Physics 214b-2008 Walter F. Smith Assignment 9

Due: 4 pm Wednesday 4-9-08

Reading: Townsend 6.1-6.3

Exercises All problems are group problems, unless otherwise marked; you are encouraged to work in small groups on these.. For the individual problems, you are not allowed to consult any other students, but you may ask me for help.

5.2 Hint: use integration by parts.

5.11 Work this problem using bra-ket notation – do not write any integrals.

9A. (Adapted from Griffiths 3.3) In class, we defined a Hermitian operator to be one with the property for all y. Show from this that for all f and g. Hint: first set , then set .

9B. In class Friday, we will show that for any observable . Use this to show that . (This is Ehrenfest’s theorem, which we proved earlier using other methods.)

9C. Resonant tunneling and molecular electronics. Consider the following potential energy:

Thus, this is two barriers each of width a, with a potential well in the middle of width b. We will use this as a model for the type of experiments that I do in my research lab, in which we measure the electronic properties of molecular assemblies. The regions to left and right represent the metal electrodes, the well in the middle represents the molecular assemblies, and the two barriers represent the tunneling barriers which almost always occur at a contact between a molecular assembly and a metal electrode. As you will see in this problem, by varying the energy of the electrons coming out of the electrodes relative to the energy levels in the molecular assembly, we can map out the energy levels of the molecular assembly.

a. Assuming a wave incident from the left, find the transmission coefficient for this structure in terms of and . You will definitely want to use Mathematica for this. The output can get pretty long, so you have to use Simplify right away; here’s an edited version of what I did, including my final result.

b. Let a = 5 Å, b = 2 Å, and , where is the ground state energy for an infinite well of width b. Use the exact values of physical constants that I have e-mailed to you. Enter the definitions for k and k into Mathematica (in terms of the energy), then make a clickable plot of the transmission coeffient as a function of energy from 0 to 8, with the vertical scale going from 0 to 1. To the right, I show how I made this plot. Note that the PlotRange command sets the vertical axis scale. The combination of the ClickPane command and the Dynamic command makes your plot clickable , so that each time you click with the mouse, the line below “Dynamic[pt]” gets updated with the coordinates. (Edmond Rodriguez taught me how to do this.) This makes it easier to explore your plot quantitatively. (Print out this plot and the code leading up to it.)

c. What you see may surprise you. Comment on the energy at which the transmission coeffient first begins to increase substantially – why should you expect this rise?

d. At energies higher than the initial peak, you should see a pattern of peaks and dips. Comment qualitatively on what might be causing this. Hint: think about constructive interference.

e. Make a new plot, covering the energy range from 0 to 2.1 , and with the vertical range going from 0 to 0.05. You should see a surprising newly-apparent feature. (This was present on the previous plots, but not visible because of the scales.) The feature is actually taller than it appears – it is so narrow that the full height is not being displayed. Zoom in on the feature by decreasing the energy scale until it is covering a range of 10^‑20 J, and increase the vertical scale as needed so that the entire feature is displayed. At what energy does this feature appear? (Print out this plot.)

f. For problem 8C. part b, you got (or should have) an energy of 2.467_-18 J. Comment on the ways in which the potential energy for that problem is similar to the potential energy for this problem.

g. Speculate qualitatively on what gives rise to the feature you saw in part e, and why this effect is called “resonant tunneling”.

h. Change the width of the barriers to 4 Å (keeping b = 2 Å), and re-do the plot using the same scales. Print out this plot, and comment on the ways in which it differs from the version with the 5 Å barriers.

i. Repeat part h with barrier width of 2 Å; you will probably need to rescale your plot to show the entire feature. What you should be seeing reflects the “broadening” (in energy) of the “virtual state” within the well as it is more strongly coupled to the outside world. This is a universal feature of quantum systems. It can be observed, for example, in the broadening of the energy levels of an atom when it sticks to a solid substrate. Measure the “full width at half maximum” or “FWHM” of the feature, i.e. the width of the peak at half its maximum value. (The easiest way to do this is to set the PlotRange to go from 0 to half the maximum value.)

j. Keep the barrier width at 2 Å. If you think about the results above, you should realize that there should be another interesting feature at lower energy. First calculate the energy at which this feature should appear, then zoom in your plot on this energy range. (This feature is so narrow that it’s invisible unless you know exactly where to look; that’s why you didn’t see it in part e.)

k. Measure the FWHM of this feature, and explain qualitatively why it is so different from part i.