# Time-independent Klein-Gordon PDE

Given the KG PDE:

$$\psi_{tt} - \psi_{xx} + m^2 \psi = 0.$$

Wikipedia describes the time-independent variant of this as just setting $$\psi_{tt}=0$$.

My question is this:

For the Schrödinger equation, the time independence is achieved by setting $$i\psi_t = E\psi$$, is it legittimate to consider setting $$\psi_{tt}=E^2 \psi$$ in the KG equation rather than $$0$$? Why is it preferencial to set it to $$0$$?

• To get time independent Schrödinger you never use $\imath\psi_{t}=E\psi$ rather you use separation of variables. – Alberto Navarro Jan 11 at 22:16
• @Alberto Navarro I see. Would it still be possible to extract an energy value from the time independent equation? Or would it strictly require time due to it being relativistic? – Jepsilon Jan 12 at 1:17