In my next quantum mechanics post, I’ll be talking about rotation matrices. My derivation of these mathematical objects will use some equations from trigonometry, the addition and subtraction formulas for sines and cosines. These are the sort of things one finds on the inside front cover of a trigonometry textbook, so if you’re not curious where anything comes from, that would satisfy you; however, if that’s what you find satisfactory, there’s precious little point waking up in the morning, so I’d like to give a little back story.

The addition and subtraction formulas give you the sine and cosine of the sum (or difference) of two angles, provided you know the sines and cosines of the angles themselves. Geometry tells us the sine and cosine of 45 degrees, by looking at an equilateral right triangle (whose internal angles are 45, 45 and 90 degrees). By looking at a 30-60-90 triangle, we can get the sines and cosines of 30 and 60 degrees. With all this information in hand, we’d like to get the sine and cosine of, say, 60 – 45 = 15 degrees, or 60 + 15 = 75 degrees.

One can extract these formulas out of a geometric argument, in the fashion of Euclid, but geometric arguments (while they lend themselves to spiffy pictures) tend to involve a certain amount of chicanery. One must find the proper “construction lines,” inscribe and circumscribe the correct circles and so forth. If one sees a geometric proof and, six months later, wishes to recover the result, remembering the necessary diagrams and manipulations can be quite the challenge.

I say “one must find” and “if one sees,” but really, this is *me* we’re talking about: I can see the proof, and I’ll remember that the final answer involves sine of this and cosine of that, but I’ve learned better than to trust my memory at getting all the plus and minus signs in the right places. (Talking to other people with college degrees in physics and math makes me suspect I’m not alone.) So, to contribute to the general welfare of the world, I’m going to go through the process I run through every time I need to use the addition and subtraction formulas. I’ve got it down to about fifteen seconds of pencil work, which I can do in the margin of my notebook, and I get all the damn minus signs in the right place.

The price we pay for having an easy-to-remember, foolproof derivation of these formulas is that we must venture beyond the ordinary “real” number line, out into the complex numbers. Rather than going over what *should* be a basic part of any grade school education, I’ll just point to Mark Chu-Carroll’s introduction.

We start with Euler’s Formula, which relates trigonometric functions to exponentials of “imaginary” numbers:

[tex]e^{i\theta} = \cos\theta + i\sin\theta.[/tex]

The best treatments I’ve seen in print on this relationship are chapter 22 of Feynman’s *Lectures on Physics,* volume one (1965) and for a complementary approach chapter 8 of Conway and Guy’s *Book of Numbers* (1995). Euler’s Formula works for any angle, so we might as well take the angle to be the sum of two other angles *A* and *B.* Naturally,

[tex]e^{i(A + B)} = \cos(A + B) + i\sin(A + B).[/tex]

But by the basic properties of exponentials,

[tex]e^{i(A + B)} = e^{iA} e^{iB},[/tex]

so by applying Euler’s Formula to each factor,

[tex]e^{i(A + B)} = (\cos A + i\sin A)(\cos B + i\sin B).[/tex]

Next, we multiply the two binomial factors, using the “FOIL” or “First, Outside, Inside, Last” approach we all learned in grade school. This gives two terms (from the “F” and the “L”) which are real numbers,

[tex]\cos A \cos B – \sin A \sin B,[/tex]

and two terms (“O” and “I”) which carry a factor *i*:

[tex]i(\sin A \cos B + \sin B \cos A).[/tex]

For two complex numbers to be identical, the “real” parts have to be equal and the “imaginary” parts have to be equal. We’ve written the *same complex number* in two different ways, so we can *match* the real parts, giving us the nice and pretty result,

[tex]\cos(A + B) = \cos A \cos B – \sin A \sin B.[/tex]

We can match the imaginary parts just as well, giving us

[tex]\sin(A + B) = \sin A \cos B + \sin B \cos A.[/tex]

It was the study of angles and rotations which made the complex numbers “click” for me. Being able to do quick and straightforward derivations of useful and pretty results is a nice thing, and incorporating the complex numbers into one’s panoply of armaments certainly makes life easier. It also has the quite considerable virtue that it allows students to one-up their teachers, as Isaac Asimov’s autobiography, *In Memory Yet Green* (1979) explains:

On September 27, 1938, I registered for my fourth year at Columbia. I was taking integral calculus now and Sidney was taking more sociology while I was doing that. After calculus, I would go around to the sociology class, where the professor held court after the lecture was over. When that was done, Sidney and I would have lunch.

It made for dull listening, generally, for I never have been impressed by the soft sciences. On October 10, I found the sociology professor (his name was Casey) had made a table on the board in the course of his lecture in which he divided people into rationalists and mystics. Under mystics he had listed mathematicians.

I studied that for a while and then, even though I was not a class member, I interrupted the postlecture session by saying, “Sir, why do you list mathematicians as mystics?”

He said, “Because they believe in the reality of the square root of minus one.”

I said, “The square root of minus one is perfectly real.”

He said, “Then hand me the square root of minus one piece of chalk.”

I said, “The cardinal numbers are used for counting. The so-called imaginary numbers, like the square root of minus one, have other functions. If you had me a one-half piece of chalk, however, I’ll hand you a square root of minus one piece of chalk.”

Whereupon Casey promptly broke a piece of chalk in half and handed it to me with a smile. “Now your turn,” he said.

“Not yet,” I said. “That is one piece of chalk you’re handing me.”

“It is half a regulation length of chalk.”

“Are you sure?” I said. “Will you swear it is not 0.52 times a regulation length or 0.48 times that?”

By now Casey realized it was time for hard logic if he was to win the argument, so he decided that since I was not a member of the class, I would have to leave the room at once. I left, laughing rather derisively, and after that I waited for Sidney in the hall.

Next time, I’ll employ the addition and subtraction formulas in working out rotation matrices, which will lead us into Lie algebras. After that, we’ll get into how rotations act on quantum-mechanical states and what we can say about quantum systems which are rotationally symmetric, which will be our prerequisite to solving the hydrogen atom via supersymmetry.

That’s the way I remember trig identities.

Are you going to talk about matrix exponentials?

Matrix exponentials will probably come after the infinitesimal generators of rotations. . . I’m not sure yet how much of the exponentials I need to discuss.

I’ve never been too impressed with the way in which Asimov tells that story. Granted, he was doing it off the top of his head, but still, it makes him seem like he’s just being a jerk.

Had I been in the same situation and been quick enough on my feet, I would have asked for negative one pieces of chalk, so that I may square root it for him. If he says that is impossible, then he’s denying the reality of negative numbers! (Which was, of course, a perfectly respectable thing to do well into the 19th century.)