Probability density of photons

[Gavroy]

hi...does anybody know here...how you can calculate the probability density of photons?

 

[mathman]

hi...does anybody know here...how you can calculate the probability density of photons?

What is the context?

 

[Dickfore]

Probability density of what physical quantity pertaining to photons?

 

[Gavroy]

of position

 

[Dickfore]

uniform.

 

[Rap]

hi...does anybody know here...how you can calculate the probability density of photons?

What do you have to begin with? If you have the intensity of electromagnetic radiation as a function of frequency, you can do it. You can use the intensity at a given frequency to calculate the energy density per unit frequency and then divide by Planck's constant to get the photon density per unit frequency, and then integrate over frequency. I don't remember the equation relating intensity to energy density, but I could look it up if you need it.

 

[Antiphon]

(1/2 epsilon0 * |E|^2 + 1/2 mu0 * |H|^2)/hf = N (photons/second/cubic meter)

 

[A. Neumaier]

hi...does anybody know here...how you can calculate the probability density of photons?

There is no such thing, since photons do not have a position operator.

See the entries ''Particle positions and the position operator'' and ''Localization and position operators'' in Chapter B1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#B1

 

[Rap]

There is no such thing, since photons do not have a position operator.

See the entries ''Particle positions and the position operator'' and ''Localization and position operators'' in Chapter B1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#B1

Thanks for the link - but what would you say is being calculated by dividing the spectral energy density at some frequency by Planck's constant times that frequency?

 

[A. Neumaier]

Thanks for the link - but what would you say is being calculated by dividing the spectral energy density at some frequency by Planck's constant times that frequency?

It is frequency dependent, hence cannot be ''the'' number density. The only meaningful measure of the ''amount of photon stuff'' in some region S is the integral over S of the energy density.

 

[Demystifier]

hi...does anybody know here...how you can calculate the probability density of photons?

See Secs. 3.4 and 4.1 of
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
This is a very general probabilistic interpretation of relativistic many-particle states, which includes photons.

 

[Rap]

It is frequency dependent, hence cannot be ''the'' number density. The only meaningful measure of the ''amount of photon stuff'' in some region S is the integral over S of the energy density.

Yes, but for a small frequency interval df, would it not be the number density of photons with frequency f to f+df?

 

[A. Neumaier]

Yes, but for a small frequency interval df, would it not be the number density of photons with frequency f to f+df?

For essentially monochromatic light, you can pretend that it is by defining it in this way, but what's the use of it?

The energy density (intensity) is very useful in optics, and sufficient without having an additional dubious concept of number density that is already meaningless for white light.

 

[A. Neumaier]

See Secs. 3.4 and 4.1 of
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
This is a very general probabilistic interpretation of relativistic many-particle states,

One shouldn't do physics like Procrustes http://en.wikipedia.org/wiki/Procrustes to match one's own preferences.

The proposed Bohmian interpretation as a density in space-time is incompatible with the standard Schroedinger view, since the wave function of a Schroedinger particle cannot be normalized such that the space-time integral is 1.

which includes photons.

The paper doesn't address this (and the associated problem of gauge invariance) at all.

 

[Rap]

For essentially monochromatic light, you can pretend that it is by defining it in this way, but what's the use of it?

Well, I wasn't considering "use" but I'm trying to understand the implications of there being no position operator for photons. When you say "pretend that it is", where does the pretense fail? I mean, what logical inconsistency results from assuming it is a photon density?

The energy density (intensity) is very useful in optics, and sufficient without having an additional dubious concept of number density that is already meaningless for white light.

White light has a certain energy density (U) as a function of frequency (f). Integrate U(f)/hf over all frequencies and you have the photon density.

 

[Gavroy]

White light has a certain energy density (U) as a function of frequency (f). Integrate U(f)/hf over all frequencies and you have the photon density.

sorry, but how do you get an energy density function of photons in space? is there no easy formula to calculate the density?

 

[Demystifier]

... interpretation as a density in space-time is incompatible with the standard Schroedinger view, since the wave function of a Schroedinger particle cannot be normalized such that the space-time integral is 1.

This technical problem is easy to overcome, similarly to plane waves (momentum eigenstates) in standard QM. You can normalize it in a large but finite spacetime box, or you can use a more rigorous rigged-Hilbert-space techniques.

The paper doesn't address this (and the associated problem of gauge invariance) at all.

The paper presents a general discussion of arbitrary spin. The gauge invariance is not discussed explicitly, but it is relatively easy to do: Instead of summing over spin indices, you sum over the physical (2 for photons) polarizations.

 

[A. Neumaier]

Well, I wasn't considering "use" but I'm trying to understand the implications of there being no position operator for photons. When you say "pretend that it is", where does the pretense fail? I mean, what logical inconsistency results from assuming it is a photon density?

For a useful probability interpretation one needs a representation of the wave functions as psi(x) where x is space position and momentum and angular momentum are represented in the standard way, so that |psi(x)|^2 can be viewed as a particle density, and the result is covariant under Euclidean motions.

But this is impossible. See the entries ''Particle positions and the position operator'' and
''Localization and position operators'' in Chapter B1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#position

So any construction of a number density (and there are a number of attempts in the literature) either fails to lead to a Schroedinger picture (and hence a proper probabilistic view) or violates basic transformation properties under rotations. Both make it practically useless.