## Question re: wavelngth

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sgt york
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### Question re: wavelngth

Is there a theoretical limit to how short the wavelength of electromagnetic radiation can be? As we're talking theoretical, assume unlimited power and technology.

If there is a limit, what would it be?

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### Re: Question re: wavelngth

If there wasn't a limit, then the wave would have infinite energy, which as far as I know, isn't theoretically possible.
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### Re: Question re: wavelngth

well, you cant have an infinite energy (0 wavelength) photon, but you can always have a photon with more energy than some other photon. The limit of this would be an infinite energy photon, but there is no finite number at which wavelength must stop.
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### Re: Question re: wavelngth

danpilon54 wrote:well, you cant have an infinite energy (0 wavelength) photon, but you can always have a photon with more energy than some other photon. The limit of this would be an infinite energy photon, but there is no finite number at which wavelength must stop.

...that we know of.
(Assuming QED permits photons of every wavelength, which I don't know. Certainly in classical E&M, there is no limit to the wavelength of a photon.)
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### Re: Question re: wavelngth

I'm neither an E&M student nor someone acquainted with QM, but methinks the Planck Length comes into play at some point, surely?
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### Re: Question re: wavelngth

The Planck length arises from mixing the constants of quantum mechanics and general relativity together. This means that it doesn't appear in any theory that doesn't include some form of quantum gravity. (So, for example, the Planck length doesn't show up in QED.) Currently, no one knows what happens on the Planck scale.

If there wasn't a limit, then the wave would have infinite energy, which as far as I know, isn't theoretically possible.

This is a common error in reasoning: allowing photons to have arbitrarily large finite energy doesn't necessitate allowing photons of infinite energy. However, no one is certain whether there is a limit to the amount of energy a photon can have.
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### Re: Question re: wavelngth

Well.... classical EM certainly has no such limitation, however, for a photon a wavelength has a definite associated energy E ~ 1/l. So a photon of wavelength of the order of magnitude of its Schwarzschild radius should be the limiting case, but I'm to lazy to work out what precisely that is right now.
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### Re: Question re: wavelngth

No, there is no limit on the wavelength of a photon. Consider the following situation:

Assume there is a minimum limit to the wavelength of a photon. Observe a photon with this wavelength moving right in some reference frame S. Then observe it from a reference frame that is moving with some speed to the left S'. In S', the photon will be observed with some shorter wavelength than the minimum. Hence the minimum wavelength cannot exist.

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### Re: Question re: wavelngth

thats a good point. As you approach the speed of light, the doppler effect can cause photons to approach infinite frequency. Since no frame is preferred to another, and you can get arbitrarily close to the speed of light, a photon can have arbitrarily high energy.
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### Re: Question re: wavelngth

Owehn wrote:
If there wasn't a limit, then the wave would have infinite energy, which as far as I know, isn't theoretically possible.

This is a common error in reasoning: allowing photons to have arbitrarily large finite energy doesn't necessitate allowing photons of infinite energy. However, no one is certain whether there is a limit to the amount of energy a photon can have.

Saying that "well, there's always a frequency that's greater than any frequency you can think of" sounds a lot like infinitely large frequency to me.
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### Re: Question re: wavelngth

Someone whose name is mathmagic should know better than that. There's a natural number bigger than any number you can think of, right? Is there an infinitely large one?
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### Re: Question re: wavelngth

skeptical scientist wrote:Someone whose name is mathmagic should know better than that. There's a natural number bigger than any number you can think of, right? Is there an infinitely large one?

Maybe it's just the realist in me, but for some reason I have trouble applying and analogizing ZFC to physical phenomena.

Intuitively, I would analyze the situation this way:

E = hc/l

Where E is energy, h is Planck's constant, and l is wavelength.

The limit of E as l goes to zero is infinity. It is absurd to think that anything in the known universe can possess infinite energy. Therefore, there must be a limit on the size of the wavelength.

Apparently, this is wrong...
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### Re: Question re: wavelngth

There's no limit to the energy that a photon can possess, but there is a limit to how far high energy photons can travel before scattering off something else and creating particles.

The GZK limit is a theoretical prediction that photons with energies greater than 6x1019 eV would interact with the cosmic microwave background radiation to form pions before traveling 50 million parsecs (~163 million light-years).

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### Re: Question re: wavelngth

mathmagic wrote:
skeptical scientist wrote:Someone whose name is mathmagic should know better than that. There's a natural number bigger than any number you can think of, right? Is there an infinitely large one?

Maybe it's just the realist in me, but for some reason I have trouble applying and analogizing ZFC to physical phenomena.

What Skeptical Scientist referred to was the fact that you can make a photon of any given, finite wavelength. It's true they can't have infinite wavelength (or zero wavelength), but that doesn't mean that for any value greater than zero, I can find a photon with that frequency...
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### Re: Question re: wavelngth

just to throw the LHC amongst the pigeons, a DC signal can be modelled as an AC signal with infinite wavelength
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### Re: Question re: wavelngth

sgt york wrote:Is there a theoretical limit to how short the wavelength of electromagnetic radiation can be? As we're talking theoretical, assume unlimited power and technology.

If there is a limit, what would it be?

The emission of photons of various wavelengths by a blackbody is an inherently probabilistic process. (Planck's law)

Technically, a blackbody has a chance to emit every possible wavelength. It's just that the probability of a photon of (nearly) infinitely short wavelength is (nearly) infinitely 0... so it (nearly) never happens.

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### Re: Question re: wavelngth

mathmagic wrote:The limit of E as l goes to zero is infinity. It is absurd to think that anything in the known universe can possess infinite energy. Therefore, there must be a limit on the size of the wavelength.

Apparently, this is wrong...

Yes, this is wrong. Whether or not such a limit exists, your reasoning is faulty. Were your reasoning true, it would also mean there is some cosmic speed limit strictly less than c, because any matter moving at c would have infinite energy, which is clearly impossible. But that doesn't mean there is any speed limit less than c. It just means that c itself is not achievable.

In the same way, the fact that a 0-wavelength photon would have infinite energy simply means that a wavelength of 0 is impossible. *Not* that there is some other magical limit strictly greater than zero.
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### Re: Question re: wavelngth

gmalivuk wrote:
mathmagic wrote:The limit of E as l goes to zero is infinity. It is absurd to think that anything in the known universe can possess infinite energy. Therefore, there must be a limit on the size of the wavelength.

Apparently, this is wrong...

Yes, this is wrong. Whether or not such a limit exists, your reasoning is faulty. Were your reasoning true, it would also mean there is some cosmic speed limit strictly less than c, because any matter moving at c would have infinite energy, which is clearly impossible. But that doesn't mean there is any speed limit less than c. It just means that c itself is not achievable by something with mass.

Fix'd your post, because that's what I'm getting at with what I was thinking with the wavelength limit.

gmalivuk wrote:In the same way, the fact that a 0-wavelength photon would have infinite energy simply means that a wavelength of 0 is impossible. *Not* that there is some other magical limit strictly greater than zero.

Before it was discovered that the speed of light was the universal speed limit, it was thought that there was no upper bound on speed. How is this case any different?
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### Re: Question re: wavelngth

Mathmagic, your whole argument rests on the premise that if photons can have arbitrarily small nonzero wavelength, they can also have zero wavelength. If you deny this premise, it's perfectly possible to have photons of arbitrarily small nonzero wavelength without allowing photons of zero wavelength (and in fact, perfectly reasonable, since it's easy to imagine a wave which has any arbitrarily short wavelength, but it's difficult to imagine what could even be meant by a "wave" with a "wavelength" of zero).

What makes you think that this premise is true? We've given you other examples (which you accept) of things which contain arbitrarily large/small elements - e.g. the natural numbers, or the positive real numbers - but don't contain any infinite/zero elements. It's true that those examples are mathematical rather than physical, but I don't see why your premise should be true for physical objects when it is obviously false for mathematical ones, and I certainly don't think it should be accepted as a given.
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### Re: Question re: wavelngth

mathmagic, you might be able to use that argument to conclude that there is an actual limit on how energetic photons are (in our frame, assuming a finite net energy in the known universe), but it doesn't give you a theoretical limit on how energetic a photon could be.

It is absurd to think that anything in the known universe can be infinitely far away from us. Therefore, there must be a limit on the how far away from us things are. Sure, there is, we can't see anything that's ridiculously far away from us. That doesn't mean that the laws of physics provide some bound on how far away from us things could be.

Note the OP's phrasing:
sgt york wrote:Is there a theoretical limit to how short the wavelength of electromagnetic radiation can be? As we're talking theoretical, assume unlimited power and technology.
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### Re: Question re: wavelngth

skeptical scientist wrote:Mathmagic, your whole argument rests on the premise that if photons can have arbitrarily small nonzero wavelength, they can also have zero wavelength. If you deny this premise, it's perfectly possible to have photons of arbitrarily small nonzero wavelength without allowing photons of zero wavelength (and in fact, perfectly reasonable, since it's easy to imagine a wave which has any arbitrarily short wavelength, but it's difficult to imagine what could even be meant by a "wave" with a "wavelength" of zero).

I don't see how my argument hinges on a zero-wavelength wave. All I'm saying is that as the wavelength approaches zero (i.e. gets VERY small), the energy gets very large. There is a finite amount of energy in the known universe. I would imagine that this energy would dictate the limit of the wavelength of a photon. Sure, it would take a very small wavelength to even get CLOSE to this energy, but according to you, you can get arbitrarily small without getting a zero-wavelength without any consequences.

The debate whether the amount of energy in the universe is fixed or is changing is another question in itself.

PRE-EDIT:

antonfire - Well, sure. If you're talking purely theoretical with unlimited energy, then obviously there would be no limit.
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### Re: Question re: wavelngth

What about the argument that a given photon appears to have arbitrarily large or small (but finite) energy as observed from different reference frames? This is directly related to the fact that "the total amount of energy in the universe" isn't a well-defined quantity.
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### Re: Question re: wavelngth

mathmagic wrote:All I'm saying is that as the wavelength approaches zero (i.e. gets VERY small), the energy gets very large. There is a finite amount of energy in the known universe. I would imagine that this energy would dictate the limit of the wavelength of a photon.

Okay, but that's not what you said before:
The limit of E as l goes to zero is infinity. It is absurd to think that anything in the known universe can possess infinite energy. Therefore, there must be a limit on the size of the wavelength.
Saying that "well, there's always a frequency that's greater than any frequency you can think of" sounds a lot like infinitely large frequency to me.
That would require the premise "if photons can have arbitrarily small nonzero wavelength, they can also have zero wavelength," which I don't believe is true.

Additionally, there may be a finite amount of energy in the known universe, but there may be higher energy densities outside the known universe. Finally, if we are allowed to consider arbitrary inertial reference frames, and special relativity correctly describes what goes on in all of them, then there's some reference frame in which the wavelength of the photon is blueshifted to something as small as you like (but still none in which the wavelength is zero).
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### Re: Question re: wavelngth

mathmagic wrote:I don't see how my argument hinges on a zero-wavelength wave. All I'm saying is that as the wavelength approaches zero (i.e. gets VERY small), the energy gets very large. There is a finite amount of energy in the known universe. I would imagine that this energy would dictate the limit of the wavelength of a photon. Sure, it would take a very small wavelength to even get CLOSE to this energy, but according to you, you can get arbitrarily small without getting a zero-wavelength without any consequences.

Okay, then if you'd said, like, *anywhere* before this, that you were hinging your argument on the fact that we can only observe a finite amount of energy, perhaps people would agree with you more. But as it is, most of your previous arguing along this line seemed to be something like .999... != 1.

(It's worth noting that by your same argument, there *is* some limit to how fast something can go, which is strictly less than c, because to go any faster would require more energy than is available in the entire visible universe. I'm going to go ahead and say that in both of these cases it is not a *theoretical* limit that you're talking about.)
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### Re: Question re: wavelngth

Since light, as far as I know, results from moving charges, shouldn't there be a limit on frequency related to our not being able to move charges faster than c?

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### Re: Question re: wavelngth

alterant wrote:Since light, as far as I know, results from moving charges, shouldn't there be a limit on frequency related to our not being able to move charges faster than c?

Even were light purely a wave, no. Mathematically, as long as amplitude decreases along with wavelength, you can have an arbitrarily short wave without requiring anything to ever move above a set maximum speed. (Consider the function x2 sin(1/x), which crosses the x-axis infinitely many times in any interval containing zero, but whose derivative never exceeds 1.)
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### Re: Question re: wavelngth

Oh yes, I see what you mean about there being no limit on frequency related to the speed of the charge.

In that case it boils down to what others have said - that the limit of wavelength is zero (but we can never get to zero itself). Whether there is a more practical limit set by the amount of energy in the universe is another, less interesting issue, unless we plan on putting all the energy of the universe into the questionable project of making one really, really energetic photon. And even then the wavelength wouldn't be quite zero.

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### Re: Question re: wavelngth

gmalivuk wrote:
alterant wrote:Since light, as far as I know, results from moving charges, shouldn't there be a limit on frequency related to our not being able to move charges faster than c?

Even were light purely a wave, no. Mathematically, as long as amplitude decreases along with wavelength, you can have an arbitrarily short wave without requiring anything to ever move above a set maximum speed. (Consider the function x2 sin(1/x), which crosses the x-axis infinitely many times in any interval containing zero, but whose derivative never exceeds 1.)

Of course, light isn't purely a wave, and QM imposes a lower limit on amplitude, since you can't drop below a single photon. And now we're at the point where all of our intuitions go to hell.
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### Re: Question re: wavelngth

you cant drop below a single photon, but that doesnt mean a single photon cant have arbitrarily high or low energy. Yes it is true that light of a certain frequency must contain a minimum amount of energy, but it is not true that there is a minimum or maximum amount of energy contained in 1 photon, unless of course you consider the amount of energy in the universe.
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### Re: Question re: wavelngth

danpilon54 wrote:you cant drop below a single photon, but that doesnt mean a single photon cant have arbitrarily high or low energy. Yes it is true that light of a certain frequency must contain a minimum amount of energy, but it is not true that there is a minimum or maximum amount of energy contained in 1 photon, unless of course you consider the amount of energy in the universe.

Right. SS's response was to my assertion that, were light purely a wave formed by moving charges, the speed limit wouldn't be violated by any wavelength, because amplitude could shrink accordingly. Since light is quantized, though, the amplitude cannot get arbitrarily small.
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### Re: Question re: wavelngth

Your argument is based on the assumption that the length observable is a classical special relativistic quantity, while my argument was quantum gravitational (it therefore fails on two accounts). The length should be an observable and there might well be a smallest non zero eigenvalue on the appropriate operator. A boost symmetry would deform the eigenvalue though. Even in classical GR your argument would seem to imply that arbitrarily high matter density could be achieved which we know not to be the case (though I fail to see precisely how your argument fails on this point right now).
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### Re: Question re: wavelngth

In classical mechanics, when kinetic energy tends toward infinite, so does speed. Relativity says that in fact, it tends toward c.
I am not an expert, but I think length below Planck wall is supposed to have no meaning, and maybe it is a limit the same way c is. If this the case, a photon with infinite energy would have a wavelength of Planck length or something like that.
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### Re: Question re: wavelngth

idobox wrote:In classical mechanics, when kinetic energy tends toward infinite, so does speed. Relativity says that in fact, it tends toward c.
I am not an expert, but I think length below Planck wall is supposed to have no meaning, and maybe it is a limit the same way c is. If this the case, a photon with infinite energy would have a wavelength of Planck length or something like that.

Two things can be separated by less than a planck length, it's simply that for the universe's purposes they occupy the same space. Same for planck time - if two events are separated by less than a planck time, the universe treats them as simultaneous.

As far as I know, this is purely due to Uncertainty constraints. If you were able to differentiate them any more finely, you'd violate uncertainty.
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### Re: Question re: wavelngth

Xanthir wrote:
idobox wrote:In classical mechanics, when kinetic energy tends toward infinite, so does speed. Relativity says that in fact, it tends toward c.
I am not an expert, but I think length below Planck wall is supposed to have no meaning, and maybe it is a limit the same way c is. If this the case, a photon with infinite energy would have a wavelength of Planck length or something like that.

Two things can be separated by less than a planck length, it's simply that for the universe's purposes they occupy the same space. Same for planck time - if two events are separated by less than a planck time, the universe treats them as simultaneous.

As far as I know, this is purely due to Uncertainty constraints. If you were able to differentiate them any more finely, you'd violate uncertainty.

Sorry, but I'm going to have to say [citation needed]. If something possesses more than the Planck energy, would the universe treat it as if it had infinite energy?
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### Re: Question re: wavelngth

Owehn wrote:
Xanthir wrote:
idobox wrote:In classical mechanics, when kinetic energy tends toward infinite, so does speed. Relativity says that in fact, it tends toward c.
I am not an expert, but I think length below Planck wall is supposed to have no meaning, and maybe it is a limit the same way c is. If this the case, a photon with infinite energy would have a wavelength of Planck length or something like that.

Two things can be separated by less than a planck length, it's simply that for the universe's purposes they occupy the same space. Same for planck time - if two events are separated by less than a planck time, the universe treats them as simultaneous.

As far as I know, this is purely due to Uncertainty constraints. If you were able to differentiate them any more finely, you'd violate uncertainty.

Sorry, but I'm going to have to say [citation needed]. If something possesses more than the Planck energy, would the universe treat it as if it had infinite energy?

[citation provided]
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### Re: Question re: wavelngth

There are a lot of ifs in that paragraph. I'd say instead that we don't know what happens at the planck scale, not that the universe does or doesn't meaningfully have intervals of that size. In fact, if you follow up the citations from that section, you'll find that John Baez agrees with me that you should take thought experiments like the one Wikipedia describes with a grain of salt.
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### Re: Question re: wavelngth

Owehn wrote:
Xanthir wrote:
idobox wrote:In classical mechanics, when kinetic energy tends toward infinite, so does speed. Relativity says that in fact, it tends toward c.
I am not an expert, but I think length below Planck wall is supposed to have no meaning, and maybe it is a limit the same way c is. If this the case, a photon with infinite energy would have a wavelength of Planck length or something like that.

Two things can be separated by less than a planck length, it's simply that for the universe's purposes they occupy the same space. Same for planck time - if two events are separated by less than a planck time, the universe treats them as simultaneous.

As far as I know, this is purely due to Uncertainty constraints. If you were able to differentiate them any more finely, you'd violate uncertainty.

Sorry, but I'm going to have to say [citation needed]. If something possesses more than the Planck energy, would the universe treat it as if it had infinite energy?

More like it would just treat it as if it had more than Planck energy, without actually knowing what that energy is in a definite, discreet way.

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### Re: Question re: wavelngth

Owehn wrote:There are a lot of ifs in that paragraph. I'd say instead that we don't know what happens at the planck scale, not that the universe does or doesn't meaningfully have intervals of that size. In fact, if you follow up the citations from that section, you'll find that John Baez agrees with me that you should take thought experiments like the one Wikipedia describes with a grain of salt.

I will accept these qualifications.
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