Although chemists are supposed to learn quantum mechanics in physical chemistry, its standard formalism using wave functions in the real space and the Schrödinger equation is not very convenient for theoretical vibrational spectroscopy. This chapter provides the bra–ket notation and the creation and annihilation operators of a harmonic oscillator, which are more convenient for the computation. In addition, the density operator is introduced to allow for analytically obtaining spectra with finite bandwidth. The quantum–mechanical expression of polarizability is also derived. This chapter does not cover all important concepts and theorems in quantum mechanics, but instead deals with fundamental ideas and equations most necessary to perform the theoretical calculations of the IR, Raman, and sum frequency generation spectra of molecular systems according to Chaps. 2 and 3 .