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April 30, 2008

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I can't say that I've understood everything in the series on QM, but it has been immensely useful for me in beginning to understand it. And, in general, most of what I've read of OB I've found useful - especially the comments, because I find myself "catching up" with a lot of these ideas, and I have a tendency when I'm catching up to the ideas of someone more intelligent than me to not find fault where I would if I had a better grasp of the ideas. Though I'm still not very far along on the path to being a rationalist, I know that it's a path I've been trying to walk my whole life, despite the fact that much of it was spent stumbling and tripping through religion, popular politics, and arguments that were more about proving who was "right" rather than finding out what was right. I'm glad to have found yet another resource for walking the path, especially one as useful as this one. I haven't commented here before, but I just thought I'd toss in that I really appreciate the writing you do here (and yours as well, Robin) and I'm glad that I stumbled across this blog.

Hrm... I wonder if there are other basies with local behavior other than the usual positional one?

If yes, does what we see as decoherence automaticlaly "look decoherent" in that basis too?

Psy-Kosh: in the basis of eigenvalues of the Hamiltonian, not only is the equation local, but nothing even moves.

Chris, forgive me if this is a foolish question, but wouldn't the components corresponding to eigenvalues of the Hamiltonian change only by a constant complex factor, rather than not changing at all?

Chris, Eliezer: Yeah, at least last what I recall studying, the time development for Hamiltonian eigenvectors basically has them spinning around the complex plane (with the rate of rotation being a function of the eigenvalue. In fact, I believe it is directly proportional)

Actually, this discussion leads me to wonder something: What properties does a matrix have to have such that its eigenvectors form a complete basis?

The eigenvectors of a matrix form a complete orthogonal basis if and only if the matrix commutes with its Hermitian conjugate (i.e. the complex conjugate of its transpose). Matrices with this property are called "normal". Any Hamiltonian is Hermitian: it is equal to its Hermitian conjugate. Any quantum time evolution operator is unitary: its Hermitian conjugate is its inverse. Any matrix commutes with itself and its inverse, so the eigenvectors of any Hamiltonian or time evolution operator will always form a complete orthogonal basis. (I don't remember what the answer is if you don't require the basis to be orthogonal.)

It would be a pleasure and a treat to join the recent discussion on QM, especially the Ebborian interlude, but I cannot afford the dozens of hours of study and reflection it would take to get to the point where I could actually contribute to the discussion.

If I ever find myself with the luxury of being able to study QM, this blog or the book that comes from it is where I would go first for written study material. (I'd probably need a reliable mathematical treatment, too, but those are easy to find.)

Physics is the study of ultimate reality!

QM is in my humble opinion humankind's greatest achievement.

Stephen: Thanks!

And yeah, I was wondering what the answer was if I don't necessarally demand them to be orthognal, just that I require them to span the space.

Anyways, am right now reading through Down with Determinants. Maybe that'll have the answer in there.

(Actually, the part which I get to, at least for finite dimensional spaces, is already effectively in there: The number of distinct eigenvalues has to equal the dimension of the space. Of course, the question of what has to be true about a linear operator for _that_ to hold is something I'm wondering. :))

"The number of distinct eigenvalues has to equal the dimension of the space."

That may be a sufficient condition but it is definitely not a necessary one. The identity matrix has only one eigenvalue, but it has a set of eigenvectors that span the space.

Stephen: whoops. Just realized that and came here to post that correction, and you already did. :)

I don't really follow a lot of what you've written on this, so maybe this isn't fair, but I'll put it out there anyway:

I have a hard time seeing much difference between you (Eliezer Yudkowsky) and the people you keep describing as wrong. They don't look beyond the surface, you look beyond it and see something that looks just like the surface (or the surface that's easiest to look at). They layer mysterious things on top of the theory to explain it, you layer mysterious things on top of physics to explain it. Their explanations all have fatal flaws, yours has just one serious problem. Their explanations don't actually explain anything, yours renames things (e.g. probability becomes "subjective expectation") without clearing up the cause of their relationships -- at least, not yet.

Psy-Kosh, Stephen: A finite-dimensional complex matrix has a complete basis of eigenvectors (i.e. it is diagonalizable) if and only if every generalized eigenvector is also an eigenvector. Intuitively, this means roughly that there are n independent directions (where n is the size of the matrix) such that vectors along these directions are stretched or shrunk uniformly by the matrix.

Try googling "jordan normal form", that may help clarify the situation.

I don't know the answer in the infinite-dimensional case.

Roland, make that two. Though this mooshed my head.

I used to really enjoy thinking about how weird QM was. Look! The little photon goes through both holes at the same time! Not really any more though, it's starting to seem a little bit...ordinary.

Which is a good thing, of course.

Quick question - since you can't integrate over a single point, does that preclude the existence of any 'motionless' particle? Anything that ceased to have an appreciable (Planck-length?) amplitude spread would, in effect, not be there? That would chime with the transform duality thingy between location and velocity.

Hope I get chatting to someone who thinks in terms of quantum/classical dualities at some point, purely so that I can use the line "you're very clever, old man, but it's all amplitudes, all the time."

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