'These children are talking over a cup–and–string "telephone". The displacement of the string is completely described by a wave function y(x,t). In an analogous way, a particle is completely described by a quantum–mechanical wave function Ψ(x.y,z,t).
Schrodinger was the first scientist to write down a wave equation. The eigen values of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom, the energy levels were found to be in accord with Rydberg’s Law.
It was initially much less knowledge on what is the wave function of the equation. After much debate, the wave function is now accepted to be a probability distribution. The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wave function gives the probability of finding the particle at a certain position.
The solution to this equation is a wave that describes the quantum aspects of a system. However, physically interpreting the wave is one of the main philosophical problems of quantum mechanics.