I have to use separation of variables on the 3-D Klein-Gordon equation:
$ c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 $ where $ \Psi (r,t) = \psi (r) T(t) $
The problems asks:
A) Use the separation of variables technique to separate th time dependence first, i.e., let $ \Psi (r,t) = \psi (r) T(t) $ and call $-\omega^2 $ the separation constant for the time equation. Write the solution.
B) In Cartesian coordinates use the separation of variables technique on the resulting $ \psi (r) $ equation, defining the separation constants as $ -k_{x}^2, -k_{y}^2, -k_{z}^2 $ in the usual way so that each k has units of length. Obtain the PDE in Catesian Coordinates
C) Apply the following boundary conditions to the solutions: the solution must be zero at $ x=0 $ and $x=a$, $y=0, y=b, z=0, z=c$. Show the new restricted from of the solutions to the PDE and the relationship between $\omega^2 $ and $ k^2=k_{x}^2+k_{y}^2 + k_{z}^2 $