Page "Quantum mechanics" Paragraph 31

The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value.

Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states.

This gives rise to the " wave-like " behavior of quantum states.

As it turns out, analytic solutions of the Schrödinger equation are only available for a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives.

Even the helium atom-which contains just one more electron than does the hydrogen atom-has defied all attempts at a fully analytic treatment.