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I'm looking at the units of the Klein-Gordon equation $$u_{tt} - c^2\Delta u = -\frac{m^2c^2}{\hbar^2}u. $$ Disregarding the units of $u$, which are the same everywhere and so cancel, I get $seconds^{-2}$ for both terms on the LHS, but $meters^{-2}$ on the RHS. What am I doing wrong?

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You are missing a $c^2$ on the RHS, equation should be

$$ \color{red}{\frac{1}{c^2}}u_{tt} - \Delta u = -\frac{m^2 c^2}{\hbar^2}u $$

Or equivalently

$$ u_{tt} - c^2\Delta u = -\frac{m^2 \color{red}{c^4}}{\hbar^2}u $$

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