Topic: Modeling Time and Velocity Using Integers in Relation to the Real Numbers

[PeterDonis]

wasn't that Feynman who demonstrated that even simple reflection of light can be modeled by quantum mechanics

Yes, using the path integral. But that model still does not include energy levels of electrons in atoms or molecules. It just puts in a value by hand for the probability of a photon of light getting scattered instead of transmitted through a given thickness of a material. The probability depends on the index of refraction of the material, and his simple model just used empirically measured values for the index of refraction of different materials (like glass).

A more complete model would be able to predict what the index of refraction would be for a given material, from the properties of its atoms or molecules. I believe such models exist, but I don't know enough about them to know what specific properties of the atoms or molecules they use. The fact that scattering is continuous in light frequency (i.e., all frequencies of light get scattered, not just particular ones) indicates to me that the energy levels of electrons in the atoms or molecules are not involved, even in a more complete model; if they were, we would expect only certain frequencies of light to be scattered, which is not what we observe.

 

[jerromyjon]

I believe such models exist

I'm curious to see any papers along these lines? I found this one searching for "cause of refractive index" but it doesn't have a PDF... what opens it?
http://arxiv.org/abs/1201.0522

 

[Ott Rovgeisha]

Yes, using the path integral. But that model still does not include energy levels of electrons in atoms or molecules. It just puts in a value by hand for the probability of a photon of light getting scattered instead of transmitted through a given thickness of a material. The probability depends on the index of refraction of the material, and his simple model just used empirically measured values for the index of refraction of different materials (like glass).

A more complete model would be able to predict what the index of refraction would be for a given material, from the properties of its atoms or molecules. I believe such models exist, but I don't know enough about them to know what specific properties of the atoms or molecules they use. The fact that scattering is continuous in light frequency (i.e., all frequencies of light get scattered, not just particular ones) indicates to me that the energy levels of electrons in the atoms or molecules are not involved, even in a more complete model; if they were, we would expect only certain frequencies of light to be scattered, which is not what we observe.

Well, this contradiction of the two models seem to be a bit too staggering... If we know about atoms and molecules: we "know" more or less what they are, then if a molecule is polarized, then in my humble point of view, we should certainly consider the idea that electrons really do get further away from the nucleus and they cannot do that without gaining energy..

If the idea, that an external electric field causes the shift in all of the energy levels could be further and more vigorously elaborated, then maybe something logical could come from there; because in my stupid, limited view: we should not pretend that molecules are something that they are not. Of course models simplify things and are useful, but they should not lead to downright basic contradictions: if electron's can't have any arbitrary energy value, then this must be taken into consideration. If a molecule does polarize, then sure as hell it WILL get further from the nucleus, therefore sure as hell gaining energy.

One idea might be (and I emphasize that this is just an idea); that when an external field is applied /changing or non-changing/, then electrons seem to be able to change their energy sort of continuously as long as it is within the lower most possible and upper most possible energy discrete energy levels. Of course, only an idea...

Somehow, we seem to try to fool the nature if our models contradict each other so grossly.. Or maybe is nature fooling us, or we are trying to fool it.. A beloved guy you quote said that nature cannot be fooled...hmm.. Iwonder, I wonder...

 

[Drakkith]

Well, this contradiction of the two models seem to be a bit too staggering... If we know about atoms and molecules: we "know" more or less what they are, then if a molecule is polarized, then in my humble point of view, we should certainly consider the idea that electrons really do get further away from the nucleus and they cannot do that without gaining energy..

I seriously doubt there's a contradiction. It's far more likely that the actual, full answer is very complicated and no one who's been to this thread has been able to explain it.

 

[PeterDonis]

this contradiction of the two models seem to be a bit too staggering

There's no contradiction. Different models are useful for different purposes. Is it a contradiction that, when doing chemistry, we model atomic nuclei as single particles, even though we know they are really composed of nucleons? Or that for many purposes we model nucleons as single particles, even though we know they are really composed of quarks? Is it a contradiction that, when doing the kinetic theory of gases, we consider molecules as single particles, even though we know they are composed of atoms, and that we don't make a detailed model of the changes in the distances between the atoms, even though we know there are such changes?

It is simply impossible for every model we use to include every known aspect of every object. Physics is too complicated and there are too many possible aspects for a single model to include them all. You always have to pick and choose, deciding which aspects are important for a particular problem and which are not. That doesn't mean the other aspects aren't there, or that your model somehow contradicts other models, used for other purposes, that include them. It just means some aspects aren't important for a particular model.

In the case of atoms and molecules scattering light, a detailed model of how the electric field of the light affects the individual electrons in each atom is simply not important enough to be worth doing; we can already make accurate predictions with a simple model of an atom or molecule as a single object that can be polarized. That doesn't mean we don't think the electrons change at all; it means the changes in the electrons aren't worth including in the model for this particular purpose.

If we know about atoms and molecules: we "know" more or less what they are, then if a molecule is polarized, then in my humble point of view, we should certainly consider the idea that electrons really do get further away from the nucleus and they cannot do that without gaining energy..

What do we gain by considering it? If you think you can make more accurate predictions by constructing such a model, with all its added complications, go ahead. If you're just trying to say that you think it happens, that's fine as far as it goes, but just saying that does nothing towards actually using that knowledge to make predictions, which is what scientific models are for.

if electron's can't have any arbitrary energy value, then this must be taken into consideration

The individual electrons do have quantized energies. And the energy levels do change when an electric field is applied. Nobody is denying this.

What you are apparently failing to understand is that the standard model of scattering of light by atoms and molecules simply doesn't go to that level of detail, because (a) it doesn't need to to make good predictions, and (b) going to that level of detail would make the model much, much more complicated. It's much simpler to just model the atom or molecule as a single object that can be polarized.

If your concern is that all frequencies of light can be scattered, even though the electron energy levels are discrete, that is because, once again, the light does not induce any jumps of electrons from one energy level to another. All the light does is impose an external field on the atom. There is no requirement that an external field imposed on an atom must assume discrete values, or that it must cause discrete changes in the total energy of the atom. If you do the Stark effect experiment in a lab, you can vary the external field continuously, and that will cause a continuous change in the electron energy levels.

when an external field is applied /changing or non-changing/, then electrons seem to be able to change their energy sort of continuously as long as it is within the lower most possible and upper most possible energy discrete energy levels. Of course, only an idea...

And an incorrect one. See above.

 

[conquest]

Hello,

Sorry for disregarding what seems to be a lengthy discussion about rather interesting topics, but I just thought I would put my 2 cents in on the question in the OP. Of course this question has already been answered in a myriad of ways, but I felt it was worth noting what seems to be the main misunderstanding of the OP. This has to do with, as was pointed out, the continuity of velocity (coming from the continuity of time) in the particular system referred to in the OP.

The main question is (as later rephrased): what is the next velocity after 0. The problem here is that we model velocity and time using the real numbers. One particularly interesting mathematical property is that they are uncountable. Uncountable means "unlistable" as explained here
This is why it makes no sense to ask for the next velocity when we model time by the real numbers: It is inherent in the real numbers that they do not posses a notion of next (since this would imply a list of them existed).

What we can do however is model time using the integers (these are the quintessential listable=countable things). So how does one go about doing that? Well we simply cut the time into little pieces, for instance seconds. Then we are of course also obliged to change our notion of the velocity at a certain second. We have some options for this like: the maximal velocity in that second or the average velocity in that second.

Suppose we pick the first then we can answer the question: what is the next velocity after 0 m/s?
A: Whatever velocity we have at t=1s. (I am actually to lazy to do an actual computation :P)