Homework 6.

edits:
 5:00 PM, Feb 19: [problem 3 clarified. 7 edited. 7 is interactive.]
11:00 AM, Feb 18: [problem 1 note added for clarity.]
Do all these problems seem worth your while? What is your favorite and least favorite? If you were to eliminate one problem, which one would it be?

[Kinetic energy.] 
1. Consider a 2 square well system: 2 identical wells of width L spaced a distance S apart. If you put the origin of your coordinate system exactly between the wells, then the wave-function in that center region is of the form: \(A_1 cosh(x/a_1)\) for the ground state and \(A_2 sinh(x/a_2)\) for the 1st excited state. You are given that: \(A_1=0.2 (\frac {2}{L})^{1/2}\), \(A_2= (\frac {5}{4})A_1\) and \(a_1= a_2=0.2\) nm, and the region separating the 2 wells is 0.4 nm wide.
a) Sketch the ground state and 1st excited state (over all regions). Pay particular attention to the center region and make your sketch reasonably consistent with the numbers given for the pre-factors and length scales. 
b) For the ground state, calculate the contribution to the kinetic energy integral from just the center region (that is, integrating from x = -0.2 nm to +0.2 nm). (To make your result readable by the graders, and to you, please leave your answer in terms of \(A_1\) and \(a_1\) and a numerical result for the definite integral over \(cosh^2(x/a_1)\), which will have units of nm. (By the way, those nm will cancel the units of \(nm^-1\) from \(A_1^2\) .) There will, of course, be other constants that always are present for a K.E.. )
c) Do the same thing for the 1st-excited state. [Again, please leave your answer in terms of \(A_1\) and \(a_1\) and a numerical result for the definite integral.]
d) Summarize/discuss your results. What does this suggest about the overall K.E. for these states?
e) (extra credit) Plot the integrand in each case; show the area under it and discuss.


[States of multiple well systems.]
2. Consider a system of 4 equally-spaced identical wells. Suppose that for a single isolated well \(\psi_1\) is an energy eigenstate with energy \(E_1\).
        When there are 4 wells, it turns out you can make 4 linearly independent, orthogonal energy eigenstates from the single well state \(\psi_1\). Using a notation where an electron in the left-most well is (1,0,0,0) and an electron in the state \(\psi_1\) in the right-most well is (0,0,0,1), one can write the ground state as approximately: \(\frac {1} {\sqrt{4}} (1,1,1,1)\)
a) Create 4 orthogonal states using this notation (with nodes that respect the symmetry of the system).
Let's call these states: \(\psi_{1,1}\), \(\psi_{1,2}\), \(\psi_{1,3}\) and \(\psi_{1,4}\).
Let's call their energies: \(E_{1,n}\), n=1 to 4.
b) Order these states according to their energies (lowest to highest).
c) Sketch a plot of each state. While this is just a sketch, be careful to keep the wave-fucntions close to zero in the regions between the wells.
d) How many nodes does each state have? Are they symmetrically located?

3. a) Continuing our exploration of a system of 4 equally-spaced identical wells. Do the same thing as the previous problem, but construct your states from the single-well state \(\psi_2\) (instead of from the ground state, \(\psi_1\)). Does that make sense?
b) What are the lowest and highest energy states in this group of 4?
( Let's call these states: \(\psi_{2,1}\), \(\psi_{2,2}\), \(\psi_{2,3}\) and \(\psi_{2,4}\).
Let's call their energies: \(E_{2,n}\), n=1 to 4.)
b) What are the lowest and highest energy states in this group of 4?
[This problem may be more difficult than it looks.]

[Time dependence.]
4. Consider a 2 square well system: 2 identical wells of width L spaced a distance S apart.
a) Show that you can make a state where the electron wave-function is mostly on the left hand side at t=0 by using a superposition state that combines the ground state and 1st excited state of the double well. Include sketches. What is that superposition state?
b) Guess the expectation value of x for this state. Is it zero? Why or why not?
c) Factoring out an overall phase factor, \(e^{-iE_1t/\hbar}\), show that there is a time, not too far from t=0, at which the phase relation between the states is reversed. What time is this? Guess the expectation value of x at this time?
d) As the wells get far apart, how does that effect the relationship between E_1 and E_2 ? How does that effect the time that you got in part c)?
e) How can the electron get from one side of the well to the other when the potential between the wells is higher than the electron energy (in either state). Discuss the relationship between: well proximity, E_1 and E_2, and how long it takes the electron to get from one side to the other.

5. For the 1D harmonic oscillator:
a) Calculate the expectation value of \(x^2\) for an electron in the superposition state that is an equal mix of \(\psi_1\) and \(\psi_3\).
b) Show in a graph how this oscillates as a function of time. What are the minimum and maximum values of \(\langle x^2 \rangle\) ?
c)  What does this imply about the kinetic and potential energy (expectation values) as a function of time for this state??? Discuss and graph.

[Selection rules.]
 6. An oscillating electric field in the x direction produces an oscillating potential energy \(V_{E+M} (\vec{r},t) \propto E_x\, x\, cos(2\pi f t) \). The strength of a transition between two eigenstates in the presence of this "perturbing potential" is proportional to the integral over all 3D space of the integrand \( \psi^*_i (\vec{r}) (E_x \, x)\, \psi_f (\vec{r}) \propto \psi^*_i (\vec{r}) (x) \psi_f (\vec{r}) \), where \(\psi_i (\vec{r})\) and \(\psi_i (\vec{r})\) are the initial and final states, respectively. [Note added: There is an x in that integral that comes from the direction of the electric field. Don't overlook the x.]
a) Given this information, which transitions from the ground state to a first-excited state are possible and which are forbidden?  For each of the possible (allowed) transitions, what polarization of light do they require? (Polarization refers to the direction of the electric field in this case.)
b) Extra credit: Starting from the initial state \(\psi_{21x}\), which transitions to 2nd excited states of the form \(\psi_{30-}\) and \(\psi_{31-}\) are possible? (In other words, leaving out d states, i.e. states of the form \(\psi_{32-}\).)
c) extra credit: Starting from the initial state \(\psi_{200}\), which transitions to 2nd excited states of the form \(\psi_{30-}\) and \(\psi_{31-}\) are possible?
(These are dipole selection rules. They refer to transitions associated with an oscillating electric dipole moment, which are the strongest transitions.)

[Time dependence of a free electron.] This problem involves a superposition state for a free electron expressed as an integral. It is a wave-packet.
This is an interactive question. Some participation is required in order to provide the motivation to finish this complex but interesting problem.
7. In 1D, for an electron in a flat potential, U(x) =0, the energy eigenstates can be written as:{cos(kx), sin(kx)}, k= 0 to infinity.
a) Show that these are energy eigenstates and determine their energies.
b) sin(kx) and cos(kx) have the same energy, and it turns out to be convenient to combine them to form a new basis of states {e^ikx} k = -infinity to infinity. Show that these are momentum and energy eigenstates. How does their momentum and energy depend on their k value?
Although these states are not normalized, one can create a simple normalized state from a superposition of these states, e.g.,
\(\psi(x) =\frac{1}{(\sqrt{2 \pi} a)^{1/2}} e^{-x^2/(4a^2)} =\int_{-\infty}^{+\infty} \phi(k)\, e^{ikx} \, dx\),
where \(\phi(k)\propto e^{-k^2a^2}\).    [In more detail, \(\phi(k)= \frac{\sqrt{a}}{(2 \pi)^{1/4}}  e^{-k^2a^2}\).]
c) Plot \(\psi(x)\). Plot \(\phi(k)\).  (Two different plots, two different scales and axes.) What does \(\phi(k)\) represent?
d) Important: How could you incorporate time dependence into this superposition state?  In other words, without calculating anything, what would you put into the integrand of  \(\psi(x) =\int_{-\infty}^{+\infty} \phi(k)\, e^{ikx} \, dx\) to get an integral expression for \(\Psi(x,t)\)?

Now let's consider a different state: one with a k distribution centered at positive k. This will make a state that moves (to the right) as a function of time.  Consider the same expression as above, but with  \(\phi(k)= \frac{\sqrt{a}}{(2 \pi)^{1/4}}  e^{-(k-k_o)^2a^2}\), where \(k_o\) is a positive number (corresponding to a positive momentum).
With this \(\phi(k)\) the wave-function now has a new phase factor \(e^{ik_o x}\). That is, \(\psi(x) =\frac{1}{(\sqrt{2 \pi} a)^{1/2}}e^{ik_o x} e^{-x^2/(4a^2)}\). This wave-function (wave-packet) will move to the right as a function of time (at a speed \(\hbar k/m\)). To show that we need to be able to put time dependence into this state in a systematic and correct manner.
e) What is the integral expression for \(\psi(x)\)?   How would you put time dependence into the integral expression for \(\psi(x)\) to get an integral expression for \(\Psi(x,t)\)?
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Let's stop here for now and we will put the rest of this problem on next week's HW. Does that sound like a good plan?
When we work on this further, we will define a unit-less quantity proportional to time,
\(T=\frac {\hbar \, t} {2ma^2}\). This makes the exponential time dependence more manageable.

...
\(\Psi(x,0) = =\int_{-\infty}^{+\infty} \phi(k)\, e^{ikx} \,  dx\),

This integrates to:

where

which can be written as:

The key thing to seeing and understanding the phenomenology of this moving wave-packet is to simplify the notation and parameters involved, e.g., by creating a unit-less time, T, and using variables that make sense to start with: a for length (the width of the packet at t=0 is proportional to a) and \(k_o\) for the center of the packet...
e) Considering Psi(x,t), where is it centered at T=1? What is its width at T=1?
f) Discuss/describe the phenomenology of this wave packet (as a function of time).