Introduction: Why does cobalt appear different from elements like copper and bismuth? Part of the underlying reason behind these differences in appearance lies in the shape and properties of orbitals of these atoms, or regions of space in which electrons have the highest probability of being found. Orbitals often differ in what are known as quantum numbers, or numbers that act as values in theoretical expressions to represent quantized properties of electrons in atoms of different elements. Quantum numbers are especially important in lasers, in that lasers work by moving electrons to high-energy orbitals, then by producing photons as electrons move back down to lower-energy orbitals.

The four quantum numbers are as follows: (1) the principal quantum number (n), which relates to energy levels of electrons; (2) the azimuthal quantum number (l), which relates to orbital angular momentum and the shape of the orbital; (3) the magnetic quantum number (m), which describes the unique quantum states of specific electrons; and (4) the spin quantum number (s), which describes the spin of the electrons involved in an atom. The azimuthal quantum number specifically can be used to define an orbital as being s (spherical), p (three dumbbells), d (four dumbbells), and f (eight dumbbells). The number of electrons differ in each of these specific orbital shapes.