Issued Wed, Oct 29 Due Wed, Nov 5

Exam II will be Thurs, Nov. 6 at 7:30 PM. It will cover Gas. Ch. 1-6, with most emphasis on new material (since the first exam.)

(Required reading for this week: Ch. 7)

There are HINTS for this homework.

1) You have a particle in a harmonic oscillator with normalized eigenfns (So, ). Say there is some other observable B, with associated operator B, which has normalized eigenfns and eigenvalues .

(So, ).

Suppose you solved for the eigenfunctions of B, and found the lowest two are given by .

Now take an electron in the harmonic oscillator, and measure the quantity B. Suppose you happen to get . Then you go and measure its energy:

i) What energies can you possibly measure? With what probabilities?

ii) Say this energy measurement gives you , and then you measure quantity B again right away. What is the probability that you will find ?

ii') Say instead you measure energy, put don't pay any attention to what you get, then you measure quantity B again right away. What is the total probability that you will get ? (The answer is different than in ii !)

iii) If you measure B and get , and then measure B again right away, what would you get?

But if you measure B, and get , then measure energy, then measure B again (as described in part ii or ii' above), you don't always get again!

How can this be? Do you think B commutes with H? (Briefly discuss.)

2) Consider a system where :

a) Use Gas 6-64 to show that .

(This is a kind of screwy mixed up position-energy uncertainty principle)

b) For stationary states (i.e, wave functions that are exact eigenstates of energy) the above uncertainty principle doesn't tell you anything. Why not?

(over ->)

3) A particle is in a harmonic oscillator, with .

a) In general, find and . (Express your answers in terms of m, , and . .)

b) Suppose your particle begins (at t=0) in the ground state, .

Is momentum, , conserved for all time?

Does the particle's position, , ever change with time?

Does the particle's energy, , ever change with time?

Hint: Think about using symmetry arguments. I claim you don't need to do any integrals for this part.)

b') Repeat part b, if you start (at t=0) in the first excited state, .

Extra Credit: 3c) Repeat part b if you start in a mixture of the first two states:

(The next problem is not extra credit, only 3c above was...)

4) Gas 6-12a only.

Note, Gas. asks you to solve the equations you get for x and p simultaneously. That means explicitly find <x>(t) and <p>(t)!

You should assume your particle starts localized at some given <x>(0), with some given <p>(0), at time 0, and write your answer in terms of those 2 givens (and the other constants in the Hamiltonian, m, , and , of course)