quantum mechanics - Derivation of momentum operator

From a video lecture on quantum mechanics at MIT OCW I found that you didn't need to know Schrödinger's equation to know the momentum operator which is $\frac{\hbar}{i}\frac{\partial}{\partial x}$. This can be derived from a 'simple' wave function of the type

$$\psi = Ae^{i(\boldsymbol{\mathbf{k}}\cdot \boldsymbol{\mathbf{r}}- \omega t )}$$ Where we require eigenvalues for $\mathcal{\hat{p}}$ to be $\hbar k$ My questions are:

• I understand the complex notation is for convenience. Since it's a complex exponential it'll give us a real and imaginary wave. Does the imaginary part have any physical significance? Are we to interpret this as two waves in superposition in the complex plane?


• As we derive the expression for $\mathcal{\hat{p}}$ for this specific function, how does it guarantee that this is indeed the $\mathcal{\hat{p}}$ for every other arbitrary wave function? Can it be derived besides using the wave function I mentioned?