In 1927, Erwin Schrödinger proposed that any electron or particle that possesses wave properties can be described by a function, represented by the Greek letter psi, $\psi$, called a wave function or state and contains all the information that is possible to know about that quantum system.
Schrödinger's equation, $\hat{H}\psi=E\psi$, is a differential equation whose solution gives us the wave function of the system and its energy. In this equation, $\hat{H}$, represents the Hamiltonian operator, whose expression for a one-dimensional system is:
\begin{equation} \hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V \end{equation}
where $\hbar=h/2\pi$ and V is the potential to which the particle is subjected.
The physical meaning of the wave function is given by its module squared, called probability density, $|\psi|^2$, and related to the probability of finding the particle in a given area of space.