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The Klein-Gordon equation directly uses a more naive relativistic energy guess of squared.
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But since this is quantum mechanics, we feel like making into the "momentum operator", just like in the Schrodinger equation.
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But we don't really know how to apply the momentum operator twice, because it is a gradient, so the first application goes from a scalar field to the vector field, and the second one...
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So we just cheat and try to use the laplace operator instead because there's some squares on it:
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But then, we have to avoid taking the square root to reach a first derivative in time, because we don't know how to take the square root of that operator expression.
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So the Klein-Gordon equation just takes the approach of using this squared Hamiltonian instead.
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Since it is a Hamiltonian, and comparing it to the Schrodinger equation which looks like:
taking the Hamiltonian twice leads to:
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We can contrast this with the Dirac equation, which instead attempts to explicitly construct an operator which squared coincides with the relativistic formula: derivation of the Dirac equation.
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