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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 $0$?

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    $\begingroup$ To get time independent Schrödinger you never use $\imath\psi_{t}=E\psi$ rather you use separation of variables. $\endgroup$ – Alberto Navarro Jan 11 at 22:16
  • $\begingroup$ @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? $\endgroup$ – Jepsilon Jan 12 at 1:17

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