Do photons have finitely many wavelengths?

[yury]

Hi all,

If the energy states of electrons are quantized, are there finitely many wavelengths that a photon can have within, say, the visible spectrum? If so, how many?

Thanks,
Yury

 

[ZapperZ]

Hi all,

If the energy states of electrons are quantized, are there finitely many wavelengths that a photon can have within, say, the visible spectrum? If so, how many?

Thanks,
Yury

Your question has an implicit assumption that all light is created only via atomic transition. This is not correct. Light can be created via molecular vibrations, and by accelerating charges. I could make a stream of electrons going through a section where they jiggle up and down, and viola! I have generated light/em radiation. Or, make a bunch of electrons move in a circular path, and I get more em radiation. The light that you get out of an ordinary light bulb has no "discrete", quantized lines. It means that the spectrum is continuous, because the thermal vibrations that created the light can have a continuous vibration spectrum.

With those in mind, it means that, classically, there's no limit to what frequency/energy that light/em radiation can have.

Zz.

 

[yury]

Thanks for the quick reply!

Is there some way to prove that these spectra are actually continuous? For example, when we jiggle the electrons in your example, can we actually jiggle them by any amount we want, or will it be limited to some combination of the energies of the other particles in the system?

Yury

 

[CarlB]

The universe is only finite in size, and has been in existence only a finite length of time, so because of the Heisenberg uncertainty principle, it is impossible to know whether the energies of photons are continuous or discrete. As far as we can tell, they are continuous.

For example, the energy given off by an atomic transition depends on the velocity of the atoms, by doppler shifting. But we also cannot for certain know whether the set of all possible velocities is finite or continuous. Theory assumes it is continuous and theory works well, but a lot of textbooks will talk about quantum mechanics wave functions "in a box" and then let the dimensions of the box go to infinity. The box is okay, but the infinity is incompatible with our universe.

Carl

 

[ZapperZ]

Thanks for the quick reply!

Is there some way to prove that these spectra are actually continuous? For example, when we jiggle the electrons in your example, can we actually jiggle them by any amount we want, or will it be limited to some combination of the energies of the other particles in the system?

Yury

I can jiggle them by putting a series of alternating magnets. Then the frequency of light that I get depends on (i) the speed/energy of the electrons passing through these magnets and (ii) the spacing of these magnets. In principle, I can set the spacing and speed/energy to anything I want (up to c, of course).

This, by the way, is what is done in every synchrotron light sources throughout the world. Light of various wavelength (you can practically dial in the wavelength/freq that you want) can be generated using insertion devices call wigglers and undulators. Just from the names themselves, you can already guess what they do.

Zz.

 

[yury]

Interesting! So when a particle is accelerating, does it do so discretely? If it didn't then we'd have our proof right?

Yury

Edit: just to clarify, this was a reply to Carl

 

[yury]

In principle, I can set the spacing and speed/energy to anything I want (up to c, of course).

Well, that solves that :). Thanks Zz.

 

[CarlB]

Interesting! So when a particle is accelerating, does it do so discretely? If it didn't then we'd have our proof right?

Unfortunately, the steps in an acceleration would be too small to observe, except possibly for very very small accelerations. But there is some evidence for discreteness in acceleration, and that is the MOND gravity theory. In that theory, very small accelerations are modified in such a way as to increase them as follows:

a_M = \sqrt{a_N a_0}

where a_M is the MOND acceleration, a_N is the Newton (or Einstein in low velocity limit) acceleration, and a_0 is a constant. The value of the constant acceleration is about the acceleration that gives one the speed of light if one accelerates for a length of time equal to the age of the universe. A set of links to the hundreds of MOND papers is organized here:
http://www.astro.umd.edu/~ssm/mond/litsub.html

The square root function in the MOND formulation reminds one of the square root function that enters into quantum mechanics in the "quantum Zeno's effect" or "paradox". Zeno's paradox in quantum mechanics is that the behavior of systems is modified when they are repeatedly measured. That is, it's an effect that comes from repeated collapse of the wave function.

If one believes that measurement is a natural part of the passage of time, then one must suppose that there must be modifications to the usual time dependence of Schroedinger's equation. This is something that the quantum Zeno effect does. There are hundreds of articles written on the quantum Zeno effect, here are 266 hits from arXiv:
http://www.google.com/search?hl=en&q=site:arXiv.org+quantum+zeno

The quantum Zeno effect comes from the fact that in quantum mechanics, things that you calculate tend to be quadratic, that is, they have the form <a|M|a> so the state "a" enters into the calculation twice. But decays are linear (that is, exponential decay) and this requires that "a" enter the calculation only once. (I.e., the probability of decay has to be proportional to the probability that the decaying object has not yet decayed and therefore be linear). This is all very strange, but it has been verified by experiment as the above links will show.

The result of this is that the actual probability of decaying cannot be exponential for all time, but instead must be quadratic at very early times. The above articles will explain this more completely. But basically, the idea is that the probability of decay must be quadratic which is incompatible with a purely exponential decay. Or see this thesis chapter:
http://research.imb.uq.edu.au/~m.gagen//pubs/thesis/03chapter2.pdf

In the above thesis, in equation (2.3), the author obtains the result that the probability of decaying in the infinitesimal time interval \Delta t is proportional not to \Delta t as would be required for exponential decay, but instead is proportional to (\Delta t)^2. Now if one assumes that measurement happens spontaneously at some slow rate, and if one also assumes that velocities are discrete, then one expects to find very low accelerations to be anomalous in that the first principle calculated rate should be quadratic, but the observed rate will be linear. In other words, there should be a square root relationship between very small accelerations and more normal accelerations.

Now given the obvious similarity of the MOND correction to gravity and the quantum Zeno effect, you'd think that the literature would be filled with papers suggesting that the MOND effect is a result of quantizing gravity at very low accelerations and ending up with a quantum Zeno (actually anti-Zeno in this case) effect. But the only paper I've ever seen suggesting this is a short note I wrote 3 years ago and didn't publish (well except on the net):
http://brannenworks.com/PenGrav.html

At this time, I'm working on rewriting the foundations of QM from a density matrix point of view http://www.DensityMatrix.com . I have not got to the point of looking at the quantum Zeno effect from this point of view because I am mostly interested in effects that have to do with the internal degrees of freedom of point particles, and therefore I don't have to deal with the passage of time or measurement issues. But I have a suspicion that the quantum Zeno effect would be a good thing to analyze from the density matrix perspective.

Carl