Klein-Gordon for a massless particle

[jabers]

So I'm trying to find a solution of the Klein-Gordon equation for a massless particle. I reached the Klein-Gordon from the total energy-momentum equation. Then for a massless particle i get to this equation:
$${ \partial^2 \psi \over \partial t^2 } = c^2 \nabla^2 \psi$$How do I solve for psi? I was thinking about trying the Frobenius method, but I'm not sure how to do that. Any help would be appreciated.

Also how do I make my typed latex display in the latex format on this forum?

 

[vela]

Try separation of variables.

LaTeX tutorial: https://www.physicsforums.com/showthread.php?t=546968

 

[jabers]

so for a one dimensional wave equation the equation is:

[tex] { \partial^2 \psi \over \partial t^2 } = { c^2 \nabla^2 \psi} = c^2 {\partial^2 \psi \over \partial x^2} [/tex]

Right?

I don't see how I could use separation of variables. If it were something like this:

[tex] { \partial x \over \partial t } = { x } [/tex]

Then I could say:

[tex] {\frac{1}{x} \partial x } = {\partial t} [/tex]

And then integrate but I don't see what you mean. What do you mean?

The only thing i could see doing is this:

[tex] {\partial^2 \psi \partial^2 x} = c^2 {\partial^2 \psi \partial^2 t} [/tex]

And integrating twice? But that doesn't really make sense to me.
I didnt think you could separate variables to solve a differential equation when there were double derivatives in the equation.

 

[vela]

Different type of separation of variables. See, for instance, http://en.wikipedia.org/wiki/Separation_of_variables#Partial_differential_equations.

You'll probably find a more illuminating discussion of the method in a math methods textbook.

 

[jabers]

Cool, thank you.

 

[Morgoth]

this equation looks pretty much like the wave-equation, for a wave moving with a speed v=c.
So I guess you can think for solutions like cos or sin, or better exponential.

If for example you say:

Ψ(r,t)= e^{i(-k•r+ωt)} which is a spherical wave, you can see how this thing behaves in your equation.