# What is the frequency spectrum for a single photon?

1. Jul 4, 2013

### phys12345

According to the Einstein light-quantum hypothsis, the photon energy is given by E = hbar*ω. If taking the photon as a very short light pulse, then the frequency spectrum is extremely wide in terms of Fourier-transform theory. How to understand the frequency ω in E = hbar*ω if taking the photon as a very short light pulse?

2. Jul 4, 2013

### chill_factor

A very short light pulse is a bunch of photons emitted at the same time. It is not a single photon. Photon number is related to intensity. Also, photon number is not conserved for most sources. You need a Fock state source to think about photon number, but then you lose phase information.

However, yes, there is a frequency spread in single photon sources. The single photons would then behave statistically.

3. Jul 4, 2013

### phys12345

Thanks. But they say nowadays a single photon can be produced and detected.
"there is a frequency spread" for a single-photon packet ---- Is that what you mean?

"The single photons would then behave statistically." ---- Is Fourier-analysis not applicable or what?

A single photon in free space has a space length?

Last edited: Jul 4, 2013
4. Jul 4, 2013

### DiracPool

A single photon is associated with a single frequency, which is its energy. What is a Fourier analysis gonna do for you?

5. Jul 4, 2013

### phys12345

What I mean is that if the single photon can be taken a finite length pulse, its frequency spectrum is extremely wide according to the Fourier transform. Then the photon pulse is made up of many monochromatic plane waves --- of course, this is a classical understanding.

6. Jul 4, 2013

### Staff: Mentor

You are thinking classically. A photon is not a finite length pulse of classical EM. A photon is a Fock state, and there isn't a classical analoge to a Fock state like there is for a coherent state.

7. Jul 4, 2013

### phys12345

They say a single photon can be produced and detected nowadays. If just for a single photon, "coherence" is not applicable, I guess. Probably there is no concept about the size of photon and it cannot be taken as a short light pulse or wave packet. But I see some literature where the photon is taken as a short light pulse, which really confused me.

8. Jul 4, 2013

### vanhees71

A single photon is represented by any one-particle Fock state with definite norm. Thus it can have any frequency spectrum you like. Of course you cannot in any way think about it in classical terms as has been stressed before in this thread. Photons are pretty complicated beasts, not only mathematically (because of gauge invariance you have a quite complicated construction of the positive-norm Hilbert space, e.g., in terms of the Gupta-Bleuler formalism or you write it in a completely fixed gauge known as the radiation gauge, but then the representation is not manifestly Poincare covariant anymore) but also concerning the physical intuition. Contrary to the case of massive particles, e.g., there is no proper operator for the "position of a photon".

9. Jul 4, 2013

### Avodyne

As vanhees71 points out, a single photon can be in a superposition of energy eigenstates, just like an electron. Thus it is possible to create a single-photon wave packet with finite extent in time (and space).

I don't think it's all that complicated. In radiation gauge (AKA Couloumb gauge or transverse gauge), everything is pretty straightforward. Spacetime translation invariance is manifest. Only Lorentz invariance is hidden.

See Mandel & Wolf's text on quantum optics for details.

Last edited: Jul 4, 2013
10. Jul 4, 2013

### Cthugha

This is very old terminology where the modes of the spectral decomposition were termed photons.

As has been pointed out, this has absolutely nothing to do with the more "modern" concept of single photons which are non-classical photon number states and can have almost any spectral, temporal or spatial shape.

11. Jul 5, 2013

### vanhees71

There is no energy eigenstate of (free) photons. There is a generalized eigenstate, which is the plane wave, but that's not square integrable. You have to smear over a finite energy range to get a true single-photon state.

12. Jul 5, 2013

### DiracPool

Holy Fock! I guess so, I need to get with the times. I know what I'm doing this weekend :tongue:

13. Jul 5, 2013

### Staff: Mentor

Yes, unfortunately there are a lot of examples of people saying the quantum word "photon" when they actually mean the classical idea of a "light pulse". It is sloppy, incorrect, and misleading.

14. Jul 5, 2013

### Naty1

You can tell from the above there is no simple answer...more discussion here:

How Big is a Photon??
https://www.physicsforums.com/showthread.php?t=657264

but no distinct precise answer.

One thing we can say is that in the Standard Model of particle physics, photons are point like particles. In free space they exist as infinite length waves, but there is no true 'free space'...there is always something to interact with....

We don't even have consensus on whether the universe consists of waves or their quanta, particles....I posted something about that in the linked thread, from Carlo Rovelli....

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