A new and better quantum theory was developed in the 1920's by an Austrian and a German physicist, Erwin Schrodinger and Werner Heisenberg. Their wave mechanics is mathematically intimidating, and almost seems to view the world as a set of solutions of differential equations. It succeeded in explaining the structure of multielectron atoms, the structure of the Periodic table, and the theory of bonding between atoms in molecules - impressive accomplishments for any theory. The picture of atoms and molecules that resulted is essentially that which we use today. We will make use of the results of wave mechanics - energy levels and atomic structure - without going through the mathematics that led to those results.

As in the simpler Bohr theory, the energy of an electron in an atom is restricted to certain values, or is quantized. Three quantum numbers instead of one are required to describe an electron, and they are designated n, l, and m. The average distance of the electron from the nucleus depends primarily on n, which is called the principal quantum number. The geometry of bonding around the atom depends primarily on quantum number l, called the orbital-shape, or azimuthal, quantum number. The energy of an electron in an atom is a function of n and to a lesser degree of l. The orbital-orientation, or magnetic quantum number, m, describes how the electron's orbit is oriented in space relative to some external reference such as a magnetic or electric field. In the absence of such fields, all m states for a given n and l value have the same energy.