Deriving momentum operator in quantum mechanics I am just wondering how we actually derive the momentum operator in quantum mechanics.
As what I have learnt, we claim that for a free particle, its probability amplitude should be constant everywhere. Therefore we could write $$\psi = Ae^{ip_\mu x^\mu} \, .$$
Then, it can be seen that our momentum operator should be $\hat{p}^\mu = i\partial ^\mu$ (note that I have used the signature $(1,-1,-1,-1)$).
How do we know that $p_\mu$ in the wavefunction is actually the 4-momentum (because our assumption is only that probability amplitude is constant and nothing else, therefore the wavefunction could be exponential of any pure complex number). Secondly, we derive this operator for only free particle. How do we know this will work for a general wavefunction?
 A: The argument is the same in non-relativistic or relativistic QM.


*

*Plane waves have definite momentum, and they are of the form as indicated: $\psi\sim A e^{-i (
p x-\omega t)}$ (in 1+1).

*Because they have definite momentum, one postulates an operator $\hat p$ such that
$$
\hat p\psi = p\psi\, .
$$

*Using the explicit form of $\psi$ one sees that
$$
\hat p\mapsto i \frac{\partial}{\partial x}
$$
The same argument holds for the time component of your $p_\mu$.

*Having found $\hat p$ for the plane wave, one postulates that this must remain true for arbitrary potentials.  


An alternate (but not completely independent since this inspired Schrodinger) approach is to realize that, in the Hamilton-Jacobi formulation of classical mechanics, the momentum operators are mapped to partial derivatives w/r to the conjugate positions, i.e. $p\to \frac{\partial }{\partial x}$ in HJ.  This remains true irrespective of the potential, supporting the postulate that $\hat p\mapsto i\partial_x$ ought to remain true irrespective of the potential.
Ultimately, these "guesses" are validated a posteriori by the solutions to the Schrodinger equation.
A: The $p_\mu$ in the exponential is the true momentum 4-vector. It is a compact form to write the 4 different spectral values of the 4 momentum operators, while the latter are the generators of the Lie algebra of the spacetime translations subgroup of the restricted Poincare group. The spectral equation $\hat{P}_{\mu} \psi = p_\mu \psi$ is valid in the (uniparticle rigged) Hilbert space which carries an irreducible representation of the translations subgroup. The translations subgroup in flat 4D spacetime or generally the restricted Poincare symmetry is known/proven to be an exact symmetry only in case of free quantum field theories in 4D. 
