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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

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

?