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

[Dinis Oliveira]

Well, there are many ways to define it...
You can just say that it is a part of a constitute that elegantly builds up the the idea between symmetry and conservation in nature, as beautifully shown by ms Emmy Noether.

Or you can equally say that kinetic energy is a body's "intrinsic ability to move relative to other bodies": if a body does not have kinetic energy, it does not move at all: it has no ability to move. If it has LITTLE kinetic energy, it moves uniformly, slowly, but forever if there are no other fields or bodies hindering that.. If it has lots of kinetic energy (this intrinsic ability to move), it moves uniformly, fast and also forever, if there are no other bodies that hinder this. If it GAINS kinetic energy, its speed is growing and it moves faster and faster, exactly as long as it is gaining the kinetic energy; afterwards it will just continue to move uniformly at the speed it has reached.

But this idea that it is meaningless to say that a body has velocity because its atoms are jiggling around is not a good idea.. Because if the center of mass is standing still, we can say that the body does stand still, just vibrating chaotically on atom's scale: but vibration like this is movement relative to the center of mass of that body, which is more or less still (i hope). In a way it is true that a body is never at rest, but that does not exclude the idea of kinetic energy...

If the atoms of a body have mass and are jiggling around they have their own kinetic energies. But if they move in a way that the center of mass of the body moves, then the body gains energy. This energy can come from work if I push the body. Kinetic energy means mass with speed. But speed needs a referential. So kinetic energy is a property of a mass that depends on other masses it has the "potential" to collide with?

 

[jfizzix]

If a ball is held at rest on a slope and then released what is it's next velocity? How can it's velocity change from nothing to something ?If the change from zero is infinitesimally small would this contradict the Quantum Theory as it's change of energy would be continuous.

As far as we know, we can treat velocities are real numbers. There is no discrete succession of velocities from rest for the same reason that there is no smallest number still larger than zero.

 

[PeterDonis]

how would you define a quantum oscillator?

By the fact that its Lagrangian is the same as the Lagrangian of an oscillator. In other words, the word "oscillator" is used because the same math applies; it does not imply that a quantum oscillator is a tiny, tiny ball on a spring, or that it is anything physically even remotely similar to a ball on a spring. It just happens to be describable by the same math.

How discrete are those energy levels of electrons IN A MOLECULE?

Just as discrete. They are different specific energies, because the electron bound states in a molecule are different from the electron bound states in a single atom, but they're still discrete.

blue sky is explained by scattering on many frequencies and an interesting note: they say that even INDIVIDUAL molecules scatter blue and green and other short wavelengths. How is this connected or not connected to special energy amounts that electrons can ...have..?

Scattering of light by gas molecules does not change the internal state of the molecules; the electrons stay at the same energy levels. So there is no connection between the frequencies of light that are scattered and the energy levels of electrons in the molecules.

 

[Ott Rovgeisha]

This doesn't work because kinetic energy is coordinate dependent. You, standing at rest on the Earth's surface, have zero kinetic energy relative to the (rotating) Earth; but you have nonzero kinetic energy relative to an inertial frame in your vicinity. You have even more kinetic energy relative to the Sun, and still more relative to the center of the Milky Way galaxy. So kinetic energy can't be an "intrinsic" property you have.

Of course it works. Maybe the word "intrinsic" is not the best here, but most certainly you can define kinetic energy as bodies ability to move relative to other bodies. Of course it depends on your choice of other bodies relative to who you are measuring it. Never argued that point..

 

[PeterDonis]

most certainly you can define kinetic energy as bodies ability to move relative to other bodies.

I've never seen this definition in a textbook or scientific paper. Do you have a reference? "Ability to move" seems vague to me.

 

[Ott Rovgeisha]

By the fact that its Lagrangian is the same as the Lagrangian of an oscillator. In other words, the word "oscillator" is used because the same math applies; it does not imply that a quantum oscillator is a tiny, tiny ball on a spring, or that it is anything physically even remotely similar to a ball on a spring. It just happens to be describable by the same math.
Just as discrete. They are different specific energies, because the electron bound states in a molecule are different from the electron bound states in a single atom, but they're still discrete.
Scattering of light by gas molecules does not change the internal state of the molecules; the electrons stay at the same energy levels. So there is no connection between the frequencies of light that are scattered and the energy levels of electrons in the molecules.

No connection? So, electrons seem to be oscillating due to the electromagnetic radiation... It becomes rather bizarre, since electrons seem to have formed the so called "chemical bond" . Please do not preach me that chemical bonds are not bonds but quantum mechanical aspect of electrons, but still, what is going on during scattering? They due model it as re-radiation of electromagnetic fields by the molecules; but as you well now, charged particles that accelerate, can only radiate electromagnetic radiation.

So what is going on there?

I am not afraid to ask this question. I find that there are some aspects about these things that are never been made totally clear, but this is important.

 

[PeterDonis]

electrons seem to be oscillating due to the electromagnetic radiation

What kind of "oscillations" are you talking about? The oscillations involved in Rayleigh scattering (see below) are not quantum oscillations and have nothing to do with the "quantum oscillator" aspect of electrons. Which kind of oscillation do you want to talk about?

It becomes rather bizarre, since electrons seem to have formed the so called "chemical bond"

How is this related to the other things you are asking about? Scattering of light by molecules doesn't have any effect on the chemical bonds in those molecules.

what is going on during scattering?

http://en.wikipedia.org/wiki/Rayleigh_scattering

charged particles that accelerate, can only radiate electromagnetic radiation

Yes. That's what happens in scattering; as described in the link above, the gas molecules get polarized by the oscillating electric field of the light, and that makes them radiate. (Note that this is a purely classical model; the light is modeled as classical waves, not photons, and the molecules and their electrons are modeled as classical particles oscillating in a classical electromagnetic field.) What's the problem?

 

[Ott Rovgeisha]

What kind of "oscillations" are you talking about? The oscillations involved in Rayleigh scattering (see below) are not quantum oscillations and have nothing to do with the "quantum oscillator" aspect of electrons.
How is this related to the other things you are asking about?
http://en.wikipedia.org/wiki/Rayleigh_scattering
Yes. That's what happens in scattering; as described in the link above, the gas molecules get polarized by the oscillating electric field of the light, and that makes them radiate. (Note that this is a purely classical model; the light is modeled as classical waves, not photons, and the molecules and their electrons are modeled as classical particles oscillating in a classical electromagnetic field.) What's the problem?

Well i do not know.. Maybe I am just an idiot.. But... When you say that electrons stay at the same energy levels in scattering...this raises a question, because, when you say (correctly) that a gas molecule polarizes, then how can it do that without its electrons changing the distance from the nucleus and therefore changing their energy levels?

I am aware that scattering is not explained via quantum oscillations, but for starters, the question is then: how does everything change? :D

At one point the molecule has so-called chemical bonds: electrons described as quantum phenomena,
at the same time, they are described as just particles that oscillate and polarize the molecule due to the electric field..
So they seem to do different things at the same time and this doesn't seem to add up too clearly.

And the dumbest question of them all, again: how can a molecule polarize without electrons going say further away from the nucleus?
See...stupid questions.. but I warned ya...

You realize why, it is such a challenge to teach those things to people? To make them wonder...think..
Often there seems to be no clearness in concepts and when one brings this up to all mighty scientists and educators, they tend to get offended.

Another thing is: people tend to box things and not to see connections. I do realize that every idea must not necessarily be connected to the other, but still...
It would be nice to show how one thing can be deduced from another or how and why they CANNOT BE deduced, or at least SEEM not to deductive.

 

[PeterDonis]

When you say that electrons stay at the same energy levels in scattering...this raises a question, because, when you say (correctly) that a gas molecule polarizes, then how can it do that without its electrons changing the distance from the nucleus and therefore changing their energy levels?

Once again: the model that says the gas molecule polarizes is a classical model. It doesn't say anything at all about electron energy levels; that's a quantum concept. In this model, the molecule is just a particle which can be polarized so that it's a little electric dipole; the model does not even include internal parts of the atoms (like electrons) or quantum phenomena (like energy levels).

If you wanted to ask the question of whether the energy levels of the electrons change as a result of the atom being polarized, you would have to construct a quantum model of the atom in an electric field. This kind of model has been constructed, in order to explain phenomena like the Stark effect, which is the change in spectral lines of an atom when it is in an external electric field. In this model, yes, the energy levels of the electrons change (as they must in order for the spectral line to change).

However, there are at least two good reasons why such a quantum model of the atom is not used to explain scattering of light. First, it's not needed, because, as I said before, scattering of light by atoms is continuous: there are no discrete spectral lines or other discrete phenomena involved. All frequencies of light are involved. So the quantum aspects of the electron energy levels are simply not involved.

Second, the model used to explain the Stark effect assumes a static electric field; but the electric field of the light in scattering is oscillating. That would make a quantum model of the atom scattering light much more complicated than the one used to explain the Stark effect.

At one point the molecule has so-called chemical bonds: electrons described as quantum phenomena,
at the same time, they are described as just particles that oscillate and polarize the molecule due to the electric field..
So they seem to do different things at the same time and this doesn't seem to add up too clearly.

That's how models in science work. You use different models for different purposes. You don't use a much more complicated quantum model for phenomena that don't involve the quantum aspects. That doesn't mean the molecule doesn't have chemical bonds or that the electrons aren't in energy levels when the molecule scatters light. It just means you don't need to model those aspects to make correct predictions about the scattering of light.

If you were to insist on including all aspects of the physics in every model, then your model of molecules with chemical bonds and electrons in energy levels is incomplete too. The nuclei of the atoms have internal parts; why aren't those included? Why do we just assume that each nucleus is a single particle with a particular positive charge when we model the energy levels of the electrons? Does that mean the nucleus somehow changes form, so it's a single particle some times and a system with internal parts other times (like in a nuclear reaction)? No, of course not. It just means that, when we're modeling the energy levels of electrons, the internal structure of the nucleus doesn't come into play, so we don't include it.

how can a molecule polarize without electrons going say further away from the nucleus?

It can't. But, as above, that doesn't mean you have to include that detail in every model in which the molecule is included.

 

[Ott Rovgeisha]

However, there are at least two good reasons why such a quantum model of the atom is not used to explain scattering of light. First, it's not needed, because, as I said before, scattering of light by atoms is continuous: there are no discrete spectral lines or other discrete phenomena involved. All frequencies of light are involved. So the quantum aspects of the electron energy levels are simply not involved.
.

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

 

[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)