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Introductory Survey

First strand: the thermodynamics and statistical physics of heat radiation

Max Planck and the (possible) discovery of quantum discontinuity.

Planck's work of 1900 is properly characterized as the beginning of quantum theory. Planck sought to give a molecular-electrodynamic account of the distribution of energy over the different frequencies in black body (=cavity) radiation. He found that he could match latest results of the experimentalists, Lummer and Pringsheim, by adopting the statistical methods of Boltzmann. This was a bitter pill for Planck, who had resisted these methods. They reduced his beloved second law of thermodynamics to a result with very high probability only. Essential to the calculation was the assumption that Planck's molecular electric resonators could adopt energies in units of h x (resonant frequency). There is a continuing debate in the literature over whether Planck recognized that this was the beginning of the end of classical physics; or if he thought of it as a temporary computational expedience.

Raleigh and Jeans and the Rayleigh Jeans' Law

Whatever Planck may have thought, Rayleigh (1900) and Jeans (1905) were quite clear that classical analysis of heat radiation yielded a disaster. If the equipartition theorem of classical physics was applied to the infinitely many degrees of freedom of radiation, it accrued an infinite total energy density. Jeans initially suspected that the problem lay in the approach to equilibrium. He supposed that the radiation field was constantly drawing heat energy from ordinary matter, but after a while, it was doing it so slowly that the disequilibrium was not noticeable.

Einstein and the light quantum

In his celebrated annus mirabilis of 1905, Einstein asserted clearly the failure of classical analysis of heat radiation. He noted that the entropy of high frequency heat radiation depended on volume in the same manner as did an ideal gas. He proposed the heuristic hypothesis that the energy of high frequency heat radiation was divided into spatially localized quanta of size hxfrequency.

Einstein and specific heats

The classical analysis of solids yields the expectation that their specific heats are constant. However measurement shows that they reduce to zero as the temperature approaches absolute zero. Einstein (1906) showed that this decline in specific heats is what one should expect if the energy of the components of matter can only adopt discrete energies spaced by the familiar unit, h x (resonant frequency).

Einstein and wave-particle duality.

In 1905, Einstein had shown that in some circumstances it was useful to treat the energy of heat radiation as spatially localized. In others, the wave character was important. So which should we pursue? Might one eclipse the other? In 1909, in an ingenious analysis Einstein showed that fluctuations in heat radiation could be accommodated only by an expression that is the sum of two terms. One clearly has a wave origin and the other a particle origin. Both conceptions were needed.

1911 Solvay Conference, the necessity of quanta

This conference marked a major meeting of the central figures in the emerging quantum theory. It set Poincaré to work on the problem of whether the discontinuities now routinely discussed were merely expedients, an easy way to get to empirical results. Or were they necessities in the new theory. In works of his last year (1912), Poincaré demonstrated that the phenomena of heat radiation could only be explained if one assumed quantum discontinuity. Paul Ehrenfest produced similar results and, in the same paper, coined the term "catastrophe in the ultra-violet."

Planck's second theory

Whatever Planck may have thought in 1900, he was now slow to adopt a radical departure from classical physics. His "second theory" of 1912 sought to localize the discontinuities in the exchange of energy between matter and radiation. Those exchanges would happen discontinuously, while matter and radiation otherwise remained classical.

Einstein's A and B coefficient derivation of the black body radiation spectrum

In this work of 1916, Einstein derived the black body spectrum by modeling a stochastic dynamics for the emission and absorbtion of radiation by sources. A novelty was his recognition that that an energized source could be stimulated to emit radiation by the presence of an ambient radiation field. This process is the basis of LASERs. The processes Einstein described become those developed by quantum electrodynamics.

Compton Effect 1923

Einstein's idea of the light quantum remained a minority view, disliked and avoided notably by Bohr. Compton's measurements of the scattering of x-ray, however, were accommodated easily in a computation that made essential use of the light quantum.

Second strand: atomic spectra

Bohr's 1913 Trilogy.

This work marked the beginning of a new approach. Nineteenth century spectroscopy had provided a wealth of empirical data on emission spectra from excited gases. The difficult was to account for the distinctness of the spectral lines in the data and the simple numerical patterns to which they conformed. In his 1913 model, Bohr pictured an atom as orbited by electrons confined to orbits with angular momentum h/2π. Emission spectra resulted when an electron jumped down to a lower orbit.

Sommerfeld 1915

Sommerfeld extended Bohr's analysis into a more general theory incorporating the notion of the quantum of action. Sommerfeld's analysis applied to any periodic motion in a classical phase space with canonical coordinates (p_i, q_i). The allowed motions were those whose action integrated over a period was a whole number n multiple of h: nh = ∫p_i dq_i.

Its Development

The Bohr-Sommerfeld theory proved immensely fertile and produced a rich and detailed account of emission spectra. In 1916, Sommerfeld could show that a relativistic correction to his 1916 theory allowed for the slight splitting of lines associated with the new constant he introduced, the "fine structure constant." The theory could also accommodate the splitting of lines due to an electric field (Stark effect) and some of the splitting of lines due to a magnetic field (Zeeman effect).

Two principles augmented the theory:

Ehrenfest's adiabatic principle provided a means of discerning how to quantize novel systems. It held that a reversible adiabatic transformation of an allowed motion preserves its allowed status. It proved compatible with Sommerfeld's general condition.

Bohr's correspondence principle was more than the uncontroversial condition that quantum systems become classical for high quantum numbers. It related the magnitude of Fourier coefficients in a classical analysis to transition probabilities between stationary states for small quantum numbers.

Spin

By the early 1920's, it was clear that an extra quantum number was needed beyond the three that naturally accrue to the Bohr-Sommereld theory. Driving experiments were the anomalous Zeeman effect and the Stern-Gerlach experiment, which deflected silver ions through an inhomogeneous magnetic field. The efforts to include this number eventually led Goudschmit and Uhlenbeck to attach the new number to an extra degree of freedom in electrons. It was initially a purely formal maneuver that became the first of the "internal" degrees of freedom so important to the later development of quantum theory.

Convergence

In the early 1920s, the ideas sketched remained a loose collection of notions now known as "the old quantum theory. They could not properly be caslled a theory since they mixed contradictory classical and quantum notions in a structure that was, literally construed, logically inconsistent. Empirical anomalies were mounting. Helium had long been a difficult case and it became apparent that the emission spectrum of helium conflicted with what the best applications of the not-quite theory predicted.

BKS Bohr Kramers Slater Proposal 1924

This was an attempt to come to grips with how charges interact without adopting the light quantum. It posited a collections of virtual fields that mediated the interactions, while respecting energy conservation only statistically. This last featured attracted much criticism and proved to be the proposal's undoing through the experiment of Bothe and Geiger.

De Broglie Waves

Einstein had shown that electromagnetic waves have a particulate aspect. De Broglie reversed the idea to suggest that electrons, that is, particles, also have a wave character. Perhaps electrons adopt the discretely spaced stationary states of the old quantum theory since they are the harmonics of the corresponding waves.

Dispersion theory, matrix mechanics, wave mechanics

These developments bring us to the final assembly of the "new quantum theory," whose exploration will be the focus of this seminar. It came out of developments in dispersion theory (scattering) that lead up to Heisenberg's 1925 "Umdeutung" = reinterpretation. Heisenberg's reinterpretation was rapidly developed into matrix mechanics by Born and Jordan; and developed into an operator calculus by Dirac; and given the now familiar Hilbert space formulation by Hilbert, Nordheim and von Neumann by 1927. Combining de Broglie's wave notions and Heisenberg and Born's matrices, Schroedinger gave his immensely fertile account of quantum states as waves in 1926.