A quantum mechanical model is necessary to accurately understand the spectroscopy of atoms and molecules. While the classical mechanical model provides some insight into the spectral characteristics of materials and a useful basis for semi-empirical modeling of amorphous solids and liquids, it fails to capture a variety of observed spectral features such as rotational lines in gaseous spectra, band structures of crystalline solids, and quantized emission spectra of atoms.

The quantum mechanical treatment accurately models these effects by characterizing particles such as electrons in an atom or molecule through wave functions. Such wave functions exist only at discrete energy levels, and interaction with optical radiation involves the allowed transitions between such energy levels. In this way, the spectral characteristics of emission and absorption depend on the specific energy levels of the atom or molecule, the dipole moment associated with the energy level between which the system transitions, and the state populations.

According to the Born-Oppenheimer approximation, a complex molecule can be analyzed by decomposing the wave function and energy levels into atomic, vibrational, and rotational components. Therefore, the mathematical treatment in this chapter separately addresses these individual components to elucidate characteristic features, and then discusses the more complicated case of aggregate spectral properties. While this chapter only scratches the surface of the field of atomic and molecular spectroscopy, the insights gained are of great importance in understanding the observed spectral properties of real materials, as well as the effects of the atmosphere on hyperspectral measurements in a remote setting.