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[itex] \frac {d \left< E_{k} | \Psi \right>}{dt} = \frac{\partial \left< E_{k} \right|}{\partial t} \left| \Psi \right> + \left< E_{k} \right| \frac{\partial \left| \Psi \right>}{\partial t} = \frac{i}{\hbar} \left< E_{k} | H | \Psi \right> - \frac{i}{\hbar} \left< E_{k} | H | \Psi \right> = 0[/itex]

and thus probability amplitude for energy [itex]E_{k}[/itex] should be time independent in any state [itex]\left| \Psi \right>[/itex].

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# Time independence of probability amplitudes

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