It is always stated the quantum operators for ${p}$ and $E$ are the ones we´re familiar with (the operator for energy, $E=i\hbar\frac{\partial}{\partial t}$ and the momentum operator, $p=-i\hbar\nabla $). But where do these operators come from? I understand that substitution of these operators in the standard momentum-energy relation amounts to the appearance of the Schrödinger equation (or the Klein-Gordon equation in the relativistic case, were, by the way, the $E$ and $p$ operators stay the same).

But in the time of the development of quantum mechanics nobody knew the (by know known) form of a wave equation, so why replace the $p$ and $E$ by their now well-known operator-form?

Did they find the right operators, corresponding to $p$ and $E$ by educated guessing, by trial and error, or some other way, from which the Schrödinger equation emerged when applied to the standard energy-momentum relation?

Did they try all kinds of combinations of operators, $i$, and the Planck-constant until the right operators were found, from which the equation (the Schrödinger equation) followed which fitted the data? Or what?

It is always stated the quantum operators for ${p}$ and $E$ are the ones we´re familiar with (the operator for energy, $E=i\hbar\frac{\partial}{\partial t}$ and the momentum operator, $p=-i\hbar\nabla $). But where do these operators come from? 

I understand that substitution of these operators in the standard momentum-energy relation amounts to the appearance of the Schrödinger equation (or the Klein-Gordon equation in the relativistic case, where, by the way, the $E$ and $p$ operators stay the same).

But in the time of the development of quantum mechanics nobody knew the (by now known) form of a wave equation, so why replace the $p$ and $E$ by their now well-known operator-form?

Did they find the right operators, corresponding to $p$ and $E$ by educated guessing, by trial and error, or some other way, from which the Schrödinger equation emerged when applied to the standard energy-momentum relation?

Did they try all kinds of combinations of operators, $i$, and the Planck-constant until the right operators were found, from which the equation (the Schrödinger equation) followed which fitted the data? Or what?

It is always stated the quantum operators for ${p}$ and $E$ are the ones we´re familiar with (the operator for energy, $E=i\hbar\frac{\partial}{\partial t}$ and the momentum operator, $p=-i\hbar\nabla $). But where do these operators come from? I understand that substitution of these operators in the classical momentum-energy relation amounts to the appearance of the Schrödinger equation (or the Klein-Gordon equation in the relativistic case, were, by the way, the $E$ and $p$ operators stay the same).

But in the time of the development of quantum mechanics nobody knew the (by know known) form of a wave equation, so why replace the $p$ and $E$ by their now well-known operator-form?

Did they find the right operators, corresponding to $p$ and $E$ by educated guessing, by trial and error, or some other way, from which the Schrödinger equation emerged when applied to the classical energy-momentum relation?

Did they try all kinds of combinations of operators, $i$, and the Planck-constant until the right operators were found, from which the equation (the Schrödinger equation) followed which fitted the data? Or what?

It is always stated the quantum operators for ${p}$ and $E$ are the ones we´re familiar with (the operator for energy, $E=i\hbar\frac{\partial}{\partial t}$ and the momentum operator, $p=-i\hbar\nabla $). But where do these operators come from? I understand that substitution of these operators in the standard momentum-energy relation amounts to the appearance of the Schrödinger equation (or the Klein-Gordon equation in the relativistic case, were, by the way, the $E$ and $p$ operators stay the same).

But in the time of the development of quantum mechanics nobody knew the (by know known) form of a wave equation, so why replace the $p$ and $E$ by their now well-known operator-form?

Did they find the right operators, corresponding to $p$ and $E$ by educated guessing, by trial and error, or some other way, from which the Schrödinger equation emerged when applied to the standard energy-momentum relation?

Did they try all kinds of combinations of operators, $i$, and the Planck-constant until the right operators were found, from which the equation (the Schrödinger equation) followed which fitted the data? Or what?