Hi, I am wondering what justifies the substitution p_x -> (h/i)d/dx ? I know it is very common but I have not seen a reason for it anywhere. Why does it make physical sense?
For a plane wave, that is [itex]\psi(x) = e^{i k x}[/itex], [itex]\frac{1}{i} d/dx[/itex] extracts the wave vector or wavenumber [itex]k[/itex]of the wave. From there we use de Broglie's relation to get the [tex]\hbar[/tex].
There's rather a good mathematical reason. The quantum Lagrangian is invariant under spatial translations, which guarantees momentum conservation. As Dirac shows, d/dx is the generator of translations ( page 100 'Princples..'). Dirac calls translation 'displacement' and demonstrates that the action of the operator is to return the momentum.