in parts (a) and (b), respectively, and translate the expected 50 percent "overlap" probabilities into conditions for the unknown vector components. Solve them together with the conditions for orthogonality and normalization. The reasons why we can demand the special properties of the vector components (real, positive, etc.) given in parts (a) and (b) are explained in the Comment at the end of the problem -- this is not part of the problem to be solved by you (you did this part already in Q6S.5).