Let $P$ be the momentum operator. Susskind writes:

$$P |\Psi \rangle =-ih \frac d {dx} |\Psi \rangle$$ Then he states that this can be rewritten as $$-ih\frac {d \psi(x)}{dx}$$ Where $\psi(x)$ is the wave function belonging to $|\Psi\rangle$

It seems to me that the first equation is meaningless. How can we differentiate a state (a ket) w.r.t. $x$? (as opposed to the representation of that state in a wave function, as in the second equation).

  • Am I correct that this notation $\frac d {dx} |\Psi \rangle$ is hacky?

  • If yes, how do we correctly write down the definition of the momentum operator in terms of the ket $|\Psi \rangle$? (i.e. without choosing a basis, and thus without using a wave function)

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    $\begingroup$ Why do people read Susskind's books/watch his lectures? They are not good... $\endgroup$ – AccidentalFourierTransform Mar 9 '18 at 15:20
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    $\begingroup$ @AccidentalFourierTransform, why do you think they are not good? I have learnt a lot about QM in just 3 days. $\endgroup$ – user56834 Mar 9 '18 at 15:25
  • $\begingroup$ If you learnt a lot of QM in just 3 days from Susskind, imagine how much you would have learnt if you used a good source! $\endgroup$ – AccidentalFourierTransform Mar 9 '18 at 15:27
  • $\begingroup$ en.wikipedia.org/wiki/Bra%E2%80%93ket_notation $\endgroup$ – Kyle Kanos Mar 9 '18 at 15:28
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    $\begingroup$ @AccidentalFourierTransform I could never really get through his lectures. I know they're not designed for a "real" class, but to me they come off as having a folksy, almost forced unsophistication which tends to obscure the actual content. Not to downplay his knowledge and contributions to physics, but it's just not my style. $\endgroup$ – J. Murray Mar 9 '18 at 15:47