There is a particularly neat way to derive Schrödinger’s equation, and to justify the “canonical substitution” rules for replacing energy and momentum with corresponding operators when we “quantize” an equation.
Take a particle in a potential. Its energy is given by
$$E=\frac{{\bf p}^2}{2m}+V({\bf x}),$$
or
$$E-\frac{{\bf p}^2}{2m}-V({\bf x})=0.$$
Now multiply both sides this equation by the formula \(e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}\). We note that this exponential expression cannot ever be zero if the part in the exponent that’s in parentheses is real:
$$\left[E-\frac{{\bf p}^2}{2m}-V({\bf x})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0.$$
So far so good. But now note that
$$Ee^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=i\hbar\frac{\partial}{\partial t}e^{i({\bf p}\cdot{\bf x}-Et)/\hbar},$$
and similarly,
$${\bf p}^2e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=-\hbar^2{\boldsymbol\nabla}e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}.$$
This allows us to rewrite the previous equation as
$$\left[i\hbar\frac{\partial}{\partial t}+\hbar^2\frac{{\boldsymbol\nabla}^2}{2m}-V({\bf x})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0.$$
Or, writing \(\Psi=e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}\) and rearranging:
$$i\hbar\frac{\partial}{\partial t}\Psi=-\hbar^2\frac{{\boldsymbol\nabla}^2}{2m}\Psi+V({\bf x})\Psi,$$
which is the good old Schrödinger equation.
The method works for an arbitrary, generic Hamiltonian, too. Given
$$H({\bf p})=E,$$
we can write
$$\left[E-H({\bf p})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0,$$
which is equivalent to
$$\left[i\hbar\frac{\partial}{\partial t}-H(-i\hbar{\boldsymbol\nabla})\right]\Psi=0.$$
So if this equation is identically satisfied for a classical system with Hamiltonian \(H\), what’s the big deal about quantum mechanics? Well… a classical system satisfies \(E-H({\bf p})=0\), where \(E\) and \({\bf p}\) are eigenvalues of the differential operators \(i\hbar\partial/\partial t\) and \(-i\hbar{\boldsymbol\nabla}\), respectively. Schrödinger’s equation, on the other hand, remains valid in the general case, not just for the eigenvalues.