Bohr-sommerfeld Quantization

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wherepr is the radial momentum canonically conjugate to the coordinate q, which is the radial position, and T is one full orbital period. The integral is the action of action-angle coordinates. This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants.

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. 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 Albert Einstein. Sommerfeld made a crucial contribution[5] by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization" (German: Richtungsquantelung). This 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.

where α \displaystyle \alpha is the fine-structure constant. This solution (using substitutions for quantum numbers) is equivalent to the solution of the Dirac equation.[11] Nevertheless, both solutions fail to predict the Lamb shifts.

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.

In 1910, Arthur Erich Haas develops J. J. Thomson's atomic model in his 1910 paper[8] that outlined a treatment of the hydrogen atom involving quantization of electronic orbitals, thus anticipating the Bohr model (1913) by three years.

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. Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the Stark effect. Bose and Einstein gave the correct quantum statistics for photons.

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.

In 1924, Louis de Broglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 Erwin Schrdinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrdinger's wave mechanics developed separately from matrix mechanics until Schrdinger 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 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 p i \displaystyle p_i are the momenta of the system and the q i \displaystyle q_i are the corresponding coordinates. The quantum numbers n i \displaystyle n_i are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). 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. 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 q i \displaystyle q_i 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.

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:

kT is Boltzmann constant times the absolute temperature, which is the temperature as measured in more natural units of energy. The quantity β \displaystyle \beta is more fundamental in thermodynamics than the temperature, because it is the thermodynamic potential associated to the energy.

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 β \displaystyle \beta , at high temperatures, the average energy U is equal to 1 / β = k T \displaystyle 1/\beta =kT . 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 (see Einstein solid and Debye model).

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. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.

which determines that the angular momentum J conjugate to θ \displaystyle \theta , the polar angle, J = M R 2 θ \displaystyle J=MR^2\dot \theta . The old quantum condition requires that J multiplied by the period of θ \displaystyle \theta is an integer multiple of the Planck constant:

And m is called the magnetic quantum number, because the z component of the angular momentum is the magnetic moment of the rotator along the z direction in the case where the particle at the end of the rotator is charged.

Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the z-component of the angular momentum to be the integers l,m. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen z-axis be quantized? This seems to pick out a direction in space.

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