Monday, August 22, 2005

Quantum wierdness

One of the things I intend to do with this blog is make a bunch of handy references for things I have to explain frequently. I'm a physicist (okay, an aspiring physicist) that also has an interest in philosophy, and that means I often encounter people in various web forums who draw completely unwarranted philosphical conclusions from physics concepts they don't understand. I can't imagine how many discussions I've had of quantum mechanics and determinism. On the other hand, I certainly am not a master of all the nuances of either physics or philosophy, so I'm also hoping to learn something from this exercise.

Anyway, I had a conversation recently with someone who linked me to this page, which tells us that Quantum Physics tells us that classic logic is flawed, and that A is both A and not A, and so on.

But quantum mechanics is expressed, without contradiction, in the language of very ordinary mathematics. You can have a working understanding quantum mechanics after you know calculus and some linear algebra. Results of quantum mechanics can be derived and expressed in sophomore -level mathematics derived from the usual, totally boring axioms. There are no contradictions there. (Though some aspects of the measurement process are poorly understood.)

Quantum mechanics is the story of the wavefunction, which contains all the information about any physical system. The wavefunction usually changes in a well-behaved, predictable manner. However, the wavefunction isn't actually something we can look at, directly--the "real-world" measurements that we can make can only look at things like position or momentum or energy.

A semi-mathematical analogy, but one which actually is quite valid, is to think of vectors. Say there is a vector (just an arrow) pointing in some direction in space--say at a 30 degree incline from the horizontal, so 60 degrees from the vertical. This arrow is our wavefunction.

Now, say I'm a scientist, and I don't know anything about that arrow, but I want to find out what direction it's pointing in. However, I can't actually look at the arrow and see which way it's pointing. The world isn't built in a way that allows that kind of measurement. All my measurements can do is ask the very crude question "is the arrow pointing in the horizontal direction, or in the vertical direction?"

Now obviously in a sense our arrow is not pointing *either* in the horizontal *or* the vertical direction--it's pointing in a diagonal direction. In another sense, you could say it's pointing in both directions simultaneously--since a diagonal is just a sum of a horizontal displacement and a vertical displacement. But both of these things are rather silly things to say. The arrow in fact does have a perfectly valid, logical direction, but we can't get at that with a single measurement--and thus we get headaches. A diagonal has a certain amount of horizontal character, and a certain amount of vertical character, so when you do a measurement you have a chance of getting either "it's horizontal" or "it's vertical" as your answer. If all you knew about were horizontal and vertical arrows, this would be pretty frustrating. You end up saying things like "it's both horizontal and vertical, both A and not-A." But once you learn about diagonals, the paradox goes away.

So particles are often in these "diagonal" states that we can't measure directly. A diagonal isn't horizontal or vertical, it's a sum of horizontal and vertical components. That's pretty easy to see. Where it gets much less intuitive is when you extend the concept. Say I prepare a bunch of electrons identically, and I can measure whether they're "up" or "down". When I measure all these identical electrons, I get inconsistent results--some electrons I measure as up, and some I measure as down. What's happening is that the electron I measured wasn't really up or down, or "neither" or "both." It was in a perfectly well-defined state that could be described as a sum of up and down.

(It's important to note here that this has *nothing* to do with the uncertainty principle or our lack of knowledge of the system. The particle really is this way.)

The problem is that we think of electrons as being like really tiny little balls. Our intuition about balls is that they don't act this way--they can't be in a sum of different states. A ball has a very definite position, a very definite speed, etc. which can be measured. An electron can be in a state that's a sum of intuitively incompatible states--like an electron can be in a state that's a sum of "moving leftward" and "moving rightward." But there's no reason for our intuition about balls to actually apply to electrons. Balls make sense to us because we deal with them regularly. We don't deal with electrons regularly, so there's no reason for us to have any valid intuition about how they act. Electrons are happy to be in states that can't be measured directly, and are happy to consist of these combination "diagonal" states.

Now, much wierder things happen *after* you do a measurement--but that's for another post.