Responsible party: East End, Spherical Chicken

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It is assumed that you already know things like the course web page, the student code of conduct, etc. Just wanted to remind people that the final exam is Saturday, December 19, 8:30 - 11:30!! Mark your calendars!

Also, there were many reminders to folks to feel free to ask questions. The way I see it is that if there is something you don't understand but you've heard of it, chances are it was already covered, and the only way you are going to understand it is to ask, since it probably won't be covered again.

This course will cover (hopefully) the first six chapters of the book, with the 6th chapter being an intro to approximation methods in quantum mechanics.

The main points of today's lecture fell into three main categories:

• wave functions and what we can do with them once we find them,
• statistics,
• and the uncertainty principle.

We don't actually find wave functions yet. That's in future chapters. But we can talk about a few things about them.

• The wave function is written (in one dimension) as <math>\Psi(x,t)</math>, and is generally complex valued.
• There is the probabilistic interpretation of the wave function. <math>\Psi^*\Psi</math> is the probability density function for the particle being described. <math>\Psi^*</math> is the complex conjugate of <math>\Psi</math>. Just replace all instances of i with -i.
• Usually the first thing to do is to normalize. <math>\Psi</math> will (should) have a constant out front, <math>\Psi=Af(x,t)</math>. Since, if the particle exists, there is a 100% chance of it being somewhere, solve <math>\int_{-\infty}^{\infty}\Psi^*\Psi dx=1</math> for A.
• We can find various expectation values for various properties of the particle. <x>, <x²>, <x>², <math>\sigma</math>, <p>, etc. How this is done will be covered in the statistics section.

Recall that studying QM is a statistical process. I guess this is a consequence of the uncertainty principle.

First, there is the average, or mean. The discrete case is that the mean is <math><x>=\Sigma x_nP_n</math>. (Right? Denoted with <x>?) This translates to <math><x>=\int xP(x)dx</math> for the continuous case.

Also, remembering that <f(x)> is given as <math><f(x)> =\int f(x)P(x)dx</math> (eq. 1.18), we can easily derive formulas for <x²>, etc.

We also were given that the standard deviation is given by <math>\sigma^2=\Sigma(x_n - \bar{x})^2P_n=\Sigma x_n^2P_n - \bar{x}^2</math> for the discrete case, and <math>\sigma^2=\int(x-\bar{x})^2P(x)dx=\int x^2P(x)dx - \bar{x}^2</math>.

The steps going from the centers to the left hand sides of those equations, to me, are “wonderful tricks,” and I am going to be asking about this in the Q&A if nobody else has.

Okay, I started to mentally drift sideways a little by this point in the lecture, but we should all know by now that we can't measure both where something is and where it is going at the same time. It was pointed out that this applies not only to what we are looking at, but what we are looking with, too! Our detectors can never be perfect for the same reasons! It was mentioned that for the most part we will be focusing on energies, and that we don't really “care” about position so much. Mostly because that is what we are able to focus on, right? (Yuichi I meant say that it is because energies of the various “quantum states” have been virtually the only quantities we could measure experimentally and therefore check calculations for. The shape of wave functions have rarely been measurable until recently. However, with the development of nanostructures, this may change in the near future.)

It was asked why wave functions are called wave functions, too. What about them is a wave? Yuichi-sensee (Kubota-sensee?) came back with “Why are physicists hung up on waves to begin with?”

Well, the answer to the first question probably has to do with de Broglie and wave/particle duality. If you have a better answer, feel free to add.

The answer to the second question was a little more interesting. Because waves are interesting. Waves can interfere with each other. Most of the time we add two things together and they strictly reinforce. Waves can cancel each other out or reinforce each other. So can the wave functions for particles. Or at least they would, if the particles involved didn't follow the Pauli exclusion principle.

Yuichi - This seems to be a wonderful start and example of pretty complete note on the materials we discussed today. There may be a few things someone can add, but if the notes for the rest of the semester continue this trend, I feel we will have great lecture notes for this class. Thank you, East End. The expression for <math>\sigma</math> should be <math>\sigma^2</math> which may well be my mistake. [East End - Fixed. Thanks.]

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