Recent content by lemma28

I'm not much into optics but I guess we are saying the same thing from different perspectives (focusing and global/local minimum...)

The problem (and the misunderstanding) is that all actual rays follows the Snell's law and they produce a local minimum for the path's time, considering nearby paths. But that doesn't mean that an allowed path (that doesn't follow Snell's law but nonetheless goes from A to B) is the path...

Consider a light starting at A in media 1 and going in and out a media 2 (say shaped as a disk) with relative index of refraction n to arrive at point B (in media 1). Fermat's principle says that the path taken by the ray between points A and B is the path that can be traversed in the least...

I'm a collage teacher and I've found a very hard problem in one of my math classrooms' textbooks. It was firstly proposed as problem n. 9, back in 1995, in the "Annual Iowa Collegiate Mathematics Competition". Link is here (no solution file available in the site for that year). The text is...

I'd suggest "Number - the language of science" by Tobias Dantzig (p398). It's sort of a classic. It's not just on the invention of numbers but rather on the evolution of the concept of number from prehistoric till modern times. I found it full of interesting ideas.

No tiny-tim, I think Doodle Bob was right in pointing out that V was not properly defined. Originally it was defined as the intersection between the perpendicular bisector of the segment F1F2 and the circle. After Doodle Bob remark it should be corrected as "intersection between the...

Thanks again tiny-tim. I must admit that it was hard work to get to the circle. I've tried several other directions before... It's very time consuming to get to the right path in geometry (unless you have some sort of genial mind of course, which is not my case... ) I've just discovered...

If you call x = - {{\hbar \omega } \over {2 k_B T}} (and x \to 0 when T >> \hbar\omega/k_B) then your expression is equivalent to {{e^x } \over {1 - e^{2x} }} Utilizing the known limit \mathop {\lim }\limits_{x \to 0} {{e^x -1} \over x} = 1 you can write \mathop {\lim...

Thanks tiny-tim for the suggestion about the Dandelin spheres. They do the work in the simplest way. But I was not completely satisfied since I looked for demonstrations using "plane" geometry. So I’ve worked a little bit more on it and I think that these are the “geometric” proofs in 2...

You're right. Naturally I intended ... 2) locus of points for which the ratio between the distances from a fixed point (focus) and a fixed line (directrix) has a constant value that is larger than 0 and less than 1. ...

I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...

We have 1 particle. Classical Mechanics says: If I know the initial state I can predict with absolute certainty what position (etc.) it will have the next second. Quantum Mechanics says: I don't know the position the particle will hit. But If I know the state I can tell you the possible future...

I know that with an Hermitian operator the expectation value can be found by calculating the (relative) probabilities of each eigenvalue: square modulus of the projection of the state-vector along the corresponding eigenvector. The normalization of these values give the absolute probabilities...

Thanks. I got it!

Please help me with folllowing double integral \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {e^{ixy} dxdy = 2\pi}} (x,y, real) It came out analyzing the relation between DiracDelta and the Fourier Transform formula. (it's the reason why insert the constant...