What is the justification for the substitution p_x -> (h/i)d/dx in physics?

[gn0m0n]

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?

 

[lbrits]

For a plane wave, that is $\psi(x) = e^{i k x}$, $\frac{1}{i} d/dx$ extracts the wave vector or wavenumber $k$of the wave. From there we use de Broglie's relation to get the $$\hbar$$.

 

[Mentz114]

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.

Clever fellow, that Dirac.

 

[gn0m0n]

Thanks once again, all.