Homework #4, final version, Due Friday 4 PM *

At this point we are getting close to the culmination of our study of 1-D QM and nearing our transition to 3-D QM, which focuses on the electron in the hydrogen atom, and angular momentum and degeneracy manifolds.
This is a really critical assignment. Part I is designed to prepare you for our future discussions regarding the nature and meaning of quantum theory; part II explores the use of raising and lowering operator formalism for the harmonic oscillator potential, and thereby introduces the matrix formalism of QM.
I hope you can spend ample time on each part. The two parts are rather different, and it is probably a good idea not to try to do them both in one day. You may want to plan to at least 2 large-time-block HW sessions in your schedule for this week (if that is how you do HW). (In other words, i think it would be confusing to do both parts at the same session.)

Part I:
1) Sketch the lowest energy 4 states of the 2 finite square well system (2 wells of identical width (nm) and depth (eV)). Count the nodes of your states and make sure they follow a systematic sequence. Do the states tend to come in pairs? How do the upper and lower pairs differ?

2) [It is essential to understand this problem before we study the nature and energy of electron states in crystals (which are systems with spatially periodic potentials).]
Sketch the four lowest energy states for a 4 finite square well system (4 wells of identical width (nm) and depth (eV), all equally spaced). How does the node counting work out? Can you do this based on just the g.s. of the single well??

3) Consider and electron in the ground state of an infinite square well of width L/2 extending from just x=0 to x=L/2 for times before t=0. Suppose that at t=0 the potential suddenly changes to an infinite square well of width L extending from x=-L/2 to x=L/2.
a) How would you approach the question of the time dependence of the electron state for t greater than zero?
b) Describe roughly what happens as a function of time. (Don't spend too much time on this one; this is just a warm-up problem.)

4) (This problem will be important for our discussion of quantum measurement (of position)).
Consider an electron in the ground state of a finite square well or harmonic oscillator or some other potential. The question of measuring what people sometimes call "the probability that the electron is in a particular region of space" could be approached in the following way:: Suppose that at t=0, the potential changes to V(x)=-alpha*delta(x-x_0). As in the previous problem, the wave-function will be unchanged between t= 0- and 0+, and, for t > 0, it will began to evolve in time in a way that is different from its time dependence before the potential changed. As in the previous problem, determining the time dependence for t > 0 involves expanding the state at t=0, Psi(x,0), in terms of the energy eigenstates of the new potential, which in this case include a single bound state (the ground state) and a continuum of positive energy states (that tend to have kinks at x=0).
a) Find the integral expression for the amplitude of the ground state coefficient in the expansion, c_1. (Evaluate it if you can. This is not required. If you do, it should be less than 1 for all k.) Convince yourself that in the limit that alpha is very very large, and thus 1/k is very very small, the value of this coefficient approaches* (2/sqrt(k)) Psi(x_0, 0),
(where k is the decay rate of the delta function g,s. wave-function (which is sqrt(k) exp{-k|x-x_0|} ). If that is the case, what is the probability that the electron is in the g.s. of the delta-function potential for t>0 (which is proportional to (c_1)^2, right?) To what other "probability" is this related?
b) Describe qualitatively the behavior of the electron state as a function of time for t less than and greater than zero.
c) What would large k reasonably mean here? If k goes to infinity, i think the probablility would go to zero, but is there an in-between region where this would work? What would be a good way to define that reqion (in terms of length scales)?
d) Discuss how this could be used to establish a position measurement on the original state.
* I think this may be correct. Perhaps someone could check, confirm or correct this (by integration) ?

5) Suppose an electron is in the g.s. of a finite square well and that at t=0 a second well, of identical width (nm) and depth (eV)) suddenly appears. What happens (with regard to the electron state)? What is the time scale for that? (That is, on what does the time scale depend?) [This is a one-electron problem. There is no 2nd electron associated with the 2nd well. The potential just changes at t=0.]

6) (optional thought problem) Suppose an electron is in the g.s. of a 2 finite square well system (2 wells of identical width (nm) and depth (eV)), and that at t=0 one well suddenly disappears. What happens?

7) a) Sketch the g.s. and 1st-excited state of a 2 delta function system (2 attractive wells of identical strength). b) Assume that the g.s. has the form A cosh(kx) in the region between the delta functions, and that the 1st exc. state has the form A sinh(kx) in the region between the 2 delta functions. Show that the transcendental equations for k-value corresponding to each state can be expressed in a form wherein tanh(kd/2) interesects 2 different functions of k, one of the form k/(2k_0 - k) and the other of the form (2k_0/k) - 1.
c) Graph those 3 functions together. Indicate which intersection corresponds to the g.s. and which to the 1st x.s.? Which has a larger k value? Which has a larger energy?
c) Show, from that graph or another one, that in the limit where d, the separation between the delta functions, becomes very large the two solutions coalesce (to what value of k?), and that as d becomes too small, the 2nd solution disappears at a critical value of d (what is that critical value of d? Where is the solution/intersection just before that is reached.). Describe, explain or discuss these results.

8) [A technique related to this problem (perturbation theory) will be relevant to our study the energy and nature of electron states in crystals. This problem also has an interesting juxtaposition of length scales.]
Consider an electron in a potential that consists of an infinite square well with a delta-function in the center.
a) For given values of V_0, L and alpha, calculate the energies of the g.s. and the first excited state. [hint: this is probably less difficult if you move the edge of the well to x=0 and put the delta function in the center of the well at L/2. That way your edge b.c. (psi=0) is automatically satisfied, and you can focus on the discontinuity issue in the center.]
b) Now let's consider the case where the delta function is weak. Then we can look at the effect of the delta-function on the energies of the infinite square well eigenstates to linear order in alpha. What inequality, related to length scales, could define "weak"?
c) For these two states, show that for small alpha (positive or negative) the difference between the actual energy and the energy of the pure inf. sq. well eigenstate approaches the expectation value of the delta-function potential in that eigenstate. [Try expanding the value of k around pi/L to linear order in alpha, and then calculate the lowest order (linear) effect of that additional small delta k term on the energy (using the relation between E and k from the Schr. eqn.)

9) Suppose {B}n is a set of orthonormal basis vectors which span an N-dimensional inner-product space. a) How many basis vectors are there? b) How would you express an arbitrary vector, V, in this space in terms of these basis vectors?

10) a) Show by matrix multiplication that (1,1) and (1,-1) are eigenvectors of the matrix:
|1, 1 |
|1, 1 |
What are their eigenvalues?
b) Find the eigenvector with the largest eigenvalue of the matrix:
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
What makes you believe (show) that it is an eigenvector?
What makes you believe (show) that it has the largest eigenvalue?
c) What is the trace of this matrix?
d) (extra credit) Find an eigenvector with the smallest eigenvalue.


Part II:

11. Calculate the commutator [a,a+] from the fundamental x,p commutator relationship.

12. Express x in terms of a and a+.

13. Using raising and lowering operators and Dirac formalism:
a) calculate the expectation values of x and x^2 for each (energy) eigenstate of the harmonic oscillator.
b) do the same for p and p^2.
c) graph these as a function of n.
d) Imagine doing the same calculation for the infinite square well eigenstates. It would not be fun. Don't do it!
e) How does the product delta x * delta p depend on n. (Discuss or graph it if it looks interesting.)

14. (Perhaps we should move this problem to next week. What do you think?)
For a particle moving in two dimensions one can define a and a+ operators in terms of x and px and y and py.
a) What is the angular momentum in 2D (about the axis perpendicular to the 2D plane)?
b) Express that in terms of your 2 sets of a’s and a+’s.