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How to derive or justify the expressions of momentum operator and energy operator?

It has been noted here, for instance, that

${\bf F } = \frac{d \bf p }{d t}$ is true in all contexts.

Likewise, in notable contexts it is apparently true that

${\bf F } = - \nabla \Phi := - \frac{d \Phi}{d \bf r}$.

Is this, in a nutshell, a sufficient and valid justification for setting (in the corresponding suitable contexts)
the momentum operator as

${\bf \hat p } :=$ proportional to $-i \nabla := -i \frac{d}{d \bf r}$

and setting the (potential) energy operator as

$\hat \Phi :=$ proportional to $i \frac{d}{d t}$

and both with the same constant of proportionality, $\hbar$, whereby

$ {\bf \hat F } = \frac{ d }{d t}[ - i \hbar \frac{d}{d \bf r} ] = - \frac{d}{d \bf r} [ i \hbar \frac{d}{d t} ] \sim \frac{ d^2 }{d t d \bf r } = \frac{ d^2 }{d {\bf r } d t } $

?