units of the (unreduced) Planck constant. For this reason, the Planck constant was often called the quantum of action. In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way. The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant. This quantization condition is often known as the Wilson–Sommerfeld rule, proposed independently by William Wilson and Arnold Sommerfeld. Examples Thermal properties of the harmonic oscillator The simplest system in the old quantum theory is the harmonic oscillator, whose Hamiltonian is: The old quantum theory yields a recipe for the quantization of the energy levels of the harmonic oscillator, which, when combined with the Boltzmann probability distribution of thermodynamics, yields the correct expression for the stored energy and specific heat of a quantum oscillator both at low and at ordinary temperatures. Applied as a model for the specific heat of solids, this resolved a discrepancy in pre-quantum thermodynamics that had troubled 19th-century scientists. Let us now describe this. The level sets of H are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule: a result which was known well before, and used to formulate the old quantum condition. This result differs by from the results found with the help of quantum mechanics. This constant is neglected in the derivation of the old quantum theory, and its value cannot be determined using it. The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a Boltzmann weight: kT is Boltzmann constant times the absolute temperature, which is the temperature as measured in more natural units of energy. The quantity is more fundamental in thermodynamics than the temperature, because it