Practice midterm problems & notes added weds 9 pm.

Notes added  (also see below for some added solutions related discussion): In addition to an in-class midterm there will be a short take-home due the night of the midterm.

This is a draft of some practice midterm problems. The level of difficulty here tends to be on the higher side; I would recommend going for mastery of the less difficult problems first. The midterm will not be quite this difficult or this long, but it will be difficult and I expect it to take the full 1hr and 45 min class period.  This is just a bunch of problems without detailed design. The midterm will have some straightforward problems and some more difficult ones. Being comfortable with the straightforward problems is very important!



I would like to also mention that there are posts here going back to early January, many of which remain relevant. (They can all be found via the blog archive on the right.) That said, starting problems, working problems and thinking about problems with just bank paper and your one page of notes is the best way to prepare imo. Unless you are stuck on some particular thing, reading produces only an illusion of preparation. In the end you have to be able to solve problems on your own.

Graphing will be featured on the midterm. Also quantum length scales (see problems 5 and 9 below), as well as the calculations of expectation values. Think about the relationship between confinement length scale (position uncertainty) and kinetic energy. Be able to recast the uncertainty principle in terms of a relationship between confinement length (position uncertainty) and kinetic energy.

Here are some partial answers to consider:
5. Well, for the H atom the emergent quantum length scale is \(\hbar^2/[m q^2/(4 \pi \epsilon_o)]\). So what does that tell us? What is it trying to say? Note that the \(q^2/(4 \pi \epsilon_o )\) is in that denominator. That is the strength of the interaction, right? So the stronger the interaction, the smaller the length scale. That \(q^2\) is a measure of the strength of the attraction between the electron and proton. The proton is pulling the electron toward it. And the \(\hbar^2\), in the numerator, is resisting that. Does that make sense?

For the harmonic oscillator,  the emergent quantum length scale is sort of similar in that in depends on \([\hbar^2/(m k)]\), where k is a measure of the strength of the HO potential that is trying to confine the electron, and \(\hbar^2\) is resisting that (like with H atom). Really the emergent quantum length scale for the 1D HO is, you may recall, \( a = [\hbar^2/(m k)]^{1/4}\).  Well, I didn't talk about m, but I hope you get the idea.

6. \(- q^2/(4 \pi \epsilon_o (4a))\), (This minus sign is important.)
Note also that \(q^2/(4 \pi \epsilon_o ) = 1.44  eV-nm\).

7. To evaluate the K.E., first use k to calculate a, then use a to calculate the K.E..

 I may add more later here in terms of commentary or other problems. Please recheck this site