Notes and guide to Midterm.

This midterm will focus on the material we have covered since the last midterm.
That includes free particle propagation, the 2D HO and the hydrogen atom system. There won't be a question on scattering on this midterm. Also, you will not be asked to do any of the difficult k space integrals that you encountered in studying the propagating Gaussian wave packet that was made up of free-electron plane-wave energy eigenstates.
For any graphing you do, it will say to include labels and scales. For the vertical axis that means actual numbers and their units. However, for the horizontal axis, putting the scale in terms of "a", the Bohr length scale, a, is just fine for graphing problems involving wave-functions where there is a length scale, a, in an exponential (such as the wave-functions for an electron in a hydrogen atom potential).
note added: r can never be negative in spherical coordinates. When you graph something as a function of r, don't go negative!

On your one page reference sheet that you bring to the exam, please bring the following:
Any integrals that you have encountered. That includes many integrals of the form:
\(\int_0^\infty r^n e^{-r/a} dr\) and \(\int_0^\infty r^n e^{-2r/a} dr\),
as well as
\(\int_0^\pi sin(\theta) \, d\theta = 2\); \(\int_0^\pi sin^2(\theta) \, d\theta = \pi/2\) ; \(\int_0^\pi sin^3(\theta) \, d\theta = 4/3\) ; \(\int_0^\pi cos^2(\theta) sin(\theta) \, d\theta = \pi/2\)
If an integrand is just a function of r, then you can skip from
\(d^3 \vec r = r^2 sin\theta d\phi d\theta dr \) to \(4 \pi r^2 dr\) without showing any work.
Remember that theta goes from 0 to pi, whereas phi goes from 0 to 2*pi. This goes along with:
\(x = r sin\theta cos\phi\), etc.  This is standard notation in all your physics classes, but I have the impression it may have been different in some math classes? Anyway, it is good to be clear on that.
Bring algebraic expressions for all wave-functions that we have discussed.
No graphs of any kind are allowed.
Value for parameters including \(\hbar^2/m\), a=.053 nm, c, etc.
Jacobian and del^2, etc in spherical coordinate for 2 (cyl) and 3 dimensions!
Also, some things you brought to the last midterm such as \(\hbar^2/m; \:\: \hbar\), etc could be useful (necessary). What else would you like to bring?


Emphasis will be on the 2DHO and the hydrogen atom electron states.  Why do we study energy eigenstates? What are the ground states and 1st excited states like for these two systems? What is sp2 bonding?  What is the origin of the Bohr length scale?

Summary and points of emphasis:
In HW 5 we studied the traveling wave packet. We modeled an electron moving with a speed \(\hbar k_o/m\) using a superposition of energy eigenstates. Most of that involved complex integrals which would not be suitable to ask about on an in-class test. Our most important result, however, was that the energy of the free-electron could be divided into two parts: one associated with motion that had a familiar classical form, and the other associated with confinement. That was really the most salient and important result from HW5 and the only part you would need to review and understand for the midterm.

In HW6 we studied the 2DHO. In class we examined the nature of the energy eigenstates of an electron in a 2DHO potential. We showed how they can be constructed as products of 1D HO energy eigenstates. We saw how they all have the same exponential factor, \(e^{-r^2/2a^2}\), which makes the WF get small and approach zero asymptotically as r gets large (compared to a). (a is the same quantum length scale that emerged for the 1DHO.)  Particular characteristics of individual eigenstates in the basis we chose for them included:
a state proportional to x,
a state proportional to y,
(those are two first excited states);
a state proportional to xy,
a state proportional to x^2-y^2,
a third state involving r^2,
those are three 2nd excited states.
Then in the HW you were asked regarding a particular made-up state: is it an energy eigenstate? 
The best way to approach that problem, if you have already studied the energy eigenstates of the system in question, is to look for familiar features in the algebraic expression for the state and try to recognize energy eigenstates you have learned about. Then endeavor to write the state you are given as a superposition of those energy eigenstates. That takes full advantage of what you have already learned. This is why (the main reason) we study energy eigenstates so much. If we had not already found the energy eigenstates, then the question about a general state: "is this an energy eigenstate?" would be really difficult. However, if you already know the energy eigenstates of the system, that is really a question of what sort of superposition one needs. The linear algebra underpinnings of quantum theory are really powerful and it is nice to take full advantage of that whenever you can. Learning how and when to take advantage of that is a key part of what you learn in this class.
     Before a state is written as a superposition of energy eigenstates, it would be very hard (impossible) to write its time dependence. However, once you have expressed a state as a superposition of energy eigenstates, it is straight forward to write the formal expression for the time dependence of the state in terms of the energy of the individual energy eigenstates involved in the superposition. This is an important thing to understand.

In HW7 we studied the energy eigenstates of the hydrogen atom system. Also, in class on Tuesday May 14. We examined 1st-excited energy eigenstates of the form x, y and z, which are very much analogous to the 1st-excited states of the 2DHO.  Additionally, there was a 4th 1st-excited state. There is a video on these states and how they can be combined, e.g., to form an sp2 basis.  Also, notes from the Tuesday May 14 class. Also, there is a video lecture from our missed class on the ground state and how it gets its size. It is important to view and understand that. Both of these videos are really important.

In class last Tuesday we studied electron spin. There is a video on spin. Spin will probably not be asked about on the midterm.

Note added, May 27: What does "roughly" mean? If you are asked to roughly calculate...
as in, for example, "Roughly what is the numerical value of ...", or
"roughly what is its value..",
That means that within about 10% is fine. Like just get a numerical result that is accurate to about 10%, and if it is 15% off that is going to be okay too. You don't have to spend time getting a precise result, just something accurate to about 15% or so.

note added: r can never be negative in spherical coordinates. When you graph something as a function of r, don't go negative!

PS. There are no questions involving 2nd excited states.

PPS. You can bring any notes you like regarding sp2 states. (no graphs though)

You are expected to understand the content of all recent videos and blog posts!