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

[eddie]

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.

 

[axmls]

I don't think this has anything to do with quantum mechanics. By placing the ball onto the ramp, you've inserted potential energy into that system. When you release it, Gravity converts that potential energy to kinetic energy, and it starts to slide down. That is experimentally verifiable.

 

[eddie]

I don't think this has anything to do with quantum mechanics. By placing the ball onto the ramp, you've inserted potential energy into that system. When you release it, Gravity converts that potential energy to kinetic energy, and it starts to slide down. That is experimentally verifiable.

The fundamental question is how does the ball change from zero velocity to some velocity ie. from nothing to something .I am aware that the total energy of the system is conserved.Thanks for your reply ,but it has not answered my question.

 

[DrClaude]

would this contradict the Quantum Theory as it's change of energy would be continuous.

Energy is only quantized for bound systems. Free particles can have any energy continously. And for a macroscopic object, the spacing between energy levels in the quantized case would be so small as to be unobservable: it would look continuous.

 

[DrClaude]

The fundamental question is how does the ball change from zero velocity to some velocity ie. from nothing to something.

I don't see how this is a problem. Care to expand on what you find difficult to understand?

 

[eddie]

I don't see how this is a problem. Care to expand on what you find difficult to understand?

Hi thanks for your reply ,I'd like to repeat my fundamental question of what is the ball's next velocity after zero it does change from nothing(zero) to something.Could you tell me if kinetic energy is quantized ,thanks again.

 

[DrClaude]

Hi thanks for your reply ,I'd like to repeat my fundamental question of what is the ball's next velocity after zero it does change from nothing(zero) to something.Could you tell me if kinetic energy is quantized ,thanks again.

For a free particle, kinetic energy is not quantized. But I don't understand your obsession here with QM: the problem you describe in the OP is classical. For a quantum system, you would have to define what you mean by "held then released" and by a quantum particle "at rest."

 

[axmls]

I think the OP is stuck up on Zeno's paradox right now. I.e. What is the first velocity after being at rest? Is it .01? .001? .00000001? .000...? But if it's .000..., then that's just a rest state.

 

[russ_watters]

The fundamental question is how does the ball change from zero velocity to some velocity ie. from nothing to something ...

Sure it does -- why wouldn't it?

...what is the ball's next velocity after zero it does change from nothing(zero) to something.Could you tell me if kinetic energy is quantized ,thanks again.

As far as is known, the universe is not quantized, so there is no identifiable "next velocity". You have to pick what time interval you want to look at -- the universe doesn't decide for you.

 

[Dale]

.Could you tell me if kinetic energy is quantized ,thanks again.

Energy is quantized in bound systems, but what you describe seems to be a free system where energy would not be quantized. You would have to solve the time independent Schrödinger's equation and see if the solutions are discrete, but I don't think they would be.

 

[eddie]

For a free particle, kinetic energy is not quantized. But I don't understand your obsession here with QM: the problem you describe in the OP is classical. For a quantum system, you would have to define what you mean by "held then released" and by a quantum particle "at rest."

Thanks for your answer to my kinetic energy question.I still would like to know how the ball initially at rest can "jump" to some velocity,it is a similar problem to Zeno's paradox but I still cannot see how the ball can go from no velocity to some velocity.PS.is a "free particle" one that is in equilibrium? if so the ball in question is not a free particle and it's it's kinetic energy that I was referring to.

 

[Dale]

Eddie, you have to decide if you want an answer according to classical mechanics or quantum mechanics.

I still cannot see how the ball can go from no velocity to some velocity.

Why not? What would prevent it from moving?

 

[eddie]

Energy is quantized in bound systems, but what you describe seems to be a free system where energy would not be quantized. You would have to solve the time independent Schrödinger's equation and see if the solutions are discrete, but I don't think they would be.

Thanks for your reply ,the kinetic energy I was referring to was that of the ball.If it's increase in velocity could be infinitely small then it's kinetic energy would be continuous.

 

[Dale]

Yes.

 

[eddie]

Eddie, you have to decide if you want an answer according to classical mechanics or quantum mechanics.

Why not? What would prevent it from moving?

Nothing so what would be it's first velocity?

 

[eddie]

Nothing so what would be it's first velocity?

Why,would the answers be different?

 

[eddie]

Yes.

So what are they?

 

[DaveC426913]

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.

2] Its "first" velocity will depend on how long you wait to measure it. Since a force is being applied to it, you will have to calculate what its velocity is after a non-zero length of time. So, pick a time. 0.00000000000000000001 seconds? OK, well, after that length of time you can easily calculate its velocity based on F=ma. Too long a delay? try 1x10^-20 seconds. Shorter time = smaller velocity. But it'll always be >0 as long as the time is > 0.

 

[eddie]

Sure it does -- why wouldn't it?

As far as is known, the universe is not quantized, so there is no identifiable "next velocity". You have to pick what time interval you want to look at -- the universe doesn't decide for you.

Hi Mr Watters thanks for your excellent answer at last I've got an answer that I understand.I asked the same question of Professor Hawking years ago ,the answer that I got was that he was too busy to give me an answer.

 

[eddie]

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.

2] Its "first" velocity will depend on how long you wait to measure it. Since a force is being applied to it, you will have to calculate what its velocity is after a non-zero length of time. So, pick a time. 0.00000000000000000001 seconds? OK, well, after that length of time you can easily calculate its velocity based on F=ma. Too long a delay? try 1x10^-20 seconds. Shorter time = smaller velocity. But it'll always be >0 as long as the time is > 0.

Thank you Dave for your reply ,my question has been answered by Russ Watters.

 

[Dale]

Nothing so what would be it's first velocity?

See: http://hyperphysics.phy-astr.gsu.edu/hbase/sphinc.html

Taking that and solving for velocity we get $v=\frac{5}{7} g \sin(\theta) t$

If you want the "first" v then all you have to do is plug in $g$, $\theta$, and the "first" $t$.

 

[sophiecentaur]

It is a shame that QM was brought into this so early on. The number of quantum states for a massive ball is huge and that number would actually depend upon the mass / size of the ball. That would make it impossible to answer such a question with a definite number - plus the fact that we are really talking in terms of a 'drift velocity' - just as with electrons in an electric current. If we're not talking about individual atoms / molecules in the gaseous state, then quantum levels are really just a distraction.

 

[eddie]

See: http://hyperphysics.phy-astr.gsu.edu/hbase/sphinc.html

Taking that and solving for velocity we get $v=\frac{5}{7} g \sin(\theta) t$

If you want the "first" v then all you have to do is plug in $g$, $\theta$, and the "first" $t$.

Hi thanks for the effort you have made to answer my question ,I know the answer to what v equals at t=0 but what I'm asking is what v equals next.I think you will find there is no answer to this question at the present time ,russ watters has stated there is no "next" velocity as the energy of the universe is not quantized

 

[eddie]

It is a shame that QM was brought into this so early on. The number of quantum states for a massive ball is huge and that number would actually depend upon the mass / size of the ball. That would make it impossible to answer such a question with a definite number - plus the fact that we are really talking in terms of a 'drift velocity' - just as with electrons in an electric current. If we're not talking about individual atoms / molecules in the gaseous state, then quantum levels are really just a distraction.

Hi Sophiecentaur thanks for your reply, the partial answer was given to me by russ watters stating that the energyy of the universe is not quantized and hence there was no "next velocity" after zero time .But the problem is that there is

 

[Dale]

I know the answer to what v equals at t=0 but what I'm asking is what v equals next

I understand your question. My question back to you is "which t is next?".

 

[eddie]

I understand your question. My question back to you is "which t is next?".

That's a good question ! how about minus infinity.I think we are getting to the crux of the matter.I'm very surprised that you are still helping me with this problem ,thank you.Also thank you for not trying to baffle me.

 

[DaveC426913]

how about minus infinity.

This is a nonsensical value.

We are talking about a lapse of time after t=0. Minus infinity does not make sense.

I wonder if what you were going for was 1 / infinity. i.e. an infinitesimally small lapse of time.

 

[eddie]

This is a nonsensical value.

We are talking about a lapse of time after t=0. Minus infinity does not make sense.

I wonder if what you were going for was 1 / infinity. i.e. an infinitesimally small lapse of time.

Yes that is what I wanted to go for .

 

[Dale]

What does it mean for one element of a set to be the next element. Think about mathematical conditions that you could write down to test if something was next.

 

[Drakkith]

Several off topic posts and their replies have been deleted. I remind all members to please stay on topic and within PF rules.

 

[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 Schrodinger 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.