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?

When a particle and its anti-particle collide, they annihilate. A photon is annihilated when it collides with another photon.

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?

Anyway, I think we can both agree that nonstandard analysis is practically useless and of limited real world utility.

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.

Yes, but you can understand the confusion when dy/dx is defined thusly:

 

[math] \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} [/math]

 

I mean, that is literally saying that "dy/dx is the ratio of delta-y over delta-x as delta-x approaches 0". The right side of the equation is a clear ratio

 

While I understand that "infinitesimal" is an abstract concept (as is "infinity") which has no place in the real set, that does not seem to me to be reason enough to preclude calling dy/dx a ratio. After all, we can certainly discuss ratios of infinities without our heads exploding, why not infinitesimals?

 

Also, I will be the first to admit that I am far from an expert in calculus (mostly self-taught ), so please don't take the above statements as a disagreement with you rather than an exposition of my level of understanding. If you could provide an example where treating dy/dx as a ratio could get you in trouble, it would be greatly appreciated!

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.

What I think I know

[math]

\frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

[/math]

[math]

\lim\limits_{x \to p} (f(x)/g(x)) = {\lim\limits_{x \to p} f(x) / \lim\limits_{x \to p} g(x)}

[/math]

but NOT for [math] p = 0 [/math]

So the [math]\frac{\mathrm{d}y}{\mathrm{d}x}[/math] can't really be separated, but in many cases it is treated as if it could (many textbooks do this).

How is that justified?

I like the intuitive argument that if it were true that heavier objects fall faster, then by combining a lighter object with a heavier one by a string, the total weight is larger than the heavier object alone, and thus the combined object should fall faster. But as the lighter object by itself would fall slower, it would, by the string, slow down the heavier object and so we have some sort of a fallacy.

I was wondering if Joule's law can be derived from Maxwell's equations, and if so, how?

Hi there,

as far as I can see the solution to that DE is a parabolic, [math]y(x) = -\frac{x^2}{2} + C[/math]. According to mathworld this functional DE defines the normal distribution:

 

[math] \frac{\mathrm{d}y(x)}{\mathrm{d}x} = \frac{y(\mu- x)}{\sigma^2} [/math],

 

where [math]\mu[/math] and [math]\sigma[/math] are the mean and standard deviation, respectively.

I forgot a [math]y[/math] in the eq.

I just learned that the normal distribution is the solution to

[math]

\frac{\mathrm{d}y}{\mathrm{d}x}+yx = 0

[/math]

What problem led to this differential equation?

How does this equation lead people to think of distributions?

In a "real" world problem (physics) what could [math]y[/math] and [math]x[/math] be?

So what you're saying is that because coulomb is defined they way it is, and that causes the introduction of [math]\epsilon_0[/math] (in Coulombs law), and similar for the definition of a second in the Biot-Savart law which requires [math]\mu_0[/math] to appear, and these in turn defines [math]c = \frac{1}{\sqrt{\epsilon_0 \, \mu_0}} \neq 1[/math], then the relation has to be like that, but the energy in the fields are equal?

I most physics book I am shown perpendicular E and B "waves" along a direction of propagation of the same magnitude? I suppose this is wrong if [math]E=c\, B[/math]?

How can it be that the electric field is [math]c[/math] times greater than the magnetic field while their energy densities are the same?

What were the reasons for abandoning the "old" models?

Thank you both for your insightful answers!

The popular layman literature I tend to get my informations from are not very clear on discerning between the "old" world and the "new" world.

In fact, I did not know that there was such a distinction.

As far as I understand, an electron orbiting a proton can be thought of as a standing wave around the proton. The orbits allowed for the electron are dictated by the distances to the center that allows standing waves from its de Broglie wavelength.

If this is true, then:

• how can there be a certain probability for the electron to be outside these very precise distances?
  
• how can Heisenberg's uncertainty principle be true as well?