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I'm looking at the Dirac Equation, in the form given on Wikipedia, and (foolishly) trying to understand it.

[itex]\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = i\hbar\frac{\partial \psi}{\partial t}\,\![/itex]

So I picture a wavefunction in an eigenstate of the momentum operator in the [itex]e_1[/itex]-direction with an eigenvalue of p, and simultaneously an eigenstate of the [itex]\alpha_1[/itex] and [itex]\beta[/itex] operators with an eigenvalue of 1 in both cases. Now obviously, for this case:

[itex]\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = \left( p c +mc^2 \right ) \psi\,\![/itex]

But we know that [itex]E = \sqrt{p^2 c^2 + m^2 c^4}[/itex], so this doesn't seem to give the right eigenvalue for the energy operator on the RHS. We want the hypotenuse of a right triangle with [itex]c p[/itex] and [itex]m c^2[/itex] as its legs, not the length of a line with those two quantities as segments of the line! It seems like it might work out right if somehow they were complex and 90 degrees out of phase, but I can't see any way to get that.

What part of my brain is broken?

Thanks.

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# Why doesn't the energy come out right in the Dirac Equation?

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