Issued Wed, Apr 17 Due Wed, Apr 24

Required reading for this week: F+T Chapter 9.7, 9.8, 5.1-5.5

(Optional: Beiser 5.8, 6.1-6.5)

1. F+T 9-13

2. F+T 9-15

(For a, answer as physically as you can, and also give a mathematical reason)

(For b, use F+T eqn. 9-17)

In addition:

2c) For the proton beam only: how much energy would it need, in order for about 1% of the beam to make it through the barrier?

Note: your answer may be rough, i.e. use F+T eqn 9-18 rather than 9-17.

Do you expect the true answer (i.e. if you did use eqn. 9-17) to be bigger or smaller than what you get? Why?

3. My driveway is flat, and at the same level as the main road in front of my house (which then runs downhill towards Broadway) There is a "speed bump" stretching across the end of my driveway, approximately 1 foot wide, and about 4 inches tall, so I figure there's no real need to set the parking brake - even if the car is drifting, the bump will stop it...

For this problem, convert to mks metric units, assume a "pointlike" car (its a subcompact), and work to only one significant figure. Please express final answers as powers of ten (not powers of e).

a) Suppose I give the car a good hefty shove (getting it moving at, say, 2 mi/h which is about 1 m/s) towards the bump. What is the kinetic energy of my car? (Assume it weighs 2200 lb = 1000 kg) What is the potential energy required for it to climb over the bump? Show that my car is safe, and will not roll away.

b) Is there any worry that the car might "tunnel" through the bump and roll away down the hill? What is the likelihood of this happening, given the above numbers?

c) If I try this little stunt once a second, all day, every day, how long would I have to try before I was likely to lose my car? (Answer in seconds, and also in ages of the universe). Would it matter if I try it 10 billion times a second?

Do you think quantum tunneling is important for macroscopic objects?

4. A particle is stuck in a rectangular box, with infinitely hard walls whose edges are a, b, and c long. In class, we found that the wave function of the particle is given by: . (the n's are three independent quantum numbers, which must be positive integers)

a) Find the value of the normalization constant, A.

b) If the particle is in its ground state (so, all three n's =1), what is the probability that this particle will be found in the volume defined by

0 <= x <= a/4,

0 <= y <= b/4,

0 <= z <= c/4?

Does your answer agree with your classical intuition? Explain.

EXTRA CREDIT: F+T 9-11

(Note: F+T have different labels: comparing with my class notes, their A is my , and their D is my . Part c will be easiest if you let MMA do it...)