With the discussion of spin, spin matrices and the Stern Gerlach experiment a certain degree of clarity about what we’ve been doing all semester has descended. At least for me. The discussion of these matters is couched in terms of matrices rather than differential operators. Unfortunately I don’t think the class has had much experience with linear algebra or matrices, although they’re quite conversant with vector calculus.
If I have the time over Thanksgiving, I might try to write up a ‘Linear algebra survival guide for quantum mechanics’. Since the kids are all quite smart and adept at taking courses (or they wouldn’t be in this one), they’ve become good at multiplying matrices times matrices and matrices times vectors. Amazingly, I don’t think they could find the eigenvectors and eigenvalues of a 2 x 2 matrix without asking Mathematica to do it (another example of “What hath Mathematica Wrought?” — see the post of 22 Sep). But the linear algebra you need to know for QM isn’t very much, and when you do know it, QM becomes much clearer.
So tell me, why is matrix multiplication exactly the way it is, and why couldn’t it be any other way? Answer at the end of the post.
Particle ‘spin’ is an extremely abstract concept. It acts like orbital spin (quacks like a duck), but has no spatial extent (doesn’t walk like a duck), and it’s hard to see what this sort of thing has to do with reality. But it does in spades, and I brought in a medical MRI of the brain to show the class. It’s all based on the ‘spinning’ hydrogen nucleus and using radiowaves to flip the spin. The class was impressed.
I’ll talk about Stern Gerlach, commuting and noncommuting operators, and the collapse of the wavefunction next time. But what does spin have to do with hair? My wife used to hack me about twizzling my beard when deep in thought. When the going gets tough in class, some of the girls give their hair a real workout — ponytails get made and unmade, hair gets knotted, unknotted and twirled etc. etc. Spinning our hair seems to help us all think better, and I have no neurologic explanation for it.
The science library building where classes are held and the new science building are beautiful and little expense appears to have been spared (this is an elite woman’s college after all). 30% of the students are science majors, but I think I’ve only seen perhaps 2 – 3 blacks in the building all semester. I called administration to find out how many were enrolled and they told me the figure was 7%. In the physical chemistry course at the local branch of the state university I audited last year blacks probably accounted for 10% of the (much larger) class, but even there at least half of them were from Africa (or Haiti). I don’t know what’s going on, but I don’t think it’s healthy.
Nonetheless this is a huge change from 50+ years ago. There was 1 black in my entering Ivy league class of 700+. A few years later in another Ivy league med school there was one black in a class of 125 (he was from Nigeria). Like it or not, there is a strong hereditary component to intelligence, and if the blacks presently in college are smart enough to do the work (which they are), so were their parents and their grandparents. This country has wasted a lot of brainpower in the past.
The instructor brought up Schrodinger’s cat when discussing superposition, and I was suprised that most of the kids hadn’t heard of it. It’s a cat in a box with something that can kill it which may go off any time. Since the cat (like all of us) is made of quantum particles, it is in a superposition of alive and dead states (until we look — e.g. perform a measurement). This made no sense to Schrodinger or any one else.
Unfortunately as a neurologist I had to deal with a lot of Schrodinger’s humans — people who were basically brain dead but being kept alive by the wonders of modern medical science. Sometimes, the reason was medical-legal as in the physician strangled by her husband, that we kept alive as long as possible, because the husband’s attorney threatened to charge us with murder if we turned off the respirator. More often, it was a family member feeling guilty about their lifelong neglect of the patient who wanted ‘everything done’.
Well that’s fairly grim, but I heard a more benign use of ‘terminal case’ the other night from two musician friends who teach violin and cello. Here it means a kid who just isn’t practicing, but comes to lesson after lesson (dragged by their parents). Eventually they quit.
Finally, here’s why matrix multiplication is the way it is. Given a linear transformation from A to B (represented by matrix X — and all linear transformations can be represented by matrices and vice versa) and given another linear transformation from B to C (represented by matrix Y) the composition of the two linear transformations is just another linear transformation (from A to C) which can be represented by a matrix Z. The wierd formula for matrix multiplication gives you Z = XY. It’s the only formula which will make this work.