Exploring Heisenberg Uncertainty Principle

[vebrown]

This quote is from a physicist in another forum:

If a single photon is monochromatic then its length is infinite according to the Heisenberg uncertainty principle.

I can't believe that is true. I have seen the Heisenberg principle applied, but not like this. Can somebody shed some light :)

 

[vebrown]

Looks like I'm not going to get a response. The issue is how many wave lengths make up one photon. I always thought it was one. A couple of guys beating me up say that it is an undertermined number and that it can not be a determined number because of the uncertainty principle. I'm having trouble understanding that.

 

[selfAdjoint]

This quote is from a physicist in another forum:



I can't believe that is true. I have seen the Heisenberg principle applied, but not like this. Can somebody shed some light :)

Not that its length is infinite, rather its position is undefined, the uncertainty in its position coordinate is not a finite number.

This is because it is monochromatic, which means precisely that it has one sharp frequency, and therefore by the Einstein relation, "photon momentum is a constant times photon frequency", its momentum is sharp too. Therefore the uncertainty of the photon's position is zero.

And by the Heisenberg uncertainty relation, "uncertainty in position times uncertainty in momentum is greater than or equal to a (positive) constant", we see that we have zero times the position uncertainty not less than a positive number, which is impossible for a finite value.

 

[vebrown]

Thanks for the response; I guess I'm a little dense; I can understand the uncertainty in position, but it still seems that a single photon should have a finite wavelength when its frequency is known.

Have we come to the point where we can't think of individual photons but must only think of a statistical concentration of them ?

 

[gulsen]

It's probaby like this: you can't know anything of an individual photon with a good accuracy. Not energy, too (since $$E=pc$$ for photon).

 

[vebrown]

Ok; thanks; I think I'm getting the picture; I'll dig through the lit some more.

 

[selfAdjoint]

Thanks for the response; I guess I'm a little dense; I can understand the uncertainty in position, but it still seems that a single photon should have a finite wavelength when its frequency is known.

Have we come to the point where we can't think of individual photons but must only think of a statistical concentration of them ?

Oh, it does indeed have a finite wavelength, in fact its wavelength is sharp if it is monochromatic. Wavelength is just c divided by the frequency (c is a speed, meters per second, and frequency is in "per seconds", so if you divide the second into the first you get meters, a length).

But think it over, wavelength is not position, is it? And I think you're looking at wavelength as if it's the "size" of the photon, but this is wrong. When the photon is traveling free in a beam like this it is, quantum mechanics says and relativity agrees, a plane wave, just a classical train of waves in space with peaks spaced oine wavelength apart, and you can't identifty one photon or another in that wave train.

 

[vebrown]

Yes; I understand all of that. Agreed. We were thinking in terms of the single photon that is emitted when an electron in one atom changes energy levels. I guess it is just not useful to think in those terms.

 

[Meir Achuz]

"If a single photon is monochromatic then its length is infinite" is even true classically. You don't need QM for that. The only way to have light of, say, exactly 450 nm is to have an infinitely long beam.