Tom Mattson

No' date=' I'm not saying that. [x,y'] is the notation for the commutator. It means: [x,y]=xy-yx. It's on and off, unfortunately. That's why I'm typing everything out. Edit: Let's try it. [math][A,A^{\dagger}]=0[/math] Does that work? Edit Again: No, it doesn't.

Right. That's what [A,A+]=0 means.

Ugh. I did not mean for this impedance thing to digress into a tangent of its own, and I am now sorry I mentioned it. Yes, I know that not all of the impedance is imaginary. I am just pointing out that in some situations, people have no problem associating imaginary (or complex) values with physical quantities, while in situations the same people automatically reject such quantities. I've just started my own thread (Observables in Quantum Mechanics) along this line (not about impedance though) in the Quantum forum.

This thread is an offshoot of CPL.Luke's thread imaginary numbers. That thread is about FTL travel, tachyons, etc. Since what I have to say is about plain vanilla quantum mechanics, I didn't want to clog his thread up any more than I already have, so I started this one. Here goes. There seems to be a prejudice against physical observables taking on complex values. This prejudice is no doubt inspired by the statement in most QM textbooks that observables must correspond to Hermetian observables, which have real eigenvalues. But this is false! It is not the case that an observable must have real eigenvalues. All that is required of an operator to represent a physical observable is that its eigenvectors form an orthonormal basis. While Hermiticity certainly is a sufficient condition for this, it is not a necessary one. A necessary and sufficient condition for an operator to correspond to an observable is normality. A normal operator A is one which satisfies [A,A+]=0. After all, take two observable operators: X and Y (whose eigenvalues are the x and y coordinates, respectively). What sense is there in saying that X+iY is not also an observable? Absolutely none, since both parts are observable.

It's just an example of a complex quantity that is also observable. In circuit theory it is possible to transform the modeling equations into something called the frequency domain. When this is done all energy storing elements can be modeled as resistors, if you generalize the concept of resistance to the complex quantity called impedance. This is just a specific example of a more general idea: That if you have two observables A and B that take on real values a and b, then there is no sense in saying that the linear combination A+iB is not also observable, despite the fact that it takes on complex values a+ib.

Yes, but neither do the so-called "real numbers".

Boy oh boy, you guys couldn't find a needle in a stack of needles! Fortunately one of my hobbies is scouring the web for downloadable freebies. Voila: http://www.plasma.uu.se/CED/Book/ This happens to be the first hit if you type "electromagnetic field theory textbook" into Google, which apparently no one did!

How does the statement "4-velocity is not a physical velocity" follow from Lorentz invariance?

Incorrect. This line of discussion started out with the following false statement by you: "If the physics inall inertial frames is the same, identical, equivalent, then a tick rate in frame A is the same as the tick rate in frame B. This is what equivalence of physical law in inertial frames means to me." I did not start out with the assumption that the postulates of SR are correct. I corrected you by saying that if they are correct (that is, if inertial frames are experimentally indistinguishable), then the tick rate of a clock cannot be the same in any two different inertial frames. As for the rest of your post, it is not worth responding to. It's a pity that you didn't learn a thing from your brief stay at Physics Forums.

The deductions that lead from the assumptions of experimental indistinguishibility of inertial frames to the Lorentz transformations have been well known for over 100 years now. Don't be mad at the messenger.

That's precisely backwards. If the physics is the same in all inertial frames, then it is necessarily implied that space and time are relative.

He specified that they are both not moving, so they have to be at rest in the same inertial frame.

I still don't know what you're talking about. You're talking about observers in the same inertial frame, and asking if they are "relevent" to one another. I can assure you that this question has no answer in Special Relativity. Now if you want to know if their assignment of spacetime coordinates to events is different, then the answer is "yes", because each is at the origin of his own coordinate system. But if you are asking if their determination of spacetime intervals between events is different, then the answer is "no".

Say what?

That's not the time evolution operator. It's the total energy operator. The time evolution operator is: [math] U(t' date='t_0)=exp(\frac{-iH(t-t_0)}{\hbar}) [/math'] You should be able to do it yourself. Write down the Schrodinger equation without the i, separate variables, and look at what happens to the temporal part.

When I say that GR is more difficult than QM, I mean mathematically. Take graduate level textbooks in each subject: Wald (GR) and Sakurai (QM). When I was in school and doing homework, I could immediately start writing down the solution to all of the problems in QM, and I could usually finish them in the same sitting. Not so with GR.

David Goodstein, the Caltech professor who had the PBS series called "The Mechanical Universe", called GR "the most fiercely difficult subject in physics mathematically". I'm not so sure about it being worse than Jackson't electrodynamics, but it is sure a lot more difficult than QM.

The Schrodinger equation is complex because the time evolution operator is complex. And the time evolution operator is complex because it must be unitary, which guarantees probability conservation. In other words, if the i is missing from the Schrodinger equation, its solutions will decay instead of wave.

The spin quantum number doesn't come from the Schrodinger formalism. In fact it cannot, because spin is not a function of coordinates. But that doesn't mean that spin cannot be entered into the Schrodinger equation. Feynman pointed out that the operator for kinetic energy (p2/2m) can actually be written as follows: [math] (\sigma \cdot p)^2/2m [/math] where [math]\mathbf{\sigma}[/math] stands for a vector operator whose components are the Pauli spin matrices.

It isn't a paradox. Work it out for yourself: Pick a statement and try to derive both it and its negation from the postulates of SR. You won't be able to do it.

That is the Taylor series for exp(x), centered at zero. But OK, I've Googled up the following page: http://encyclopedia.laborlawtalk.com/Definitions_of_the_exponential_function In the calculus books I normally refer to, Characterizations 3 and 4 are normally presented as the "definition" of ex, and the series is derived from that.

Eh? Proving that a function is differentiable on R by using its Taylor series? Don't you have to assume that the function is differentiable at the center of the series in order to even produce that series? Don't you further have to use the fact that (ex)'=ex to generate the series? To prove differentiability, you should be looking at the definition of differentiability. And to calclulate the derivative of a function, you should be looking at the definition of the derivative. Once you have established those things, then you can go about talking about Taylor series.

Ah, in that case you might want to invest in a book on the logic of categorical statements. Logic is a branch of philosophy. It's also a branch of mathematics, but the math people don't usually deal with categorical statements. They usually deal with symbolic logic. That's probably why no one here knew the answer! Sorry, I had two parts of my post and I hit "send" without finishing one of them. The obverse is equivalent to the converse when looking at conditional statements. For categorical statements, use the chart in the link to see what is equivalent to what. Hey, no problemo.

The obverse is NOT the contrapositive. It is equivalent to the converse of a statement. It shouldn't take that much work if you are taking a course. Looking up the definitions in a book should be sufficient. You should be looking under immediate inferences of categorical statements. Try this: http://sask.usask.ca/~wiebeb/Immediate.html Incidentlly, this is traditionally regarded as philosophy, not math.