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At one point in Chapter 1 of Sakurai he is deriving the momentum operator in the position basis - I just don't see how he makes some of the mathematical leaps (at least leaps for me) between the following expressions

[tex]

\int dx' |x' + \Delta x'> < x' | \alpha > = \int dx' | x' > < x' - \Delta x' | \alpha > [/tex]

[tex] = \int dx' | x' > \left ( < x' | \alpha > - \Delta x' \frac{\partial}{\partial x'} < x' | \alpha > \right )

[/tex]

I guess I kind of see that the [tex] |x' + \Delta x' > [/tex] acting on the position wavefunction of alpha is the same as translating the position wavefunction to the right by [tex] \Delta x' [/tex]. Is this the logic here or is there a more mathematical way of showing it.

Now, the transition to the last equality is even less intuitive for me.

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# Confusing step in Sakurai Chapter 1

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