Here is what Wikipedia (and Google translate) has about it:
The equation named after him was postulated by Schrödinger in 1926 . The starting point was the idea of matter waves, going back to Louis de Broglie , and the Hamilton-Jacobi theory of classical mechanics. The effect ##S## classical mechanics is identified with the phase of a matter wave. As soon as typical distances are smaller than the wavelength, diffraction phenomena play a role, and the classic mechanics must be replaced by a wave mechanics.
The Schrödinger equation cannot be derived from classical physics, but is a postulate. Formally, however, the Schrödinger equation can be derived from the Hamilton function (expression for the energy) of the problem under consideration
$$
E=\dfrac{\mathbf{p^2}}{2m} +V(\mathbf{r},t)
$$
can be derived by replacing the classic quantities energy, momentum and location according to the correspondence principle with the corresponding quantum mechanical operators :
\begin{align*}
E&\longrightarrow \hat{E}=i \hbar\dfrac{\partial}{\partial t}\\[6pt]
\mathbf{p}&\longrightarrow \hat{\mathbf{p}}=-i \hbar \nabla\\[6pt]
\mathbf{r} &\longrightarrow \hat{\mathbf{r}} = \mathbf{r}
\end{align*}
Subsequent application to the unknown wave function ##\psi = \psi (\mathbf{r}, t)## gives the Schrödinger equation
$$
i \hbar \dfrac{\partial \psi}{\partial t}=-\dfrac{\hbar^2}{2m}+\Delta \psi + V\psi
$$
In the same way, the Hamilton function can be converted into a Hamilton operator.
Historically, Schrödinger started from Louis de Broglie's description of free particles and introduced analogies between atomic physics and electromagnetic waves in the form of de Broglie waves (matter waves):
$$
\psi(\mathbf{r},t) = A \,\exp\left(-\dfrac{ i }{\hbar}(Et- \mathbf{p}\cdot\mathbf{r})\right)
$$
in which ##A## is a constant. This wave function is a solution to the Schrödinger equation just mentioned
##V (\mathbf{r}, t) = 0##. In the statistical interpretation of quantum mechanics (founded by Max Born ) there is the square of the amount ##|\psi|^{2}## the probability function density of the particle to the wave function.
