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

[PeterDonis]

Yes that is what I wanted to go for .

1 / infinity is not a real number, and measurements and their results are described by real numbers. Real numbers form a continuum, and in a continuum there is no "next element" after a given element. So there is no "next instant of time" after $t = 0$, and therefore no "next velocity" after $v = 0$. The velocity changes continuously as the time advances continuously.

 

[ZapperZ]

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 ball is on a flat surface, and I come along and push on it, are you still puzzled that it moves?

Zz.

 

[MikeGomez]

Excellent answers have been provided, but I just wanted to add one more point.

0 velocity is just a random and arbitrary velocity, and it is relative to some reference frame. If you are traveling in an airplane at 500 mph, you and everything inside the plane have a velocity of 500 mph relative to the ground, yet you might also say that a cup of water on a tray is traveling at 0 velocity relative to you inside the plane. It (the cup of water) has different velocities relative to different frames, and 0 velocity is just 1 of an infinity of choices of reference frames.

So an equivalent to your question of how does an object go from zero velocity to a non-zero velocity might just as well be “How does an object go from say 10 mph to the next higher velocity above 10 mph? or “How does an object go from 2801.63 kilometers per second to some other velocity?” In the end, these and all velocities (less than c) can be considered to be at rest (zero velocity) in some reference frame.

Therefore, due to the relative nature of velocity, in my opinion you are asking in your OP about the nature of acceleration, and in particular what is the smallest increment of acceleration (if any). Perhaps viewing it in this way might help a bit.

 

[Dale]

Closed for moderation due to some recent posts.

EDIT: The thread is reopened after some cleanup. Note, the system described by the OP is not a bound system so, as described in posts 4, 7, and 10, its energy levels are not expected to be quantized.

 

[gianeshwar]

Energy is only quantized for bound systems. Free particles can have any energy continously.

Can anyone please give simple explanation.

 

[gianeshwar]

Can anyone please give simple explanation.

Got the answer at http://en.wikipedia.org/wiki/Energy_level

 

[Ott Rovgeisha]

Can anyone please give simple explanation.

Well, I don't know what did the original poster wanted to say with it, but what is usually meant by it, is the following:

if for example there is an electron, a free one, say in vacuum, it can for example gain kinetic energy in a continuous manner: it can have any value, by growing gradually. The fact that "gradually" is of course also a suspicious idea, and depends on definition, but that is what is meant.

But when an electron is in interaction with a proton in a nucleus, it seems not to be able to gain ANY value of energy, it cannot be closer or further away from the nucleus by just any distance, but very certain distances, gaining or losing energy only by very certain quantities. One can say that electron is BOUND with a proton...

But this is interesting, because usually those situations occur when there are different forces acting against each other...
Like a ball on a spring: resonance occur only at specific frequencies and you can look at it as the ball gaining and loosing a very certain amounts of energy.

 

[Dinis Oliveira]

1] Any object with mass is, in essence, always moving. It's made of atoms and atoms bounce around. It is really meaningless to say that the object's velocity is ever zero. You'd have to average the Brownian motion of every atom in it.

Then what is kinetic energy?

 

[Ott Rovgeisha]

Then what is kinetic energy?

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...

 

[PeterDonis]

when an electron is in interaction with a proton in a nucleus, it seems not to be able to gain ANY value of energy

Yes, and the key point is why this is true. It is true because, as the Wikipedia article gianeshwar linked to says, if the electron is confined to a finite region of space (as it is if it is in a bound state), then the wavelength of its wave function can only assume discrete values, corresponding to some integral number of standing waves in the finite region of space (the fact that it must be an integral number of standing waves is what makes the values discrete). The discrete allowed values of energy are a consequence of the discrete allowed values of wavelength (actually of frequency, which is determined by wavelength).

If the electron is free, it can be anywhere in space, and so its wavelength can assume any value at all. Therefore, its frequency and hence energy can also assume any value at all.

Like a ball on a spring: resonance occur only at specific frequencies and you can look at it as the ball gaining and loosing a very certain amounts of energy.

No, this is not the same. The kinetic energy of the ball on the spring varies continuously; it does not jump discretely from one value to another. The ball on the spring is a classical system, not a quantum system. The electron in an atom is a quantum system, and its energy does not vary continuously.

 

[gianeshwar]

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.

How does it explain energy asked here?
In original question as well,it was asked about next velocity and hence I think next kinetic energy.
In mathematics definitely we can easily prove that some functions do not have maxima or minima or both in certain domains despite these being bounded functions[e.g.,f(x)=x in (0,1)]

 

[PeterDonis]

you can equally say that kinetic energy is a body's "intrinsic ability to move relative to other bodies":

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.

 

[DaveC426913]

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..

That's not exactly what I was trying to say. What I was trying to say was that - in the context of the OP's question about infinitesimally small velocities - it is kind of meaningless to say a body has no velocity.

The OP is struggling with how an entire object can go from zero velocity to an infinitesimally small velocity. The velocity of an object is - at this granularity - simply sum (or average) of the velocities of the individual particles. So they're already moving.

 

[gianeshwar]

the electron is confined to a finite region of space (as it is if it is in a bound state)

Probability of finding it anywhere else is zero or near zero?
I think it must not be zero.
I am attempting to understand quantum concepts.

 

[PeterDonis]

Probability of finding it anywhere else is zero or near zero?
I think it must not be zero.

Technically, it's not exactly zero anywhere, at least not in a realistic model. (For pedagogical purposes, models are often used where the probability is exactly zero outside a finite region; that type of idealized model is what I think the Wikipedia page you linked to was referring to.) But even in a realistic model, the probability does go to zero as you go to spatial infinity, and that turns out to be sufficient to get quantized wave functions.

For an example, see the Wikipedia article on the hydrogen-like atom:

http://en.wikipedia.org/wiki/Hydrogen-like_atom

In this case, the quantized wave functions are the spherical harmonics, which are described by discrete quantum numbers; these are more complicated functions than the simple standing waves in the Wikipedia article you linked to, but they still have the same discreteness property, which is the key point.

 

[gianeshwar]

This discussion is very interesting.Thank you every participant!
I have so far never solved Schrödinger wave equation ,which I am now eager to solve.
I think I will get thrilling discrete solutions ?(Please give me some directions if possible.)
I have so far solved only simple differential equations to get continuous functions as solutions.

 

[Ott Rovgeisha]

Yes, and the key point is why this is true. It is true because, as the Wikipedia article gianeshwar linked to says, if the electron is confined to a finite region of space (as it is if it is in a bound state), then the wavelength of its wave function can only assume discrete values, corresponding to some integral number of standing waves in the finite region of space (the fact that it must be an integral number of standing waves is what makes the values discrete). The discrete allowed values of energy are a consequence of the discrete allowed values of wavelength (actually of frequency, which is determined by wavelength).

If the electron is free, it can be anywhere in space, and so its wavelength can assume any value at all. Therefore, its frequency and hence energy can also assume any value at all.
No, this is not the same. The kinetic energy of the ball on the spring varies continuously; it does not jump discretely from one value to another. The ball on the spring is a classical system, not a quantum system. The electron in an atom is a quantum system, and its energy does not vary continuously.

The spring and the ball IS sort of the same, because the TOTAL energy in A SINGLE resonance frequency IS fixed. Of course it various from kinetic to potential, but the energy itself is fixed: the energy in ONE resonance frequency. There are similarities there, of course, also differences, but again, important similarities. Again, one must be very careful in treating those things...but there are some interesting similarities which may or may not imply to some connection that we may have not been able to resolve yet.

As for confining an electron into a defined space... Well, to be honest I am not sure what that means. How do you confine an electron.. With a box? it is also made of electrons and atoms.
This is kind of abstract and not very clear for me at least.

 

[jbriggs444]

The spring and the ball IS sort of the same, because the TOTAL energy in A SINGLE resonance frequency IS fixed. Of course it various from kinetic to potential, but the energy itself is fixed: the energy in ONE resonance frequency.

The energy of a ball on a spring is not a function of resonant frequency. For a fixed ball mass and a fixed spring constant, it is a function of amplitude and can vary while frequency remains unchanged. You can add an arbitrarily small amount of energy and make the amplitude larger or remove an arbitrarily small amount of energy and make the amplitude smaller.

For a bound electron the amounts of energy you can add or remove are quantized. Only certain discrete increments or decrements are possible.

 

[Ott Rovgeisha]

The energy of a ball on a spring is not a function of resonant frequency. For a fixed ball mass and a fixed spring constant, it is a function of amplitude and can vary while frequency remains unchanged. You can add an arbitrarily small amount of energy and make the amplitude larger or remove an arbitrarily small amount of energy and make the amplitude smaller.

For a bound electron the amounts of energy you can add or remove are quantized. Only certain discrete increments or decrements are possible.

Yes of course. But still: very distinct frequency values of a wave pattern... kind of similar to an electron: very distinct energy level corresponding to a very distinct standing wave pattern.. (which seems to be a standing wave of probability, so nobody in hell hasn't figured out, what an electron really is, either bound or unbound).

But of course, you are right, there seem to be some very important differences... Although, how would you define a quantum oscillator?

BUt by the way.. How discrete are those energy levels of electrons IN A MOLECULE?

For example, 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..?

 

[jbriggs444]

Yes of course. But still: very distinct frequency values of a wave pattern... kind of similar to an electron: very distinct energy level corresponding to a very distinct standing wave pattern..

For a ball and spring there is only one resonant frequency and no dependence of energy on that frequency. Are you, perhaps, thinking of standing waves on a rope or in a bounded pool of water?

However this has little to do with the original question posed in this thread.

 

[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...