Classically, this oscillator undergoes sinusoidal oscillation of amplitude and frequency , where E is the total energy, potential plus kinetic. In equilibrium at temperture T, its average potential energy and kinetic energy are both equal to ; they depend only on temperature, not on the motion's frequency.
Quantum mechanically, the probability of finding the particle at a given place is obtained from the solution of Shrödinger's equation, yielding eigenvalues and eigenfunctions . For the one dimensional harmonic oscillator, the energies are found to be , where is Planck's constant, f is the classical frequency of motion (above), and n may take on integer values from 0 to infinity. The turn out to be real functions involving the Hermite polynomials. From equation 1, only the ground state () is populated as the temperature . The energy does not go to zero but to . The corresponding zero-point motion is a quantum mechanical phenomenon. Classically, there is no motion as . Thus, we expect that quantum mechanics predicts more motion than classical mechanics, especially at low temperature.