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I have seen two different forms of Schrödinger equation: $$i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}=\hat{H}|\psi(t)\rangle$$ and $$i\hbar\frac{d|\psi(t)\rangle}{d t}=\hat{H}|\psi(t)\rangle.$$

Are these two equations equivalent? If not, in what situations are each equation used?

Qmechanic
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TaeNyFan
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    When you write the abstract ket $|\psi(t)\rangle$ there is a single variable, so $d/dt$ would be appropriate. When you project onto the position basis you construct the wavefunction $\psi(t,x)=\langle x|\psi(t)\rangle$ and so, when writing the equation in position representation, you have more than one variable, so that $\partial/\partial t$ would be appropriate. – Gold Oct 04 '21 at 12:02
  • Possible duplicate: https://physics.stackexchange.com/q/367750/2451 – Qmechanic Oct 04 '21 at 15:38

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There is no difference. Strictly speaking the second one is correct, because the state vector is a function only of time, but physicists aren't always very careful about distinguishing partial vs single-variable derivatives. The partial derivative might have stuck because as an analogy to the wavefunction $\psi(x,t)$, for which a partial derivative is appropriate.

Javier
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