Issued Wed, Oct 23 Due Wed, Oct 30

(Required reading for this week: Ch. 7)

There are HINTS for this homework!

1a) I often write (*)

(where is any normalized wave function, and the 's form some complete orthonormal basis).

Use Dirac notation to prove that (**)

b) The above assumed that the basis vectors, , are characterized by discrete eigenvalues, labelled by (integer) n's. Redo part a when the basis set instead has continuous eigenvalues labelled by (continuous) p's.

(Please write down the analogy for equations (*) and (**), and show how the second comes from the first)

2) You have a particle in a harmonic oscillator, with normalized eigenfunctions as usual. (So, ).

Imagine there is some other observable B, with associated operator B, which has normalized wave functions and eigenvalues . (So, ).

Suppose that you were given some explicit form for B, and you solved for the eigenfunctions, and found .

Now take an electron in the harmonic oscillator, and measure the quantity B. Suppose you happen to get . Then measure its energy. (What are the possible results of this measurement, and the respective probabilities?) Right after measuring the energy, measure quantity B again. What is the overall probability that you will get again?

Discuss your result. E.g: 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, you don't always get again! How can this be? (Do you think B commutes with H?)

3) Consider a conservative system, where :

a) Use Gas 6-64 to show that for any particle, .

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

(cont.-->)

b) For stationary states (i.e, your wavefunction is an exact eigenstate of energy, or put another way, it separates into a function of x and a function of t), the above uncertainty principle doesn't tell you anything. Why not?

4) Gas 6-13. (When Gas. writes, e.g. dx/dt, he means of course d<x>/dt)

Solve the equations you get for x and p simultaneously, to find

<x>(t) and <p>(t).

(Assume your particle starts out localized at the origin, with zero average momentum at time t=0.)

5) For any conservative system:

a) Use Gas Eq. 6-68 to show that ,

where T=kinetic energy (H=T+V).

In a stationary state, show that the LHS is zero, so .

(This is the quantum version of the Virial theorem).

Use this to show that <T>=<V> for any energy eigenfuction of the harmonic oscillator.

b) Show that . Use this relation to find . State in words what "freshman physics" this tells you, in the classical limit.

c) Apply Gas. Eq 6-68 to the cases A=1, and A=H. Briefly discuss and interpret the significance of your results.

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