Midterm preparation post.

April 22, 3:30 PM
Things to bring to the exam:

 
1) equation sheet (4 pts): Throughout this post it specifies the equations and information that you will want to bring to the midterm.  (Don't put anything else on it, only the things specified.) That should all fit on a piece of regular 8.5 x 11 inch paper.* It is really important for your success on the midterm that you read through this post carefully and bring everything that it says to bring. It will be almost impossible to do well on the exam without these things and you can't ask for info you neglected to bring during the exam. When you finish your test, please hand in your equation sheet along with your work. Also, please keep the exam paper with the 4 questions on it as you will need it later in the quarter and the TAs do not need it.
      I think I forgot to mention that you can also bring the time independent Schrodinger equation, the Schrodinger wave equation and the form of time dependence for energy eigenstates. You can write those on your equation sheet also.
* In most of this post it says large index card, but you can bring a piece of paper instead.
**I think I said in class that it was worth 4 pts, so you can get 4 pts for making a nice equation sheet.

2) Calculators are fine.

3) You cannot use phones.

There are 4 questions. The involve, ISW, HO and FSW problems as well as something novel. Once you have your critically important equation sheet fully prepared, I would recommend getting a good night's sleep so that you can think creatively and intuitively.

Here is what you will see at the top of the exam paper I will give you in class. Where it says 1 a) (10),
the (10) means that problem 1 part a is worth 10 points.

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April 21, 8 :15 PM, added notes.
Number of questions: I think there will likely be 4 questions.

Size: You will be asked to calculate the size of an electron on the midterm. Size depends on the wave-function of the electron.  For any wave-function such that \(\langle x\rangle =0\) size is given by the square root of \(\langle x^2 \rangle\). In general, size is defined as \(\sqrt{[\langle x^2\rangle - \langle x \rangle^2]}\). Please don't ask what size means during the midterm because you are supposed to know that already.

Integrals: I think you will need them. I would take the section below on integrals seriously. Let's add to that section that you would also want to be comfortable with and an knowledgeable about integrals of \(cos^2(n \pi x/L)\) and \(sin^2(n \pi x/L)\) for square wells.

Numbers: You should have values for \(\hbar, \: \hbar c, \: mc^2, \: \hbar^2/m, \: m/\hbar^2\) on the large size index card you bring. In addition, it might save you time to also think about and write down values for \(\pi^{1/4}=1.33\), \(1/\pi^{1/4}\), \(\pi^{1/2}\) and of course \(\pi\) and \(\pi/2\) as well. When you know these things intimately, your calculation will move more smoothly and be more error free.

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April 20, 3 PM: To prepare for the midterm (on April 23rd), I would recommend reading everything on this blog. I will update this post as I think of additional things that are relevant. The midterm will consist of several problems, perhaps in the range of 3 to 5, depending on how long the individual problems are. (I am still thinking about it, but I thought I should do this first.) Even though I say large index card below, you can just use a regular piece of paper, written on one side only.

Also, speaking of paper, bring blank paper!!! You will write your solutions on blank paper and keep the test paper. Do not turn in the test paper with your test. 4 points off for that. Keep the test paper, and turn in your neat, well organized solutions on blank paper that you brought with your name on the first page. Don't bring weird paper or fold the corners. There will be staplers for you to use at the front of the room when you are done.

Abbreviations I will use here are:
SWE = Schrodinger wave equation
TISE = time-independent Schrodinger equation
BCs = boundary conditions
GS= ground state
1ES= 1st excited state
ISW = infinite square well
FSW= finite square well
HO=harmonic oscillator
EVal = expectation value
wf = wave-funciton
wtf= wave-time function (not really)
large index card = 1 side of a regular piece of paper.

Specific things you should be prepared to do include:
Graphing: For example, graphing wave-functions. Be prepared to sketch a rough but reasonable graph with labels and scales. For a wave-function, the vertical axis has units of \(nm^{-1/2}\) and the horizontal axis has units of nm. There will definitely be some, maybe a lot, of graphing of wave-functions or other things. If you are not feeling really comfortable with units, I would recommend that you work on that. On the midterm, and in the class, energy must be in units of eV or meV.  Do not give any answers in Joules. Lengths must be in nm. Time can be in seconds or, better, femtoseconds (fs). Are there any other unit or graphing issues you can think of? (Please comment to this post.)

Calculating numbers: You will want to feel really familiar with hbar c in eV-nm and mass as mc^2 in eV.  You will want to be able to calculate actual numbers in eV, nm and fs quickly and accurately. You should have values for \(\hbar, \: \hbar c, \: mc^2, \: \hbar^2/m, \: m/\hbar^2\) on the large size index card you bring.  Numbers and units are a very important part of physics.

Systems: You will want to be familiar with the ISW, HO, FSW, and delta-function potential states. Specifically, the 3 lowest energy eigenstates and their energies for the ISW and HO. Also, be familiar with the GS and 1st excited state for the 2nm wide, 0.3 eV deep well that we studied in HW 3. Bring A, B, k, b and E for each of those states and have a pretty good sense of what they mean and what the state wave-function looks like. On the large index card you bring, include the equations for three states and their energies for the ISW and HO, in addition to one-state, the ground state, for the attractive delta function. For the HO and delta function, you would also want to include the equation for the length parameter, or inverse length, showing how it is related to the strength of the potential.

Integrals. For all the integrals listed here, you not only want to bring them on a card, you also would want to understand them and to feel very comfortable sketching their integrands.  Understanding them also includes recognizing whether the integrand would be odd or even if you were to integrate from -infinity to infinity, and what that implies. If that seems confusing, then maybe you don't understand them yet. (Or, on the other hand,  maybe I just explained that really badly. Maybe someone can explain this better in a "peer-to-peer" comment.)
On the large size index card you bring to the exam please bring definite integrals of the form:
\(\int_0^\infty e^{-x^2/a^2} x^n dx\)
for n=0, 1, 2, 3, 4, 5, 6.
Also:
\(\int_0^\infty e^{-2x/a} x^n dx\)
for n=0 to 6,
or equivalently,
\(\int_0^\infty e^{-2bx} x^n dx\)
for n=0 to 6.
Also:
\(\int_d^\infty e^{-2bx} x^n dx\)
for n=0 to 6.
Please let me know if I have thought of all the relevant integrals that you may need based on what we have been doing for HW1-3 and in class. We can always add more and I appreciate your feedback.

Expectation value calculations: Be able to do expectation value calculations with confidence. This could include the EVal of x, x^2, p, K.E. and P.E..  You will want to be able to calculate size, even in the context where the EVal of x is not zero.

Expect something novel: I will try to come up with a problem similar to what we have been doing, but new in some way. This is to test your ability to think about quantum in addition to remembering everything we have done so far. Here is some advice for thinking about quantum. Energy eigenstates don't have to be sanctified by a book or an authority on quantum. Any function that satisfies the TISE and can be normalized is a legitimate energy eigenstate wave-function. (Boundary conditions are included as part of the TISE.) Anywhere there is an infinite wall, the wf has to be zero there. Anytime there is a finite wall, the BC are continuity and smoothness (as in HW 3). You may be asked to envision and sketch a wave-function that you have not seen before. You may be asked to guess a functional form in a novel situation. You may need to use your integrals (see above) to normalize that guess. A guess is correct if it satisfies the Schrodinger equation and the BCs, and is normalized. [A lot of physics is done by guessing wave-functions (cf. Robert Laughlin, Theory of fractional quantized hall effect; Leon Cooper, Theory of superconductivity.]

Normalization and orthogonality: Know what normalized means. You will want to understand how to used orthogonality in the context of expectation value calculations, e.g., involving mixed-states.

Mixed states: Understand the nature of a mixed state, also called a non-stationary state. Understand how it differs from an energy eigenstate in its form and time-dependence.