# Photon wavefunction and localizability

## Main Question or Discussion Point

Hi!
I've read a lot of posts about the photon wavefunction. I'd like to ask you if this statement is correct:
"It is possible to write down a wavefunction for the photon which correctly describes its dynamics. But this wavefunction cannot be interpreted as an object whose squared modulus
gives the probability to find the photon in a point of space."

## Answers and Replies

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Jano L.
Gold Member
Hello eoghan,
could you write down the wave function for the photon which you have on mind? I think there is not such thing in common theory, but I do not know for sure.
J.

I´m thinking of Landau-Peierls or Cook wavefunction

Jano L.
Gold Member
eoghan, it is hard for me to understand the meaning of the expression
"a wavefunction for the photon which correctly describes its dynamics. "

What do you mean by the dynamics of the photon? Maxwell's equations? But these say what happens to the fields, not to particles. These are too different things.

By using the term wave function, one also anticipates that some kind of Schroedinger's equation will be governing it.

However, in Cook's paper Photon Dynamics in PR, there is no wave function. He merely defines some kind of generalized fields - _operators_ psi(x), phi(x) right from the creation/annihilation operators a+,a and from plane waves in 3D space. Then he introduces some quadratic function D of them and gives it probabilistic interpretation.

Even if it turned out that his different prediction can somehow be confirmed by experiment (I do not know anything about it), those fields psi, phi are not wave functions in the standard sense, and the equation they obey is much closer to Maxwell's equations that to Schroedinger's equation.

Returning to your original question, I would say that it is possible to define many mathematical objects whose evolution is formally given by Maxwell's equations. But I would be very suspicious about interpreting them as densities of particles. The problems introducing a photon wave function maybe just mean that it is a wrong idea. Field, classical or quantum, is not a particle nor a wave function.

J.

Even if it turned out that his different prediction can somehow be confirmed by experiment (I do not know anything about it), those fields psi, phi are not wave functions in the standard sense, and the equation they obey is much closer to Maxwell's equations that to Schroedinger's equation.

Returning to your original question, I would say that it is possible to define many mathematical objects whose evolution is formally given by Maxwell's equations. But I would be very suspicious about interpreting them as densities of particles. The problems introducing a photon wave function maybe just mean that it is a wrong idea. Field, classical or quantum, is not a particle nor a wave function.
J.
That's the answer I was looking for. Thank you

kith
Photon wavefunctions are a controversial issue. There is no position operator for photons, so a wavefunction as an expansion coefficient in the position basis can not be defined. But there are various other things you can do.

First of all, you can simply find a wavefunction in the momentum representation. Also, you can rewrite Maxwell's equations in Schrödinger form, i.e. i∂tψ(r,t)=Hψ(r,t).

You can find a couple of review articles on this on google scholar.

Jano L.
Gold Member
Kith,
out of curiosity, if we have wave function in particle momenta, what prevents its Fourier transformation into function of particle coordinates?

Demystifier
Kith,
out of curiosity, if we have wave function in particle momenta, what prevents its Fourier transformation into function of particle coordinates?
Nothing, of course. But the problem is to interpret this wave function probabilistically, in a relativistic-invariant way.

There are at least two ways how it can be done.

One is to generalize 3-dimensional space probability into 4-dimensional spacetime probability, as e.g. in
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]

The other is to retain a 3-dimensional "space" probability, but on a hypersurface which at some parts may not be spacelike:
http://xxx.lanl.gov/abs/quant-ph/0602024 [Int.J.Mod.Phys.A22:6243-6251,2007]