Classical harmonic motion follows Hooke's law, F = -kx, where F is the force, k is the force constant and x is the displacement. The force is the mass times the acceleration, which is the second time derivative. As the section titlted "Vibrational solution" shows the solution to the classical harmonic oscillator is a cosine function. The natural frequency is given by:
The classical solution predicts oscillator motion that will have the greatest residence time at the turning point and the most rapid trajectory when passing through the equilibrium position. The classical solution also predicts that the potential energy will be:.
The quantum mechanical harmonic oscillator
The quantum mechanical harmonic oscillator is based on the same potential energy function as the classical harmonic oscillator. However, the Schrodinger equation involves the hamiltonian operator and has the appearance:.
The Q in this equation represents the nuclear coordinate of a normal mode. Q is a vector that describes the collective motion of the nuclei as they oscillate about the equilibrium position. See the section on normal modes to understand the meaning of collective motion..
Quantum solutions
The quantum mechanical solutions are Gaussian functions multiplied by polynomials. These are wave functions that describe the localization of the nuclei in the harmonic potential energy surface. The solutions form an orthogonal set of functions. Orthogonal functions have no overlap. The only way to couple two orthogonal functions is by means of a perturbation. In spectroscopy the perturbation is electromagnetic radiation.
The vibrational transition moment
The sheer size of the wave is a problem if we consider the absorption of light by atoms and molecules. A typical molecule might be 1 or 2 nm in size. Visible light has wavelengths between 400-700 nm. Thus, the wavelenght of light is at least 200 times larger than a typical molecule. There is no easy way to picture how light is absorbed using the wave picture. This is one reason that the particular picture is useful. In order to combine the wave and particle pictures we often speak of a wave packet.
