Free Physics (and Math) Books

Challenge: Think of any physics book that is known by its author’s last name.

OK, what is its free replacement?

A variant on this question: How much of the MIT undergraduate physics curriculum can be taught with free books? The only reasonable answer would be all of it, because we’ve had the Web for 30 years now. Sadly, the textbook business is not reasonable.

If people had decided to be useful at any point in the past 25 years, you could go to physics.mit.edu and click to download all-the-textbooks-you-need.tgz, but we got MOOCs instead. Not to mention the “open courseware” that too much of the time is just a stack of PowerPoints. Oh, and software that puts kids under surveillance so that a company can monetize their behavior. Because that’s the future we deserved, right?

There are books out there, but they peter out after you get past the first year or so, and a lot is pitched either too low or too high. Either there’s a few chapters in a big “university physics” kind of volume that wouldn’t be enough to fill a whole semester, or there’s a substantial text that’s intended for graduate students. Plenty of times, one finds a totally decent set of lecture notes that whiffs at the last step by not incorporating homework problems. If we really want institutional change, we need (among other things) more drop-in replacements for the books to which physicists habitually turn, so that we can overcome the force of tradition.

In what follows, I go through the MIT course catalogue and provide links and commentary.
Continue reading Free Physics (and Math) Books

Regarding Gender Queer

Maia Kobabe’s Gender Queer rocketing to the status of the most banned book in the United States is darkly hilarious. Yeah, nothing says “pornography” like four pages of quotations from philosopher Patricia Churchland.

It’s a memoir by someone who just doesn’t want or like sex all that much. (Representative dialogue from page 138: “I think I’m asexual.” “You can’t be, I’ve seen you lust after other people.” “Well. Yeah. But not very often and I don’t enjoy it.”) Oh, noes, three panels of mostly-clothed fooling around by two people in an affectionate, monogamous relationship that ends with them deciding that the activity was hotter in the anticipation than the actuality. That’s roughly one one-billionth as steamy as anything Famke Janssen says or does in GoldenEye.

Rolling up the Bloch Ball

In an earlier post, we discussed how to do quantum mechanics for the simplest possible quantum system, a single qubit, using expectation values. What if we want to apply quantum theory to a bigger system, like multiple qubits put together? This is where the standard mathematical language of the subject starts to pay off. It is possible to keep working with expectation values the way we were, and in some applications it is even beneficial. However, expressing what the valid set of preparations looks like is difficult to do without bringing in more of the linear algebra.

I’ve taught this to college students, after first reviewing how complex numbers work and some basics about how to manipulate matrices — adding them, multiplying them, taking the trace and the determinant, what eigenvalues and eigenvectors are.

For our own purposes, our next step will be to develop the framework in which we can consider multiple qubits together. It might not seem obvious now, but a good way to make progress is to combine our three expected values $(x,y,z)$ into a matrix, like so:
$$ \rho = \frac{1}{2} \begin{pmatrix} 1 + z & x – iy \\ x + iy & 1 – z \end{pmatrix} \, . $$
This matrix has some nice properties of the sort that we can generalize to bigger matrices. For example, its trace is 1, which feels kind of like how a list of probabilities sums up to 1. Meanwhile, the determinant is the pleasingly Pythagorean quantity
$$ \det\rho = \frac{1}{4}(1 – x^2 – y^2 – z^2) \, . $$
This will be nonnegative for all the valid preparation points. So, the product of the two eigenvalues of $\rho$ will be positive for every point in the interior; we can only get a zero eigenvalue by picking a point on the surface. Using the trace and the determinant, we can find the eigenvalues thanks to a nifty application of the quadratic formula:
$$ \lambda_\pm = \frac{\mathrm{tr}\rho \pm \sqrt{(\mathrm{tr}\rho)^2 – 4\det\rho}}{2} = \frac{1}{2}(1 \pm \sqrt{x^2 + y^2 + z^2}) \, . $$
And indeed, this will always give us positive real numbers, except on the surface of the Bloch ball where the $\lambda_+$ solution is 1 while the $\lambda_-$ solution is 0. Requiring that a matrix’s eigenvalues be nonnegative is another property we can generalize.

Another interesting thing happens if we take the square of $\rho$:
$$ \rho^2 = \frac{1}{4}
\begin{pmatrix} 1 + x^2 + y^2 + z^2 + 2z
& 2x – 2iy \\
2x + 2iy
& 1 + x^2 + y^2 + z^2 – 2z
\end{pmatrix} \, . $$
If the point $(x,y,z)$ is on the surface of the sphere, then $\rho^2 = \rho$. This will turn out to be a way to characterize the extreme elements in our set of valid preparations, no matter how big we make our matrices.

This gets us almost to the point of being able to do the quantum math for the parable of the muffins.