Issued Wed, Feb 28 Due Wed, Mar 6

Required reading: through F+T 3.7 (Beiser Ch. 5.1-5.6)

Reminder: (Optional) paper topic choice also due Wed, Mar. 6. (See instruction sheet I handed out earlier) Please hand this in separately from your h.w. Feel free to talk to me about your topic this week, if you're unsure.

1) A particle in a 1-D box has a minimum allowed energy of 2.5 eV.

a) What is the next higher energy it can have? And the next higher after that? Does it have a maximum allowed energy?

b) If the particle is an electron, how wide is the box?

c) The fact that particles in a 1-D box have a minimum energy is not completely unrelated to the uncertainty principle. Find the minimum momentum of a particle, with mass m, trapped in a 1-D box of size L. How does this compare with the momentum uncertainty required by the uncertainty principle, if we assume ?

2a) How much time is needed to measure the kinetic energy of an electron to an uncertainty of 0.10%, if the electron is known to travel at 5 m/s? How far does the electron travel in this period of time?

b) What is the uncertainty in the momentum of the electron from (a)?

What is the uncertainty in its position? Check the uncertainty principle.

c) Repeat part (a) for a 1.00 g insect. What do you conclude?

3. Verify that the uncertainty principle can be expressed in the form , where is the uncertainty in the angular momentum of a particle, and is the uncertainty in its angular position. (You may think of a particle, mass m, moving in a circle of fixed radius r, with speed v)

b) At what uncertainty in L will the angular position of a particle become completely indeterminate?

4. F+T 3-2. Assume that the potential V is time-independent, V=V(x).

5. F+T 3-6. (Use the infinite square well solutions, i.e. the particle-in-a-box equations, NOT just the uncertainty principle for this one.)

Extra Credit: (This one is more challenging than usual, but fun!) Using the results of problem 3, estimate the maximum length of time you could ever hope to balance a perfectly sharp pencil on its tip, in a vacuum, due to quantum effects. Assume a pencil weighing 10g, 10cm long.