An Introduction to Electron Orbitals

The Wave-Particle Duality of Matter and Energy
In the early 1920's, Louis de Broglie proposed that if energy is particle-like, perhaps matter is wavelike; therefore, if electrons have wavelike motion and are restricted to orbits in fixed radii, that would explain why they have only certain possible frequencies and energies. From the mass-energy equivalence (E=mc2) with that for the energy of a photon (E = hv = hc/λ) to derive a formula for the wavelength of any particle of mass "m" moving at a speed "u": λ = h/mu.
According to this equation, matter does behave as though it moves in a wave (Note: an object's wavelength is inversely proportional to its mass, so heavy objects, such as planets and baseballs, have wavelengths that are many orders of magnitude smaller than the object itself.

The Atomic Orbital and the Probably Location of the Electron
Schrödinger developed an equation which defines the energy level, shape, orientation and spin of an electrons orbital:
HΨ = EΨ
where Ψ2 is the probability density: a measure of the probability that the electron can be found within a particular tiny volume of the atom. This probability is expressed through an electron probability density diagram or more simply, an electron density diagram. If you were to plot the probability of the electron vs the distance from the nucleus it would look a y = 1/x2 function. If one were to mentally divide the "cloud" up into many different rings and plot a graph of the total probability of an electron being in a spherical layer (sum of 2) vs the distance from the nucleus, the graph would start from zero, rise to a peak then drop down and approach y=0 as x  ∞. If one were to draw a solid sphere representing where an electron would be 90% of the time it would be called a probability contour plot.

Quantum Numbers of an Atomic Orbital
There are four principle quantum numbers which define an electrons orbit, they are as follows:
1. The principle quantum number (n): is a positive integer. This number defines the size of the electrons orbit. It also defines the energy level to which the electron is at.
2. The angular momentum quantum number (L): is an integer from 0 to n-1. This number defines the shape of the orbital.
3. The magnetic quantum number (ml): is an integer from -L through 0 to +L. It defines the orientation of the orbital in the space around the nucleus and is sometimes called the orbital-orientation quantum number (Note: if L = 0, then ml = 0 and if L = 1 then ml = -1, 0 or +1).
4. Electron - spin quantum number (ms): is either +½ or -½ and simply denotes the spin the electron has.

Orbital Shapes
1s = sphere shaped and has 0 nodes.
2p = dumbbells shaped and has 1 node.
3d = four leaf clovers shaped and has 2 nodes.