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Author Topic: How does mass increase at higher speeds?  (Read 46076 times)

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #100 on: 19/01/2013 02:03:54 »
Why do you need a photon for this? A light pulse wouldn't work?
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #101 on: 19/01/2013 03:28:53 »
Why do you need a photon for this? A light pulse wouldn't work?
Who said a photon was needed? Einstein used a "burst of radiation". In any case it's sufficient, to use a photon, not neccesary. If you read the derivation I pointed to by Antippa below Eq. (8) they you'd see that it uses an atom emitting a photon. You can't have an atom emitting light, pe se
« Last Edit: 19/01/2013 09:10:42 by Pmb »
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #102 on: 19/01/2013 14:13:05 »
Ok, anyway, if you want to use the term "photon" because it's emitted or absorbed by an atom, you can't say it can be localized in flight. Either is a photon, with its quantum properties, or is a classical pulse of light; you can't mix the two in a sort of "chimeric" beast.
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #103 on: 20/01/2013 14:05:14 »
Ok, anyway, if you want to use the term "photon" because it's emitted or absorbed by an atom, you can't say it can be localized in flight. Either is a photon, with its quantum properties, or is a classical pulse of light; you can't mix the two in a sort of "chimeric" beast.
I'm afraid that you’ve made the same mistake here that you’ve made in your previous posts.  You say it can’t be done but give no proof. In this case you claim “you can't say it can be localized in flight” but don’t explain what that means or why you can’t say it and what it means not to be able to say something when in practice (i.e. in practical examples, math and all) it works just fine.

I feel like I’ll just be repeating myself here so I’m ending my contribution in this thread and am as such agreeing to disagree. That means I won’t respond to any further assertions you make even if I  know them to be wrong.

I rest easy in the knowledge that has been done successfully on numerous occasions by many physicists in many textbooks and at least some journal articles. Most notably its now being used in the new version of Exploring Black Holes. I’d like to note that one of the authors of that text, Edwin Taylor, is also co-author of the MIT Introductory Physics Series Quantum Mechanics text.
« Last Edit: 20/01/2013 14:46:04 by Pmb »
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #104 on: 21/01/2013 12:43:57 »
Ok, anyway, if you want to use the term "photon" because it's emitted or absorbed by an atom, you can't say it can be localized in flight. Either is a photon, with its quantum properties, or is a classical pulse of light; you can't mix the two in a sort of "chimeric" beast.
I'm afraid that you’ve made the same mistake here that you’ve made in your previous posts.  You say it can’t be done but give no proof. In this case you claim “you can't say it can be localized in flight” but don’t explain what that means or why you can’t say it and what it means not to be able to say something when in practice (i.e. in practical examples, math and all) it works just fine.
When I answered to the question of what is the "centre" of a system of two photons (question intended in the sense that the photons were emitted but not yet absorbed) I said that it's not possible because you can't say where a photon is exactly in flight, that is after emission and before detection.
The proof is very simple, it only needs the Young experiment with two slits and photons emitted one at a time: if you can say which slit the photon passed through, the interference pattern disappears...
 

Offline JP

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Re: How does mass increase at higher speeds?
« Reply #105 on: 21/01/2013 16:43:56 »
Lightarrow, is there any reason why you can't take a classical limit of the quantum theory to come up with classical photons like Pmb claims?  That approach is certainly valid for electrons (and explains why we have "classical" electrons" when we know they also behave like waves in the 2 slit experiment).
 

Offline AndroidNeox

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Re: How does mass increase at higher speeds?
« Reply #106 on: 21/01/2013 17:44:02 »
All matter and energy has mass. Even the kinetic energy of an object has mass. In fact, when you compress a spring, its mass increases... not detectably, but the potential energy added to the spring has its own contribution to the total mass.
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #107 on: 21/01/2013 19:09:31 »
Lightarrow, is there any reason why you can't take a classical limit of the quantum theory to come up with classical photons like Pmb claims?  That approach is certainly valid for electrons (and explains why we have "classical" electrons" when we know they also behave like waves in the 2 slit experiment).
When you take the classical limit for a photon (h --> 0) it gets zero energy, so it disappears.
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #108 on: 21/01/2013 19:12:37 »
All matter and energy has mass. Even the kinetic energy of an object has mass.
If you mean relativistic mass, ok. If you mean "mass", with this term it's usually intended "invariant" mass and then it's false, unless the object rotates around a fixed point.
Quote
In fact, when you compress a spring, its mass increases... not detectably, but the potential energy added to the spring has its own contribution to the total mass.
Correct.
 

Offline JP

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Re: How does mass increase at higher speeds?
« Reply #109 on: 21/01/2013 20:32:40 »
Lightarrow, is there any reason why you can't take a classical limit of the quantum theory to come up with classical photons like Pmb claims?  That approach is certainly valid for electrons (and explains why we have "classical" electrons" when we know they also behave like waves in the 2 slit experiment).
When you take the classical limit for a photon (h --> 0) it gets zero energy, so it disappears.

Good point.  After reading that and a bit more thinking, I believe that what's going on with these "classical photons" is two limits.  If we have a field made of photons, the classical limit does not correspond to taking h->0, but rather to taking many photons.  In this limit, you recover Maxwell's equations, but you've lost information about the behavior of individual photons.

Then you take a second limit corresponding to wavelength->0 which gets you ray optics.  So essentially, your "classical photon" is an arbitrary packet of energy assigned to propagate along a ray.  It's related to real photons only insofar as the sum over many photons gets you the classical field, which you then use to define rays.
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #110 on: 21/01/2013 22:57:47 »
Quote from: lightarrow
When you take the classical limit for a photon (h --> 0) it gets zero energy, so it disappears.
(sigh!) I'm clearly sorry that I asked. :)

That's wrong. If you were right then no classical particles exist in the classical limit. Don't forget what h physically means. I means that for every quantum mechanical particle that has inertial energy (defined as E = K + E0 = Kinetic Energy + Rest Energy) has an associated frequency given by the relationship E = hf. What does it mean to take h -> 0 for an electron? It means that there is no associated wavelength.

Recall that in classical electrodynamics one can have a very small packet/burst of radiation (which can be described by a Fourier integral) which has enegy and momentum p. The relationship between them is a non-quantum mechanical relationship, i.e. E = pc. The shape of the light pulse can be selected such that the spatial extention is small enough for all practical purposes.
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #111 on: 21/01/2013 23:02:06 »
Quote from: lightarrow
If you mean relativistic mass, ok.
You know that its relativistic mass from the context in which he used it. Its not wise to assume that everyone who uses the description that he does needs to be reminded of these facts. To assume so is condescending to the poster because we're assuming that he doesn't know what he's talking about and we need to correct them.

I recommend that if someone says "mass depends on speed/energy etc." then we simply assume that by "mass" they mean relativistic mass. Having to remind people all the time is a waste of everybody's time. People can get irritated when someone has to comment on it every time they use the concept, don't you think
 

Offline JP

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Re: How does mass increase at higher speeds?
« Reply #112 on: 22/01/2013 03:57:26 »
Recall that in classical electrodynamics one can have a very small packet/burst of radiation (which can be described by a Fourier integral) which has enegy and momentum p. The relationship between them is a non-quantum mechanical relationship, i.e. E = pc. The shape of the light pulse can be selected such that the spatial extention is small enough for all practical purposes.

Yep, and a photon is usually defined similar to a Fourier component (monochromatic plane wave) of a pulse.  It has a well-defined frequency, momentum and polarization, but no simple position representation (much as a plane wave has a well-defined frequency, direction and polarization, but exists over all space).  Just like a pulse which has a confined position can be expressed as a superposition of many plane waves, a classical beam which has a limited area in space can be expressed as a state consisting of many photons.  Once you have that state, you can invoke geometrical optics or similar approximations to make it appear like a classical particle, but you've taken two approximations: many photons->classical pulse->geometrical/partical approximation to Maxwell's equations. 

I've no doubt that you can make these two approximations in that order because I know the math fairly well.  So if "classical photon" means (essentially) classical approximation to a solution of Maxwell's equations, which are themselves a many-photon approximation to a quantized field, then yes--"classical photons" are a thing.

What I'm less sure of is this: can you go directly from the mathematical description of a photon and, by expanding in powers of h, (presumably by expanding the action?) get to a classical particle description?  Or is there something about photons that prevents you from doing this directly?  I know, for example, that photon wave-functions are controversial and unlike those of massive particles (http://arxiv.org/pdf/quant-ph/0508202v1.pdf). 
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #113 on: 22/01/2013 13:41:39 »
Quote from: JP
Yep, and a photon is usually defined similar to a Fourier component (monochromatic plane wave) of a pulse.
Huh? What is a "Fourier component (monochromatic plane wave) of a pulse."?
 

Offline JP

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Re: How does mass increase at higher speeds?
« Reply #114 on: 22/01/2013 15:06:20 »
A monochromatic plane wave is a solution to the wave equation that has both a definite direction and frequency.  You can write it as
c8531d011ab11d8c373a228910453760.gif
where k is wave vector (2*pi/lambda*direction), x is position, omega is (angular) frequency and t is time.  Each of these solutions satisfies a wave equation, so any pulse built from these satisfies the wave equation.  Since they're exponential solutions, the integral is a Fourier integral that takes a weighting function in k and omega to x and t.
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #115 on: 23/01/2013 17:24:46 »
Quote from: lightarrow
When you take the classical limit for a photon (h --> 0) it gets zero energy, so it disappears.
(sigh!) I'm clearly sorry that I asked. :)

That's wrong. If you were right then no classical particles exist in the classical limit. Don't forget what h physically means. I means that for every quantum mechanical particle that has inertial energy (defined as E = K + E0 = Kinetic Energy + Rest Energy) has an associated frequency given by the relationship E = hf. What does it mean to take h -> 0 for an electron? It means that there is no associated wavelength.
I have already replied you (in a previous post of this or another similar thread, don't remember) that a photon is different because has zero mass. For an electron, you can have non-zero momentum p even at very low speeds, because of its non-zero mass. Then De-Broglie relationship: c6a6eb61fd9c6c913da73b3642ca147d.gif = h/p tells that, in the limit h --> 0, c6a6eb61fd9c6c913da73b3642ca147d.gif = 0, that is, frequency should be infinite.
For a photon you can't do it, because its momentum p too would vanish, in the limit h --> 0.
« Last Edit: 23/01/2013 17:27:34 by lightarrow »
 

Offline AndroidNeox

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Re: How does mass increase at higher speeds?
« Reply #116 on: 23/01/2013 20:38:27 »
All matter and energy has mass. Even the kinetic energy of an object has mass.
If you mean relativistic mass, ok. If you mean "mass", with this term it's usually intended "invariant" mass and then it's false
[/quote]

Naturally I'm referring to relativistic mass, since that's what the question is about. The rest mass doesn't change because it's never in motion and has no kinetic energy. I was specifically referring to kinetic energy having mass.
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #117 on: 23/01/2013 21:19:32 »
All matter and energy has mass. Even the kinetic energy of an object has mass.
If you mean relativistic mass, ok. If you mean "mass", with this term it's usually intended "invariant" mass and then it's false
Naturally I'm referring to relativistic mass, since that's what the question is about. The rest mass doesn't change because it's never in motion and has no kinetic energy. I was specifically referring to kinetic energy having mass.
Certainly. However we should be more precise when we discuss this subject because it's very easy to make confusion. For example, saying "All matter and energy has mass. Even the kinetic energy of an object has mass" is very confusing: in the first sentence, matter has invariant mass, "energy has mass" is incorrect, since energy is "a property" of a body, and a property cannot have mass (as if I would say that a number has a colour); we should say instead that "a body which has energy has mass", but in this case is not always invariant mass...
As you see, things are not so simple.
« Last Edit: 23/01/2013 21:21:48 by lightarrow »
 

Offline AndroidNeox

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Re: How does mass increase at higher speeds?
« Reply #118 on: 23/01/2013 21:32:15 »
Certainly. However we should be more precise when we discuss this subject because it's very easy to make confusion. For example, saying "All matter and energy has mass. Even the kinetic energy of an object has mass" is very confusing: in the first sentence, matter has invariant mass, "energy has mass" is incorrect, since energy is "a property" of a body, and a property cannot have mass (as if I would say that a number has a colour); we should say instead that "a body which has energy has mass", but in this case is not always invariant mass...
As you see, things are not so simple.

Energy does have gravitational mass. Put a kilogram of matter and one of antimatter into an impregnable box, like a Schrödinger cat box, and the mass of the box (any category of mass you care to choose) will not change when the contents annihilate each other. Even if the box only contains light, the mass(es) will not change.
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #119 on: 24/01/2013 09:04:56 »
Certainly. However we should be more precise when we discuss this subject because it's very easy to make confusion. For example, saying "All matter and energy has mass. Even the kinetic energy of an object has mass" is very confusing: in the first sentence, matter has invariant mass, "energy has mass" is incorrect, since energy is "a property" of a body, and a property cannot have mass (as if I would say that a number has a colour); we should say instead that "a body which has energy has mass", but in this case is not always invariant mass...
As you see, things are not so simple.
Energy does have gravitational mass.
1. Energy *cannot* have mass, or charge, or lenght, colour, as a number cannot have mass, charge, lenght, or colour. I've already written it but you probably still haven't understood it. Energy is *a property* of a body, and *not a body itself*. What would you answer to someone who stated that his "age" has "weight"?
Please, answer this question, before stating another time that energy has mass.
2. Even when your statement is correctly written, that is: "a body which has energy
also have gravitational mass" is not exact because a photon, by itself, doesn't have gravitational mass (before you contest this, think to a physics book where this is written). Instead "a region of space which has electromagnetic energy" does have.
Quote
Put a kilogram of matter and one of antimatter into an impregnable box, like a Schrödinger cat box, and the mass of the box (any category of mass you care to choose) will not change when the contents annihilate each other. Even if the box only contains light, the mass(es) will not change.
Correct, but it doesn't confirm your statement.
By the way, there is no need of matter and antimatter and not even of light in a box,  two photons are enough, because such a system have a non-zero mass (I mean invariant mass, not relativistic mass), I have already showed it in a recent thread and in several others, during the years.
« Last Edit: 24/01/2013 09:11:36 by lightarrow »
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #120 on: 24/01/2013 13:50:43 »
Quote from: lightarrow link
That's wrong. I have already replied you (in a previous post of this or another similar thread, don't remember) that a photon is different because has zero mass.
I can see that you’re not one who is much for agreeing to disagree, huh? Okay.

Quote from: lightarrow link
For an electron, you can have non-zero momentum p even at very low speeds, because of its non-zero mass. Then De-Broglie relationship: c6a6eb61fd9c6c913da73b3642ca147d.gif = h/p tells that, in the limit h --> 0, c6a6eb61fd9c6c913da73b3642ca147d.gif = 0, that is, frequency should be infinite.
And since the product of wavelength and momentum has to remain the same in that relationship it means that it’s an improper limit since as ones going down, the other is going up.

In any case you keep forgetting that we’re treating it as a classical particle for which the relationship: c6a6eb61fd9c6c913da73b3642ca147d.gif = h/p ignored. Thus it means nothing in the classical sense to take a classical limit of a photon. JP and myself explained how it works in a classical sense.  You insist on ignoring its defining properties. It’s a classical description of a Fourier integral description of a sharp pulse of radiation. Since JP and I both have our convictions I’m wondering what it is that you hope to gain into continuing this? You certainly can’t change the textbooks or change the usage of the center of mass of a two photon system in the textbooks. As I have asked too many times now with no response – why do people insist in using the scenario I’ve described and never came to a wrong answer.

Please explain, in detail if you please, what it means when you say “You can’ do that!” when so many people do it, and very successfully too I might ad. You imply that you can’t draw a worldline of a photon in a spacetime diagram and yet physicists do it all the time. E.g. referring to finding the center of mass of two photons you claimed “You can't localize a photon, so you can't do that.” When in fact derivations based on the localizability of property of a photon that plays a crucial role in its use in the derivation, you know, the one you claimed can’t be done. While you’re at it please state in detail what you mean whey you say “You can’t localize a photon”.  Again you claimed that just because I was speaking about classical physics you claimed that I was speaking about quantum mechanics, again with no proof provided. I suggest that you take a look at the derivation again http://home.comcast.net/~peter.m.brown/sr/einsteins_box.htm just pas Eq. (8). Nothing about a quantum nature about anything is mentioned in there. Easy as cake. Over and over again you keep making the unfounded assertion that you can’t localize a photon but then never state what that means in practice. Then you later claim that you “so you can’t speak of photons”. Then you go on an claim I don’t know a lot about QM just because you didn’t understand what I wrote and you didn’t appear to go back to JPs post to see the context in which I made the statement which makes you think I don’t understand QM. As JP says and I believe him Maybe there's a reason why you can't, but based on my (admittedly limited) understanding of the Standard Model, I don't see why this would be.

 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #121 on: 24/01/2013 17:11:34 »
I can see that you’re not one who is much for agreeing to disagree, huh? Okay.
It depends. Because of my nickname, I like to talk about light and photons :), so I've studied this subject and discussed it in the forums, for several years.
Quote
Quote from: lightarrow link
For an electron, you can have non-zero momentum p even at very low speeds, because of its non-zero mass. Then De-Broglie relationship: c6a6eb61fd9c6c913da73b3642ca147d.gif = h/p tells that, in the limit h --> 0, c6a6eb61fd9c6c913da73b3642ca147d.gif = 0, that is, frequency should be infinite.
And since the product of wavelength and momentum has to remain the same
? We are making the limit h-->0, so it doesn't remain the same!
Quote
In any case you keep forgetting that we’re treating it as a classical particle for which the relationship: c6a6eb61fd9c6c913da73b3642ca147d.gif = h/p ignored.
But the point is that *you have to prove* treat it can be treated as a classical particle.
Quote
Thus it means nothing in the classical sense to take a classical limit of a photon.
Exactly! Read again what you have written here   :)
Quote
JP and myself explained how it works in a classical sense.
JP wrote this:
<<Good point.  After reading that and a bit more thinking, I believe that what's going on with these "classical photons" is two limits.  If we have a field made of photons, the classical limit does not correspond to taking h->0, but rather to taking many photons.  In this limit, you recover Maxwell's equations, but you've lost information about the behavior of individual photons.

Then you take a second limit corresponding to wavelength->0 which gets you ray optics.  So essentially, your "classical photon" is an arbitrary packet of energy assigned to propagate along a ray.  It's related to real photons only insofar as the sum over many photons gets you the classical field, which you then use to define rays.>>
 

So JP, essentially, says: OR you recover classical electromagnetism, and you have lost information about the behaviour of individual photons, OR you come up with the geometrical optics approximations and you can talk about packets of energy assigned to propagate along a ray.
But this is exactly what I've already written to you in previous posts  :)
Quote
You certainly can’t change the textbooks
Sincerely I have never read "classical photon" in a book of physics.
Quote
or change the usage of the center of mass of a two photon system in the textbooks. As I have asked too many times now with no response – why do people insist in using the scenario I’ve described and never came to a wrong answer.
Please explain, in detail if you please, what it means when you say “You can’ do that!” when so many people do it,
"So many people" are able to localize a photon in flight? Forget it...
Quote
and very successfully too I might ad. You imply that you can’t draw a worldline of a photon in a spacetime diagram and yet physicists do it all the time.
How do you draw a photon's worldline between (0,0,0,0) and (1,1,0,0)? Just for curiosity.
Quote
E.g. referring to finding the center of mass of two photons you claimed “You can't localize a photon, so you can't do that.” When in fact derivations based on the localizability of property of a photon that plays a crucial role in its use in the derivation, you know, the one you claimed can’t be done. While you’re at it please state in detail what you mean whey you say “You can’t localize a photon”.  Again you claimed that just because I was speaking about classical physics you claimed that I was speaking about quantum mechanics, again with no proof provided. I suggest that you take a look at the derivation again http://home.comcast.net/~peter.m.brown/sr/einsteins_box.htm just pas Eq. (8). Nothing about a quantum nature about anything is mentioned in there. Easy as cake. Over and over again you keep making the unfounded assertion that you can’t localize a photon but then never state what that means in practice. Then you later claim that you “so you can’t speak of photons”. Then you go on an claim I don’t know a lot about QM just because you didn’t understand what I wrote and you didn’t appear to go back to JPs post to see the context in which I made the statement which makes you think I don’t understand QM. As JP says and I believe him Maybe there's a reason why you can't, but based on my (admittedly limited) understanding of the Standard Model, I don't see why this would be.
I've already explained in simple terms why you can't localize a photon, but you don't accept it because, you say, it works only for photons described in quantistic sense; I tried to show you that this is the only description for the term "photon" and so we are in a loop...
I sincerely don't know what else I could say.
« Last Edit: 24/01/2013 17:14:29 by lightarrow »
 

Offline JP

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Re: How does mass increase at higher speeds?
« Reply #122 on: 24/01/2013 18:49:36 »
I've already explained in simple terms why you can't localize a photon, but you don't accept it because, you say, it works only for photons described in quantistic sense; I tried to show you that this is the only description for the term "photon" and so we are in a loop...
I sincerely don't know what else I could say.

This sums it up nicely: you two disagree on what you call a photon.  Pmb's photons are not at all the quantized photons of quantum optics.  If I were writing the textbooks, I'd side with Lightarrow on this and reserve photon to mean a very specific thing in quantum mechanics that has no classical analogue: the state resulting from an exitation of hbar*omega of energy in the EM field.  However, as I'm not the President of Physics, I can't do this, and I've noticed that many physicists use the term in a sense similar to what Pmb is doing, and this works out pretty well.  The problem arises when compare the properties of these different uses of "photon" and realize they're talking about two completely different things with completely different properties, and that there is no classical limit of a single-photon Fock state.

Perhaps the solution should be to elect me the President of Physics and I'll rewrite all the textbooks to clear this up?   :P
 

Offline Pmb

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Re: How does mass increase at higher speeds?
« Reply #123 on: 25/01/2013 02:32:57 »
Quote from: lightarrow
I've already explained in simple terms why you can't localize a photon, but you don't accept it because, you say, it works only for photons described in quantistic sense; I tried to show you that this is the only description for the term "photon" and so we are in a loop...
I sincerely don't know what else I could say.
If that is your response then you weren't paying attention to what I was saying. You consistently keep forgetting the approximation and what it would mean to put the photon's position vector inside the area of uncertainty according to how the wave function would average the position. I gave you an example of a pixel of 0.001 mm in width a length and when it detects the photon then its localized in that area and the location of the photon is the location of the pizel) e.g. geometric center.

You youy insist on ignoring every single thing I've said regarding approximation then there is no use to continue this conversation. Why should I post something I know you're going to igore?

While you're at it it wouln't hut you to finally state what it is you mean by saying something can or can't be localized. E.g. find a QM texbook and quote the definition of "localized" or "localize" so you won't be vauge anymore.
« Last Edit: 25/01/2013 02:50:38 by Pmb »
 

Offline lightarrow

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Re: How does mass increase at higher speeds?
« Reply #124 on: 25/01/2013 11:39:01 »
Quote from: lightarrow
I've already explained in simple terms why you can't localize a photon, but you don't accept it because, you say, it works only for photons described in quantistic sense; I tried to show you that this is the only description for the term "photon" and so we are in a loop...
I sincerely don't know what else I could say.
If that is your response then you weren't paying attention to what I was saying. You consistently keep forgetting the approximation and what it would mean to put the photon's position vector inside the area of uncertainty according to how the wave function would average the position. I gave you an example of a pixel of 0.001 mm in width a length and when it detects the photon then its localized in that area and the location of the photon is the location of the pizel) e.g. geometric center.

You youy insist on ignoring every single thing I've said regarding approximation then there is no use to continue this conversation. Why should I post something I know you're going to igore?

While you're at it it wouln't hut you to finally state what it is you mean by saying something can or can't be localized. E.g. find a QM texbook and quote the definition of "localized" or "localize" so you won't be vauge anymore.
It's you that don't pay attention to what I wrote. I've already written that "saying to have exactly localized a photon is the same mistake of saying to have perfectly localized a particle' path in a cloud chamber".
And, again, you don't need to talk of photons, if you intend to make such coarse approximations, you can talk of pulses of light.
 

The Naked Scientists Forum

Re: How does mass increase at higher speeds?
« Reply #124 on: 25/01/2013 11:39:01 »

 

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