### Quantum Probability

I took part in a brief discussion over at antimeta which reminded me that I ought to get back to a document I started writing on quantum mechanics for dummies. One of my pet peeves is that I believe there to be a little bit of a conspiracy to make quantum mechanics seem less accessible to people. Not a deliberate conspiracy - but people maintaining an aura of mystery about it that puts people off the subject. All of the fuzzy talk about quantum mechanics in the popular science press does nothing to help the situation. In particular, there is a core of quantum mechanics that I believe requires few prerequisites beyond elementary probability theory, vector spaces and complex numbers.

Anyway, I did some more digging on the web and found this course by Greg Kuperberg. The opening paragraphs almost take the words I wanted to say out of my mouth. In particular, despite the mystical mumbo-jumbo that is often written on the subject, the rules of quantum mechanics are "rigorous and clear" and "The precepts of quantum mechanics are neither a set of physical forces nor a geometrical model for physical objects. Rather, they are a variant, and ultimately a generalization, of classical probability theory." Most of all "...more mathematicians could and should learn quantum mechanics...". You don't even have to understand F=ma to get started with quantum mechanics and get to the point where you can really and truly get to grips, directly, with the so-called paradoxes of quantum mechanics such as the Bell Paradox. The strange thing is that you won't find words like this in most of the quantum mechanics textbooks. They throw you into physical situations that require finding tricky solutions to the Schrödinger equation while completely failing to give any insight into the real subject matter of quantum mechanics. Most QM books I know are really introductions to solving partial differential equations. (Remark to physicists: I bet you didn't know you could get the simultaneous eigenvalues for the energy and angular momentum operators for the hydrogen atom by a beautifully simple method that doesn't require even looking at a differential equation...) The best thing about the newly appearing field of quantum computing is that it's slowly forcing people to thing about quantum mechanics separately from the mechanics.

So even though I haven't read that course myself yet, I'm recommending that everyone read it :-) And some time I might get back to the even more elementary introduction I hope to put together.

Anyway, I did some more digging on the web and found this course by Greg Kuperberg. The opening paragraphs almost take the words I wanted to say out of my mouth. In particular, despite the mystical mumbo-jumbo that is often written on the subject, the rules of quantum mechanics are "rigorous and clear" and "The precepts of quantum mechanics are neither a set of physical forces nor a geometrical model for physical objects. Rather, they are a variant, and ultimately a generalization, of classical probability theory." Most of all "...more mathematicians could and should learn quantum mechanics...". You don't even have to understand F=ma to get started with quantum mechanics and get to the point where you can really and truly get to grips, directly, with the so-called paradoxes of quantum mechanics such as the Bell Paradox. The strange thing is that you won't find words like this in most of the quantum mechanics textbooks. They throw you into physical situations that require finding tricky solutions to the Schrödinger equation while completely failing to give any insight into the real subject matter of quantum mechanics. Most QM books I know are really introductions to solving partial differential equations. (Remark to physicists: I bet you didn't know you could get the simultaneous eigenvalues for the energy and angular momentum operators for the hydrogen atom by a beautifully simple method that doesn't require even looking at a differential equation...) The best thing about the newly appearing field of quantum computing is that it's slowly forcing people to thing about quantum mechanics separately from the mechanics.

So even though I haven't read that course myself yet, I'm recommending that everyone read it :-) And some time I might get back to the even more elementary introduction I hope to put together.

Labels: mathematics, physics

## 6 Comments:

Seems that the link to Kuperberg's course should be

http://sigfpe.blogspot.com/2006/03/www.math.ucdavis.edu/~greg/intro.pdf

Gosh, I made the same mistyping (although the href attribute of the link in the previous comment is set right). Sorry.

How about Kindergarten QM if you're looking for simplicity?

David,

Great link, thanks. Seems like I'm also not the only one to think it's weird that the almost trivial sequence of linear operations that goes into quantum teleportation took so long to discover. I don't think this is just an example of saying something is easy with hindsight - it really is easy. There are no new concepts required beyond the most elementary features of QM. Same goes for interactive-free measurement. I'm almost tempted to say that QM sits in some kind of collective blind-spot that makes us bad at reasoning about it.

In the english-speaking world the wave and hence differential equation-based approach to QM dominates, it is true, but in France the algebraic approach is dominant: see the textbook by Cohen-Tannoudji.

tzut,

That's interesting. I didn't know the French studied QM differently.

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