Also, your definition is not good for quantum mechanics, since the velocity $v$ or indeed such a thing as $x(t)$ (at least as a number) does not really exist.
And if you take the expectation values, then Ehrenfest's theorem essentially tells you it's the same as in classical mechanics.
@DanielSank : because the detection event involves the interaction between a photon wavefunction and an electron wavefunction which together result in the localisation of the former akin to an optical Fourier transform. Please ask a question for a more detailed answer with references.
@JohnDuffield Dude, I'm not asking for this explanation because I don't understand it. I'm asking because you seem to think you understand everything with some magical clarity not enjoyed by everyone else around here and I want to understand what you think you understand.
@DanielSank : I've been considering that I'm wrong for at least ten years. I remember staning in front of the mirror saying what you thought you knew is wrong. Since then I've done a lot of reading, and find that some physics which is considered to be common knowledge flatly contradicts "Einstein and the evidence".
@Danu: The subset $\mathbb{Q} - \{r\}$ for any $r\in \mathbb{Q}$ is not closed in the subsace topology of $\mathbb{Q}\in\mathbb{R}$ - it is dense (because you can get arbitarily close to any rational with rationals), so the subspace topology $\mathbb{Q}\subset\mathbb{R}$ is not discrete, so the inclusion of the discrete $\mathbb{Q}$ into $\mathbb{R}$ is not a homeomorphism.
@JohnDuffield 1) Why do you think quantum computing is "pseudoscience"? 2) why do you, John Duffield personally, think that interactions are enough to explain the so-called "measurement problem" in quantum mechanics? Are you referring to decoherence mechanisms by which the density matrix of a subsystem diagonalizes in the basis of the interaction term?
Unfortunately, if I ask question 1 on the main site it would be closed immediately.
@DanielSank : "Einstein and the evidence" is just shorthand for bona-fide peer reviewed papers and robust scientific evidence/observation. If you're at odds with that you ought to have a good reason why. Saying my textbook says different is just some other appeal to authority.
@JohnDuffield If you review the chat log far enough back you will find that I have repeatedly voiced my opinion that appeal to textbook is just another version of appeal to authority.
I believe my exact words were "Textbooks are not God" or some such thing.
@JohnDuffield Anyway, I don't understand your insistence that we stick to evidence and peer reviewed work as I don't think I've given you any reason to believe that I disagree with that (obvious) standard for scientific discussion.
@DanielSank : I don't insist we stick to it, I insist that we don't dismiss it. Now you'll have to excuse me, I have to go. Meanwhile ask your questions on the main site, I really don't think anybody will close sincere serious questions from a professional physicist. And while we're at it, why don't you ask a big question? One you'd really like to know the answer to? We could do with more of them on this site. I'm not saying you'll get an answer, but it could be interesting.
@0celo7 It's ok for the basics. It gives a good introduction to how quantum algorithms work. I recommend you read it until you start to get bored with all the details of how e.g. Shor's algorithm works.
@0celo7 Note that it's painfully out of date with respect to how error correction works. The schemes discussed within are considered barbarically inefficient.
@DanielSank have you heard of dyakonov? there is indeed a lot of "hype" in the area of qm computing... but this is not to disparage any of the many excellent scientists working in it.
@0celo7 (a bit afraid to ask here, aka "fools rushing in where angels fear to tread") duffield is interested in hawking radiation? as anomalous or contradicting some kind of established physics theory?
@DanielSank dwave, shmdwave. guilty as charged but they cannot be blamed for all the hype. "for a wild time..." try reading 1 of dyakonovs papers (of the 2 wild ones cited). and consider his credentials. do you ever read aaronsons blog? aaronson luvs him :P
@ACuriousMind The link you provided, I don't quite understand how it won't violates newton's 1st law at [3]? This kind of analysis seems to me exactly the kind of physics needs rigor mathematics (for t at the neighborhood of T). Would you mind sharing your thoughts on it? http://www.pitt.edu/~jdnorton/Goodies/Dome/
But it does seem we can't find a t such that [3] violate newton's 1st law.
@Shing Well, it very much depends on what exactly you mean by "Newton's first law", but it doesn't violate the law that a body upon which no forces act has zero acceleration.
@DanielSank it cites Dyakonov who is a mainstream scientist criticizing QM computation hype (outside of the obvious/ "easy target" Dwave marketing/ fanfare).
Spectral theorem is covered in the graduate level functional analysis course.
user54412
I'm siding with 0celo7 on this one -- I expect a first linear algebra class to cover the definition of a vector space, the idea of a basis, and how to compute determinants.
@DanielSank actually you know what really annoyed me about my undergrad linear algebra course? It didn't cover the complex cases of anything! So I had to re-examine everything upon going to QM :(
@DanielSank So, I'm studying diff geo right now. Forms and that sort of thing, and that fact is totally necessary for understanding differential forms and change of variables and what-not... but what would you do if someone asks you, "prove that the determinant of a matrix is the volume of the parallelipiped spanned [...]"? Hehe I think this is in Rudin but I'm being lazy and asking you first
user54412
@DanielSank Well, I'd teach it if I was ever called upon to teach linear algebra. But that would be a strange sequence of events to happen.
@NeuroFuzzy To be totally honest with you: I would go sit on the beach, close my eyes, and think really hard for about an hour. After that time, I would either have come up with a clear way of proving it using elementary matrices and/or induction, or I'd realize that I've forgotten all of that and give up, feeling hopelessly old.
@NeuroFuzzy Also, Mukres's book is awesome. I recommend it. I also strongly recommend Hubbard and Hubbard's book, though it may be too elementary for you at this stage.
user54412
10:57 PM
@0celo7 no one, but that's what it is -- we can find coordinates that diagonalize the metric (i.e. make it Minkowski)
The antisymmetry means that if the vectors are not linearly independent then the determinant is zero.
The linearity also means that if you double the length of a vector, the volume doubles.
Obviously the determinant of the identity matrix is 1.
Given any matrix, if the columns are linearly dependent, the determinant and volume are zero, so you're done.
If they're independent, apply elementary matrix transformations (like in Gaussian elimination) until you get a diagonal matrix. Now the determinant is the product of the diagonal elements, which is obviously the same as the volume.
Of course, you have to multiply by the determinants of the elementary transform matrices, but those are all either 1, -1, or exactly the scalar by which you scaled one of the rows of the matrix.
@NeuroFuzzy It might feel better if you spend a private moment convincing yourself that each of the three types of elementary matrices correspond to very simple geometrical transformations.
