Editing Old quantum theory (section)

=== Thermal properties of the harmonic oscillator ===
The simplest system in the old quantum theory is the [[harmonic oscillator]], whose [[Hamiltonian (quantum mechanics)|Hamiltonian]] is:
: <math>
H= {p^2 \over 2m} + {m\omega^2 q^2\over 2}.
</math>

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:
: <math>
E= n\hbar \omega,
\,</math>
a result which was known well before, and used to formulate the old quantum condition. This result differs by <math>\tfrac{1}{2}\hbar \omega</math> 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]]:
: <math>
U = {\sum_n \hbar\omega n e^{-\beta n\hbar\omega} \over \sum_n e^{-\beta n \hbar\omega}} = {\hbar \omega e^{-\beta\hbar\omega} \over 1 - e^{-\beta\hbar\omega}},\;\;\;{\rm where}\;\;\beta = \frac{1}{kT},
</math>

''kT'' is [[Boltzmann constant]] times the [[Thermodynamic temperature|absolute temperature]], which is the temperature as measured in more natural units of energy. The  quantity <math>\beta</math> is more fundamental in thermodynamics than the temperature, because it is the [[thermodynamic potential]] associated to the energy.

From this expression, it is easy to see that for large values of <math>\beta</math>, for very low temperatures, the average energy ''U'' in the harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that ''kT'' is the typical energy of random motion at temperature ''T'', and when this is smaller than <math>\hbar\omega</math>, there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all.

This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the [[specific heat]], so the specific heat is exponentially small at low temperatures, going to zero like
: <math> \exp(-\hbar\omega/kT) </math>

At small values of <math>\beta</math>, at high temperatures, the average energy ''U'' is equal to <math>1/\beta = kT</math>. This reproduces the [[equipartition theorem]] of classical thermodynamics: every harmonic oscillator at temperature ''T'' has energy ''kT'' on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to&nbsp;''k''. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times&nbsp;''k''. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3''k'' per atom, or in chemistry units, 3''R'' per [[Mole (unit)|mole]] of atoms.

Monatomic solids at room temperatures have approximately the same specific heat of 3''k'' per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the [[third law of thermodynamics]]. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature.

This contradiction between classical mechanics and the specific heat of cold materials was noted by [[James Clerk Maxwell]] in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, [[Peter Debye]] gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see [[Einstein solid]] and [[Debye model]]).