Homework #5, due Friday, 4:30 PM

Added notes: Problem 4 is particularly important; (3 is a prelude to 4). The image to the right is from a comment by Jerome about problem 4. (See comment #14 below.

The matrix problems, 5 and 6, should be computationally easy, yet very intriguing. No one is really expected to do problem 8.

1. For the 1-D harmonic oscillator:
a) Starting from the ground state wave-function, psi(x), use a+ to generate the 1st excited state wave-function and then graph it vs x.

2. Graph:
a) 2(x /a)^2 - 1
b) 2 (x/a)^3 - 3
correction added: that should say 3x instead of "3".
In what regions are each of these functions positive? negative? At what values of x do the sign changes take place?
c) Note that these are related to harmonic oscillator ground states.

3. Consider an electron moving in 2 dimensions in the potential V(x) = (1/2) k [x^2 + y^2].
a) what is the kinetic energy operator?
b) Starting with the ground state find the first three eigenstates* of H, for a particle of mass m in this potential. [Hint: If you can show that the differential equation is separable with regard to x and y, then the solutions (eigenstates) can be constructed via products of 1D HO eigenstates.]
* That is, the 3 eigenstates which have the lowest energies.

4. Using the (standard) definition of 2D angular momentum (z component), Lz= x p_y - y p_x ,
a) show that Lz can be expressed as - i hbar [ ax+ ay - ax ay+]
[hint: I think that x and y raising and lowering operators commute with each other; this is related to the separability of the diff. eq., i believe. Actually, do you need to use this? I am not sure.]

b) Calculate the expectation value of Lz for each of your eigenstates from #3

c) Being careful about commutation, calculate Lz^2 (conceptually similar to p^2, except now we are dealing with angular, rather than linear, momentum.)

d) Calculate the expectation value of L^2 (==Lz^2, in 2D) for each of your 3 states from #3. What do you learn from this? Are these expectation values zero or non-zero? Feel free to interpret speculate, propose new directions for investigation...

e) How many states are in the next "degeneracy manifold"? (That is, if you consider a 4th state, how many states are there that have the same energy and are linearly independent?) What are they? (That is, what is a spanning set for this degeneracy manifold?)

5 a) For the matrix:
| 0 1 0 |
| 1 0 1 |
| 0 1 0 |
Use intuitive methods (i.e., guess a vector and mutiply it by the matrix, ask a friend or a computer (anything NOT involving determinants)) to find the eigenvectors. Show that your putative eigenvectors are indeed eigenvectors by direct multiplication. What are their respective eigenvalues.

5 b) a) Multiply the matrix:
| 0 0 0 |
| 1 0 0 |
| 0 1 0 |
times the vector (1,0,0).
b) Do it again. (I mean with the vector that you got from a). Not the original one.) What do you learn from this?

6. a) Find the eigenvectors, and their respective eigenvalues. for each of the following matrices:
| 1 0 | | 0 1 | | 0 i |
| 0 -1 | | 1 0 | | -i 0 |

b) Express the eigenvectors of the 2nd matrix in the basis of the eigenvectors of the 3rd matrix.
Correction: Well, how about eigenvectors of the 1st matrix in the basis of the eigenvectors of the 2nd matrix. (You can do the 2nd in terms of the 3rd for extra credit, if you want.)

c) What do these matrices have in common? Do you think they like each other?? Why or why not!?
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7. Consider an electron in a 1D potential which is constant.
a) Show that exp{ikx} is an energy eigenstate with energy ___.
b) Show that it is also a momentum eigenstate with momentum hbar k.
c) Show that cos(kx) is also an eigenstate of one of these but not the other. Which one? Discuss briefly. (What is the relationship between exp{ikx} and cos(kx) ?)
d) For an intitial state, Psi(x,0), that is of Gaussian form (like the HO g.s.), calculate the time dependence of the state function, and of the expectation value of x^2.
e) For this state, show that the expectation value of p is zero, and that
f) the expectation value of p^2 is independent of time.

8) If you are really ambitious*, take the same state, multiply it by exp{ik_0 x} and then recalculate everything. I think you will likely find that the expectation of p is now finite (hbar k_0) and the the expectation value of x is now time dependent, and that you have a wave-packet moving to the right at a speed proportional to k_0/m .
*This might not be fun. Please do problem 9 first. You really don't have to do this problem.

9. Consider the 1d delta-function potential, V(x) = alpha delta(x).
Suppose that you write the part of the state function to the left of delta function as:
exp{ikx} + B exp{-ikx},
and the part of the state function to the right as
C exp{ikx},
and that you think of these terms as representing an incoming wave from the left, a reflected wave (B), and a transmitted wave (C).

a) Use the boundary conditions at the position of the delta function to calculate B and C.
b) Are B and C real or complex? What do they mean? Is there something you could graph as function of alpha that might convey some interesting information regarding them? Discuss.
c) Evaluate |B|^2 + |C|^2.
d) How are your results different for positive and negative alpha?