Planck, the Second Law of Thermodynamics, and Black‐body Radiation

Planck’s work on black‐body radiation grew out of a failed attempt to use electrodynamics to show that entropy can never decrease, i.e., that the second law of thermodynamics is not just a statistical but a strict law of nature. This original interest is reflected in his approach to the problem of black‐body radiation. Planck derived the formula for the spectral distribution of black‐body radiation from the formula for the entropy of a resonator interacting with the radiation at its resonance frequency. He initially chose an entropy formula that gave him a black‐body radiation formula proposed by Wien. Deviations from this Wien law at low frequencies led him to adopt a new entropy formula, which gives a law that reduces to the Wien law at high frequencies and to (what is now known as) the Rayleigh‐Jeans law at low frequencies. This new Planck law agreed remarkably well with all experimental data. Planck thus set out to find a derivation of the entropy formula leading to it. Although he continued to resist Boltzmann’s statistical interpretation of the second law for another decade, Planck borrowed some of Boltzmann’s techniques for this derivation. The derivation critically depends on energy elements with sizes proportional to the frequency of the radiation and Planck’s constant as the proportionality constant. Planck’s papers of 1900–01, however, leave open the question of how these energy elements are to be interpreted.

Failures

We consider three topics which proved frustratingly resistant to the methods of the old quantum theory up to the point of emergence of the quantum mechanics of Heisenberg and collaborators in late 1925. First, the old theory could not account convincingly for the superfluity of stationary states implied by the existence of the complex multiplets seen in most atomic spectra. Second, the progressively more complicated theories proposed for explaining the splittings of lines in the anomalous Zeeman effect were found to lead inevitably to glaring inconsistencies with the assumed mechanical equations of motion. Finally, there was the problem of the dual spectrum of helium, and even more basically, of the ground state energy of helium, all calculations of which in terms of specified electron orbits gave incorrect results. We relate the tangled history of the efforts to provide a theoretical resolution of these problems within the old quantum theory.

Introduction to Volume One

We provide an overview, as non‐technical as possible, of the contents of Vol. 1 of the book. Reflecting the structure of the volume, this overview consists of two parts. In the first part, we summarize the most important early contributions to quantum theory (covered in detail in Chs. 2–4). This part starts with Planck’s work on black‐body radiation culminating in the introduction of Planck’s constant in 1900. It then moves on to Einstein’s 1905 light‐quantum hypothesis, his theory of specific heats, and his formulas for energy and momentum fluctuations in black‐body radiation. After summarizing Bohr’s path to his quantum model of the atom, it concludes with Einstein’s 1916–17 radiation theory combining elements of Bohr’s model with his own light‐quantum hypothesis. In the second part we summarize our analysis of the old quantum theory (given in detail in Chs. 5–7). After a brief overview of the career of Sommerfeld, who together with Bohr took the lead in developing the old quantum theory, we review the three principles we have identified as the cornerstones of the theory (the quantization conditions, the adiabatic principle, and the correspondence principle). We then discuss three of the theory’s most notable successes (fine structure, Stark effect, X‐ray spectra) and, finally, three of its most notorious failures (multiplets, Zeeman effect, helium).

Successes

The set of principles formulated in 1915-1918, and now collectively called the old quantum theory, were successfully applied to a number of problems in atomic and X-ray spectroscopy. The three most notable successes are all associated with the Munich school headed by Arnold Sommerfeld. First, there was the derivation of a relativistic fine-structure formula which predicted splittings of stationary state energies for orbits of varying eccentricity at a given principal quantum number. These splittings were empirically verified by Paschen for ionized helium, and constituted the first quantitative confirmation of the special relativistic mechanics introduced by Einstein a decade earlier. The relativistic fine-structure formula was also applied successfully to the splitting of lines in the X-ray spectra of atoms of widely varying atomic number. Finally, the principles of the old quantum theory (in particular, the use of Schwarzschild quantization in combination with Hamilton-Jacobi methods of classical mechanics) were successfully applied to explain the first order splitting spectral lines in the presence of an external electric field (Stark effect).

Einstein, Equipartition, Fluctuations, and Quanta

After three papers on statistical mechanics, mostly duplicating work by Boltzmann and Gibbs, Einstein relied heavily on arguments from statistical mechanics in the most revolutionary of his famous 1905 papers, the one introducing the light‐quantum hypothesis. He showed that the equipartition theorem inescapably leads to the classical Rayleigh‐Jeans law for black‐body radiation and the ultraviolet catastrophe (as Ehrenfest later called it). Einstein and Ehrenfest were the first to point this out but the physics community only accepted it after the venerable H.A. Lorentz, came to the same conclusion in 1908. The central argument for light quanta in Einstein’s 1905 paper involves a comparison between fluctuations in black‐body radiation in the Wien regime and fluctuations in an ideal gas. From this comparison Einstein inferred that black‐body radiation in the Wien regime behaves as a collection of discrete, independent, and localized particles. We show that the same argument works for non‐localized quantized wave modes. Although nobody noticed this flaw in Einstein’s reasoning at the time, his fluctuation argument, and several others like it, failed to convince anybody of the reality of light quanta. Even Millikan’s verification of Einstein formula for the photoelectric effect only led to the acceptance of the formula, not of the theory behind it. Einstein’s quantization of matter was better received, especially his simple model of a solid consisting of quantized oscillators. This model could explain why the specific heats of solids fall off sharply as the temperature is lowered instead of remaining constant as it should according to the well‐known Dulong‐Petit law, which is a direct consequence of the equipartition theorem. The confirmation of Einstein’s theory of specific heats by Nernst and his associates was an important milestone in the development of quantum theory and a central topic at the first Solvay conference of 1911, which brought the fledgling theory to the attention of a larger segment of the physics community. Returning to the quantum theory after spending a few years on the development of general relativity, Einstein combined his light‐quantum hypothesis with elements of Bohr’s model of the atom in a new quantum radiation theory.

Guiding Principles

The development of the complex of assumptions and methods now referred to as the “old quantum theory” mainly took place in the first five years following the introduction of the Bohr atomic model in 1913. Three guiding principles emerged that were used, sometimes in overlapping ways, to explain the flood of spectroscopic data that needed to be explained. First, quantization rules (or conditions) were proposed to single out the allowed orbital motions of electrons in atoms. These rules were derived in various forms by Planck, Sommerfeld, and Wilson, but were put into their most general form by Schwarzschild, who recognized the underlying principle as the quantization of the action variables of a multiply periodic classical system. Second, the special role of the action variables in quantization was given convincing support by the transfer of the adiabatic principle of mechanics to quantum theory (work primarily due to Paul Ehrenfest). Third, the correspondence principle, or statement of asymptotic coincidence of quantum and classical theory in the limit of large quantum numbers, originally introduced by Bohr in 1913 as a supporting argument for his quantization of angular momentum in his theory of the hydrogen atom, was extended by Bohr and Kramers to provide selection rules and approximate intensity predictions evening the regime low quantum numbers.

The Birth of the Bohr Model

We follow Niels Bohr from his 1911 dissertation on the electron theory of metals to his 1913 trilogy on the constitution of atoms and molecules. The dissertation shows that Bohr was thoroughly familiar with the early work of predominantly German physicists on quantum theory and that he suspected that the behavior of bound rather than free electrons called for new laws of physics. During postdoctoral work with Rutherford in Manchester, Bohr learned about the alpha-scattering experiments by Geiger and Marsden that led Rutherford to suggest that an atom consists of a nucleus containing most of its mass with a cloud of electrons swirling around it. Bohr tried to infer the atomic structure in more detail from these and further alpha-scattering experiments. Bohr’s models are in the tradition of British atomic modeling of J.J. Thomson and others but Bohr also borrowed from Planck the notion that energy is proportional to frequency. These early ideas have been preserved in the so-called Manchester memorandum, a set of notes Bohr prepared for Rutherford before returning to Copenhagen in July 1912. In this memorandum, Bohr only considered the ground state of an atom and focused on chemical rather than spectroscopic phenomena. He first started thinking about excited states when he encountered models similar to his own by another British model builder, Nicholson. His interest shifted from chemistry to spectroscopy when a Danish colleague, Hansen, alerted him to the Balmer formula. Within a month of first laying eyes on Balmer’s formula, Bohr submitted the first installment of his trilogy, which contains his famous model of the hydrogen atom. In the following months he completed the trilogy, dealing with more complicated atoms and molecules and presenting results directly coming out of the research recorded in the Manchester memorandum.