Old quantum theory
[4] 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 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.[7] Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.John William Nicholson is noted as the first to create an atomic model that quantized angular momentum as[11] In 1913, Niels 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.Sommerfeld made a crucial contribution[12] by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization" (German: Richtungsquantelung).Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers.Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom.Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which 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 terms of a version of these transition matrices, creating matrix mechanics.[15] The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete.The integral is an area in phase space, which is a quantity called the action and is quantized in units of the (unreduced) Planck constant.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.Applied as a model for the specific heat of solids, this resolved a discrepancy in pre-quantum thermodynamics that had troubled 19th-century scientists.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., for very low temperatures, the average energy U in the harmonic oscillator approaches zero very quickly, exponentially fast.This reproduces the equipartition theorem of classical thermodynamics: every harmonic oscillator at temperature T has energy kT on average.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.The integral is easiest for a particle in a box of length L, where the quantum condition is: which gives the allowed momenta: and the energy levels Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force F binding a particle to an impenetrable wall.A rotator consists of a mass M at the end of a massless rigid rod of length R and in two dimensions has the Lagrangian: which determines that the angular momentum J conjugate toIn modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are quantum superpositions of the states in other orientations, so that the process of quantization does not pick out a preferred axis.The angular part of the hydrogen atom is just the rotator, and gives the quantum numbers l and m. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.For a fixed value of the total angular momentum L, the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants): Fixing the energy to be (a negative) constant and solving for the radial momentum[20][21][22]) Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing.In 1924, as a PhD candidate, Louis de Broglie proposed a new interpretation of the quantum condition.instead, He then noted that the quantum condition: counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple ofFor example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls.The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic.The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.
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