In Schrodinger Equation, we found the energy operator H can be written as below, where momentum operators show up, we need to dig further why?

Brandon starts from the translation operator, which we can infer that it’s an Unitary operator too, and Taylor expansion leads to

since momentum is the generator of the spatial change, hence the Hermitian operator in above equation must can be replaced by momentum operator p.

great physicists are keen to find the pattern and conclude all below holds

And they totally match what deduced in classical mechanics:

Then to see what if have the momentum operator on wave function psi

Now taking the pains to deduce the position perspective