The Schrodinger Equation

This equation has different solutions, each with it's own Eigen value. The Psi function Ψ is a complex quantity with real and complex parts that depend on space and upon the digital Eigen values of that solution. The Copenhagen interpretation of this complex function is that it represents the probability of finding an electron or whatever at that point in space.

The probability density for few solutions are shown on my Quantum Mechanics page.

For a central force such as the Coulomb attraction of the nucleus, these solutions are determined largely by the structure of space and time. Expressing the Laplacian operator in polar coordinates is the big trick.