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I am reading an online text on quantum computing and the author expands and re-expresses the Schrodinger equation. I am not really sure as to the intermediate steps he used or what happened to the derivative when going from the first to second equation. Thanks in advance.

$$\frac{d} {dt} |ψ(t)> = −iH(t)|ψ(t)> $$

"re-express this equation, to first order in the infinitesimal quantity dt".

$$|ψ(t + dt)> = (I − iH(t)dt)|ψ(t)>$$

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You could use a forward difference approximation for the derivative:

$$ \frac {d \psi}{dt} = \frac{\psi(t + dt) - \psi (t) }{ dt } $$

Here I am treating dt now as an infinitesimal. Insert this and solve for $\psi (t+dt) $.

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