orbit at energy E is the area of the ellipse of constant energy, The x radius of this ellipse is while the p radius of the ellipse is . Multiplying, the area is . So if a pendulum is slowly drawn in, such that the frequency changes, the energy changes by a proportional amount. Old quantum theory After Planck identified that Wien's law can be extended to all frequencies, even very low ones, by interpolating with the classical equipartition law for radiation, physicists wanted to understand the quantum behavior of other systems. The Planck radiation law quantized the motion of the field oscillators in units of energy proportional to the frequency: The quantum can only depend on the energy/frequency by adiabatic invariance, and since the energy must be additive when putting boxes end-to-end, the levels must be equally spaced. Einstein, followed by Debye, extended the domain of quantum mechanics by considering the sound modes in a solid as quantized oscillators. This model explained why the specific heat of solids approached zero at low temperatures, instead of staying fixed at as predicted by classical equipartition. At the Solvay conference, the question of quantizing other motions was raised, and Lorentz pointed out a problem, known as Rayleigh–Lorentz pendulum. If you consider a quantum pendulum whose string is shortened very slowly, the quantum number of the pendulum cannot change because at no point is there a high enough frequency to cause a transition between the states. But the frequency of the pendulum changes when the string is shorter, so the quantum states change energy. Einstein responded that for slow pulling, the frequency and energy of the pendulum both change, but the ratio stays fixed. This is analogous to Wien's observation that under slow motion of the wall the energy to frequency ratio of reflected waves is constant. The conclusion was that the quantities to quantize must be adiabatic invariants. This line of argument was extended by Sommerfeld into a general theory: the quantum number of an arbitrary mechanical system is given by the adiabatic action variable. Since the action variable in the harmonic oscillator is an integer, the general condition is This condition was the foundation of the old quantum theory, which was able to predict the qualitative behavior of atomic systems. The theory is inexact for small quantum numbers, since it mixes classical and quantum concepts. But it was a useful half-way