Orbitals
Only when we get down to atomic dimensions--as defined by Planck's constant h--does the wave nature dominate the behavior. In a hydrogen atom, for instance, the wave then can be stable only in certain resonant states--"eigenstates," a word cobbled together from German and English, with "eigenvalues" giving the energy levels. The atom is then like some musical instrument tuned to certain notes--e.g. it can sound a C or a D, but never an in-between false note. Until its note sounds, we only know the likelihood of it being this state or that.
This is quite different from the Bohr-Sommerfeld atom where (at least initially) Kepler orbits served as a model to the motion of electrons. Those Kepler orbits were classified by total energy and angular momentum (corresponding to ellipticity), but they were all flat, two-dimensional. Even now, in the popular literature, atoms are often drawn as miniature planetary systems: but that is not the correct picture.
In contrast, wave functions are 3 dimensional, spread over space. Yet their basic modes can also be classified, and interestingly, the scheme of modes turns out similar to the one based on Kepler motion, although the underlying concepts are quite different. (The mathematical tools are somewhat related to the ones used by Gauss to extract the main modes of the Earth's magnetic field--tools taken, in turn, from modes deduced for the Earth's gravity field). Nowadays the basic modes of atomic (or molecular) wave functions are often known as orbitals, patterns of regions in which the wave function is concentrated. The lowest orbitals are symmetric and spherical, but the more complex ones have multiple lobes, a bit like 3-dimensional cloverleaf patterns with various numbers of lobes.
The basic classification is still by energy--n=1 levels, then n=2 levels, etc.
In additions, the levels of asymmetry are denoted by values (0,1,2,3,...) of (lower case) L (representing angular momentum, related to asymmetry) or, traditionally, by letters (s,p, d, f....), as shown on the energy-level diagram in the preceding section. The "s" mode is symmetric, "p" has 2-lobed symmetry, then "d" , then "f", etc. A third number m exists, and visual descriptions of the peaks of the modes can be found here; of course, these are just the surfaces where the wave function is largest, from which it tapers down to the rest of space.
Orbitals are important not only when the atom jumps from a high level to a lower (unoccupied) one, but also, in an atom at its lowest level ("ground state") it turns out electrons must occupy different orbitals (except that electrons with opposed spin may double up). That idea has led to an explanation of the periodic table of chemical elements. Orbitals can also be deduced for molecules, which have more complex spectra, usually in the infra-red.
Later Developments
The above are just the basics, a preliminary reconnaissance of a new terrain,. For detail and applications one must study quantum theory systematically--with all its math, its manipulations of angular momentum and spin and so forth, followed perhaps by scattering theory, Dirac's theory of the electron, quantum electrodynamics and more.
The mathematical treatment has led to a fairly good understanding of atomic spectra, including their intensity variations--although solving for energy levels of complicated atoms requires tedious approximations. (It is the same in celestial mechanics--the tracking of a single planet around its sun is simple, but tracking multiple objects interacting with each other is hard). Quantum theory also led to an understanding of molecular spectra, of the chemical bond, of the perodic table of elements (see above), of the behavior of atoms arranged in crystals (including semiconductors, which made computers practical), of electric "superconductivity" in very cold materials, of magnetic effects on which both medical imaging (MRI) and modern magnetometry depend, of lasers and a lot more.
