Klein-Gordon for a massless particle

1. May 20, 2012

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?

Last edited by a moderator: May 20, 2012
2. May 20, 2012

vela

Staff Emeritus
3. May 20, 2012

jabers

so for a one dimensional wave equation the equation is:

$${ \partial^2 \psi \over \partial t^2 } = { c^2 \nabla^2 \psi} = c^2 {\partial^2 \psi \over \partial x^2}$$

Right?

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

$${ \partial x \over \partial t } = { x }$$

Then I could say:

$${\frac{1}{x} \partial x } = {\partial t}$$

And then integrate but I dont see what you mean. What do you mean?

The only thing i could see doing is this:

$${\partial^2 \psi \partial^2 x} = c^2 {\partial^2 \psi \partial^2 t}$$

And integrating twice? But that doesnt 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.

4. May 20, 2012

vela

Staff Emeritus
5. May 20, 2012

jabers

Cool, thank you.

6. May 22, 2012

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)= ei(-kr+ωt) which is a spherical wave, you can see how this thing behaves in your equation.