Old quantum theory

The old quantum theory is a collection of results from the years 1900–1925, which predate modern quantum mechanics. The theory was never complete or self-consistent, but was instead a set of heuristic corrections to classical mechanics. The theory has come to be understood as the semi-classical approximation to modern quantum mechanics. The main and final accomplishments of the old quantum theory were the determination of the modern form of the periodic table by Edmund Stoner and the Pauli exclusion principle, both of which were premised on Arnold Sommerfeld's enhancements to the Bohr model of the atom.
The main tool of the old quantum theory was the Bohr–Sommerfeld quantization condition, a procedure for selection of certain allowed states of a classical system: the system can then only exist in one of the allowed states and not in any other state.

History

The old quantum theory was instigated by the 1900 work of Max Planck on the emission and absorption of light in a black body with his discovery of Planck's law introducing his quantum of action, and began in earnest after the work of Albert Einstein on the specific heats of solids in 1907 brought him to the attention of Walther Nernst. Einstein solid, followed by the Debye model in 1912, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.
In 1910, Arthur Erich Haas further developed J. J. Thomson's atomic model in a paper that outlined a treatment of the hydrogen atom involving quantization of electronic orbitals, thus anticipating the Bohr model by three years.
John William Nicholson is noted as the first to create an atomic model that quantized angular momentum as. Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.
In 1913, Bohr displayed rudiments of the later defined correspondence principle and used it to formulate a model of the hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Hendrik Lorentz and Einstein. Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization". This model, which became known as the Bohr–Sommerfeld model, allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.
Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. Planck introduced the zero point energy and Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the Stark effect. Satyendra Nath Bose and Einstein developed the Bose–Einstein statistics for bosons. Einstein also refined the quantization condition in 1917.

End of old theory

In 1924, Bohr, Kramers and John C. Slater promoted what was known as the BKS theory which considered systems as quantum mechanical but the electromagnetic field as a classical field. However the theory was rejected by the Bothe–Geiger coincidence experiment.Image: Drawing_of_Sommerfeld_atom.svg |thumb|310px|The Sommerfeld extensions of the 1913 solar system Bohr
model of the hydrogen atom showing the addition of elliptical orbits to explain spectral fine structure. The circular n=3 corresponds to a higher energy orbital. n=3 has multiple orbits because of azimuthal quantum number.
Kramers prescriptions for calculating transition probabilities between quantum states in terms of Fourier components of the motion, were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in his 1925 Umdeutung paper, in terms these transition rules, later creating matrix mechanics with Max Born and Pascual Jordan.
In parallel in 1924, Louis de Broglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Einstein a short time later. In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. The Schrödinger equation developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory.
The mathematical formalism of modern quantum mechanics was developed by Dirac and John von Neumann.

Other developments

In the 1950s Joseph Keller updated Bohr–Sommerfeld quantization using Einstein's interpretation of 1917, now known as Einstein–Brillouin–Keller method. In 1971, Martin Gutzwiller took into account that this method only works for integrable systems and derived a semiclassical way of quantizing chaotic systems from path integrals.

Basic principles

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the quantization condition:
where the are the momenta of the system and the are the corresponding coordinates. The quantum numbers are integers and the integral is taken over one period of the motion at constant energy. The integral is an area in phase space, which is a quantity called the action and is quantized in units of the 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 is the thermodynamic potential associated to the energy.
From this expression, it is easy to see that for large values of, 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, 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
At small values of, at high temperatures, the average energy U is equal to. 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 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 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 3k per atom, or in chemistry units, 3R per mole of atoms.
Monatomic solids at room temperatures have approximately the same specific heat of 3k 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.