What is the PDE for photons proposed by Bialynicki-Birula?

[birulami]

Neither in Sam Treiman's http://books.google.de/books?id=e7fmufgvE-kC" I was able to find the Schrödinger equation for a photon, i.e. a particle without rest mass. The Schrödinger equation straight from Treiman's book (typos are mine, if any)

$$-\frac{\hbar^2}{2m}\Delta\Psi + V\Psi = i\hbar\frac{\partial\Psi}{\partial t}$$​

with potential $V$ and the Laplace-Operator $\Delta$ applied for all coordinates except $t$, does contain the (rest)mass $m$ in the denominator, so I guess this won't work for the photon.

How then does the equation look like to cover massless particles? Or does it not apply?

Thanks,
Harald.

 

[jostpuur]

You are not going to find the Shrodinger equation for photon anywhere in literature. Check my responses #95 and #96 in the thread What really is a photon?

 

[pam]

The wave equation for A^\mu is the equivalent SE for a photon, with A^\mu being considered the wave function of the photon. It is the zero mass limit of the Klein Gordon equation. This interpretation is in the literature.

 

[jtbell]

The Schrödinger equation [...] does contain the (rest)mass $m$ in the denominator, so I guess this won't work for the photon.

Note that the photon is highly relativistic, and the Schrödinger equation is non-relativistic. The SE contains the classical definition of kinetic energy, translated into operator form:

$$K = \frac{1}{2}mv^2 = \frac{p^2}{2m} \rightarrow \frac {1}{2m} \left( -i \hbar \frac{\partial}{\partial x} \right)^2 = - \frac{\hbar^2}{2m} \frac {\partial^2}{\partial x^2}$$

(in one dimension)

 

[birulami]

You are not going to find the Shrodinger equation for photon anywhere in literature.

Autsch. I wondered already why in one part of a book I find Maxwell's equations and a description of the photon, and in other parts of a books I find the Schrödinger equation. But a connection between the two is suspiciously absent.

Too bad.
Harald.

 

[olgranpappy]

Autsch. I wondered already why in one part of a book I find Maxwell's equations and a description of the photon, and in other parts of a books I find the Schrödinger equation. But a connection between the two is suspiciously absent.

Come on now. We can certainly write down a Schrödinger equation for a system composed of matter and photons (in Columb gauge to be specific).


\frac{i\partial}{\partial t}|\Psi> = H |\Psi>

H = H_{\rm matter} + H_{\rm free photon} + H_{\rm interaction}

H_{\rm free photon} = \sum_{k,j}a^{\dagger}_{k,j}a_{k,j} c \hbar k

where c is the speed of light and k is the wave-vector and j is one of two polarizations. And \Psi is some god-awful wavefunction.

(sorry for not putting TeX wrappers around the TeX... my TeX comes out completely different from what I wrote when I do that. Anyone else have this problem?)

 

[jostpuur]

The wave equation for A^\mu is the equivalent SE for a photon, with A^\mu being considered the wave function of the photon. It is the zero mass limit of the Klein Gordon equation. This interpretation is in the literature.

Klein-Gordon equation is not a consistent generalization of the non-relativistic Shrodinger equation. The second order (in respect to time) PDE does not become first order PDE in the low energy limit. Classical fields do not become quantum mechanical wave functions either, since they usually have different dimensions.

With the photon these problems are not so obvious because there is no non-relativistic limit anyway, but the fundamental problem with relativistic quantum theory is still there.

We can write down a relativistic SE easily as

$$i\partial_t \Psi = \sqrt{-\nabla^2 + m^2}\Psi$$

and it is easy to substitute m=0 also and get

$$i\partial_t \Psi = \sqrt{-\nabla^2}\Psi$$

but the mainstream story goes on so, that having found the relativistic SE, we next notice that it cannot be used for some locality and causality related reasons.

Come on now. We can certainly write down a Schrödinger equation for a system composed of matter and photons (in Columb gauge to be specific).


\frac{i\partial}{\partial t}|\Psi> = H |\Psi>

H = H_{\rm matter} + H_{\rm free photon} + H_{\rm interaction}

H_{\rm free photon} = \sum_{k,j}a^{\dagger}_{k,j}a_{k,j} c \hbar k

where c is the speed of light and k is the wave-vector and j is one of two polarizations. And \Psi is some god-awful wavefunction.

(sorry for not putting TeX wrappers around the TeX... my TeX comes out completely different from what I wrote when I do that. Anyone else have this problem?)

Working in momentum space is closer to hiding the problem than solving it. You wouldn't bother transforming this into the position representation?

 

[jostpuur]

(sorry for not putting TeX wrappers around the TeX... my TeX comes out completely different from what I wrote when I do that. Anyone else have this problem?)

Not now, but I remember having some problem like this once. I think I just edited it until it started working. Like through deleting and rewriting.

hmhmh... or did this happen before I had understood that one has to reload the page once after editing the equations in order to see the correctly? I'm not sure...

 

[olgranpappy]

Working in momentum space is closer to hiding the problem than solving it. You wouldn't bother transforming this into the position representation?

Nah, I'm happy with that Hamiltonian the way it is.

I'm interested in practical things such as calculating scattering amplitudes for x-ray experiments, so I do just fine in momentum space, thank you very much.

 

[QuantumDevil]

I saw an interesting attempt to find PDE for photons made by Polish physicist Bialynicki-Birula:

http://www.cft.edu.pl/~birula/publ/APPPwf.pdf
http://www.cft.edu.pl/~birula/publ/reconstr.tex
http://www.cft.edu.pl/~birula/publ/CQO7.pdf