hobz

Thanks for answering! This is a lot to take in. I did some research to be able to grasp the meaning. I have a few follow up questions if you don't mind. First. Why are smaller distances equivalent to higher energies? Second. When I look a the fine structure constant, it appears to be made up of other constants that also seem independent of energy. Third. Is it correct to say, then, that the (known) fundamental forces become of the same (or comparable) magnitude, but does not become the same force per say? Or what does "unify" mean in this context?

This is pretty interesting. Do you have some recommendations of literature dealing with this and perhaps some cue words for google seaches?

I was watching a documentary on the Theory of Everything with some lectures by Holger Bech Nielsen. In this documentary, he mentions that in the early universe, the four fundamental forces were not existing separately because of the very high temperature, but does not explain why. How can this be? I would understand it, if it were because energy associated with temperature completely dominated the other forces, making their respective forces insignificant, but it seems like this is not the case (e.g. the Placnk epoch subtitle on http://en.wikipedia.org/wiki/Timeline_of_the_Big_Bang)

Can you elaborate on how the photon wavelength relates to "seeing" electrons?

Thanks for the answers. I will look up some literature on waveguides, Klaynos. swansont: Interesting! From that perspective, there seem to be no answers to your (retorical) questions, but I presume that classical physics would have a way of predicting that microwaves would stay inside the ovens when using a grated piece of metal. What is the QM explanation? Does the photons chance of hitting a free electron vary with its energy (i.e. wavelength)?

My understanding so far. Electrons can respond to any EM wave regardless of wavelength. Ideal conductors have completely free electrons, and thus there electrons can respond to any EM wave. Now if I grate a plate of metal (which is assumed to be an ideal conductor) with holes larger than the wavelength of an incoming EM wave, how can the wave "escape"? Surely the electrons in the metal surrounding the holes respond to the incoming wave, and if so, I guess they should absorb the energy in the wave?

I find this hard to believe. Can someone confirm or deny?

Thanks for this elaborate reply! Do you know of any site or text going through, step by step, what Einstein did to derive his theory?

I was discussing with my friends the implications of neutrinos travelling faster than light (be it real or hypothetical). See http://www.bbc.co.uk/news/science-environment-15791236 So far I've got the following background knowledge: Michelson-Morley found that the speed of light was the same in all frames of reference. Einstein claimed that nothing could exceed the speed of light and formalized the theory of relativity based on that maximum speed. I'm not sure why this claim is valid. What special property of light makes this obvious? If something CAN travel faster than light, what are the implications? Isn't it just a matter of correcting [math]c[/math] in [math]\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}[/math]?

Take a look at http://en.wikipedia.org/wiki/Escape_velocity#Derivation_using_G_and_M Is this calculus, or pseudo-calculus?

If you look at http://en.wikipedia.org/wiki/Capacitance (just the stuff before the TOC) there is an example of what I have been talking about. Is this not abusing the notation?

If nonstandard analysis ties the intuitive infinitesimal calculus of Leibniz, et al, to a rigorous foundation of the standard calculus (using limits), then I would have to completely disagree with that statement. In many engineering and physics classes, informal treatment of dy/dx very often leads to treating it as a ratio, even though it strictly is meaningless or undefined. Non the less, all the physics derived from this informal basis, can be rigorously based on the nonstandard analysis, which then is everything else but useless.

Let me first state that I too am no expert (which is why I am asking people in sf). As far as I know, the concept of "infinitisimal" has been abandoned since Weierstrass gave a rigorous definition of the concept of limits and used limits to "derive" calculus. The exception is, as DrRocket says, the non-standard calcuclus, which is, as far as I know, also a rigorous method worked out by a guy in the 60's. I think the problem is that dy/dx is really a notation and not a ratio. The notation is sometimes abused by (e.g.) multiplying by dx on both sides, which has no meaning from the dy/dx limit definition. My question is, how can you justify doing it anyways? P.S. If you like the infinitisimals, I can recommend "Calculus made easy" by Silvanus Thompson. It's very informal, only relying on algebra.

Yes, it is not a ratio.

They can't be separated because [math]\mathrm{d}y[/math] and [math]\mathrm{d}x[/math] are not defined by the difference equation. [math]\mathrm{d}x[/math] is not [math]lim_{\Delta x \rightarrow 0}\Delta x[/math] as it then would be zero.