The specific heat of hydrogen gas at low temperatures was first measured in 1912 by Arnold Eucken in Walther Nernst’s laboratory in Berlin, and provided one of the earliest experimental supports for the new quantum theory. Even earlier, Nernst had developed a quantum theory of rotating diatomic gas molecules that figured in the discussions at the first Solvay conference in late 1911. Between 1913 and 1925, Albert Einstein, Paul Ehrenfest, Max Planck, Fritz Reiche, and Erwin Schrödinger, among many others, attempted (...) theoretical descriptions of the rotational specific heat of hydrogen, with only limited success. Quantum theory also was central to the study of molecular spectra, where initially it was more successful. Moreover, the two problems interacted in sometimes surprising ways. Not until 1927, following Werner Heisenberg’s discovery of the behavior of indistinguishable particles in modern quantum mechanics, did American theorist David Dennison find a successful theory of the specific heat of hydrogen. (shrink)

In 1909, Einstein derived a formula for the mean square energy fluctuation in blackbody radiation. This formula is the sum of a wave term and a particle term. In a key contribution to the 1926 Dreim¨.

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-12 particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary (...) spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation. (shrink)

We discuss the early career of John H. Van Vleck, one of the earliest American quantum theorists who shared the 1977 Nobel prize with his student Philip W. Anderson and Sir Nevill Mott. In particular, we follow Van Vleck's trajectory from his 1926 Bulletin for the National Research Council on the old quantum theory to his 1932 book, The Theory of Electric and Magnetic Susceptibilities. We highlight the continuity of formalism and technique in the transition from dealing with spectra in (...) the old quantum theory to dealing with susceptibilities in the new quantum mechanics. Our main focus is on the checkered history of a numerical factor in the so-called Langevin-Debye formula for the electric susceptibility of gases. Classical theory predicts, under very general conditions, that this factor is equal to 1/3. The old quantum theory predicted values up to 14 times higher. Van Vleck showed that quantum mechanics does away with this "wonderful nonsense" and restores the classical value 1/3 under equally general conditions. The Langevin-Debye formula thus provides an instructive example of a Kuhn loss in one paradigm shift that was regained in the next. In accordance with the expectation of Thomas S. Kuhn that textbooks tend to sweep Kuhn losses under the rug, Van Vleck did not mention this particular Kuhn loss anywhere in his 1926 NRC Bulletin. Contrary to Kuhn's expectations, however, he put the regained Kuhn loss in susceptibility theory to good pedagogical use in his 1932 book. Kuhn claimed that textbooks must suppress, truncate, and/or distort the prehistory of their subject matter if they are to inculcate the exemplars of the new paradigm in their readers. This claim is not borne out in this case. We argue that it is ultimately because of the continuity of formalism and technique that we draw attention to that Van Vleck could achieve his pedagogical objectives in his 1932 book even though he devoted about a third of it to the treatment of susceptibilities in classical theory and the old quantum theory in a way that matches the historical record reasonably well. (shrink)

One finds, even in texts by distinguished physicists, diverse enunciations of the correspondence principle. Typical is that quantum mechanics should agree with classical mechanics in some appropriate limit. Most commonly, the limit specified is that of high quantum numbers, or of large masses and orbits of large dimensions. But sometimes it is specified as mean behavior when large numbers quanta are involved, or sometimes even as just the average of quantum mechanical variables. Sometimes, the principle is even taken as a (...) prescription for replacing the classical dynamical observables with an appropriate mathematical operator. In 1918, however, Bohr proposed what he would later call the correspondence principle as a way of deriving amplitudes and polarizations of emitted and absorbed spectral lines. I will begin with Bohr's principle and trace the evolution of correspondence considerations through the 1920's, with a view as to whether in each case it is supposed to play the role of a theorem, an adequacy constraint, an inductive hypothesis or a heuristic. (shrink)

Niels Bohr’s “correspondence principle” is typically believed to be the requirement that in the limit of large quantum numbers (n→∞) there is a statistical agreement between the quantum and classical frequencies. A closer reading of Bohr’s writings on the correspondence principle, however, reveals that this interpretation is mistaken. Specifically, Bohr makes the following three puzzling claims: First, he claims that the correspondence principle applies to small quantum numbers as well as large (while the statistical agreement of frequencies is only for (...) large n); second, he claims that the correspondence principle is a law of quantum theory; and third, Bohr argues that formal apparatus of matrix mechanics (the new quantum theory) can be thought of as a precise formulation of the correspondence principle. With further textual evidence, I offer an alternative interpretation of the correspondence principle in terms of what I call Bohr’s selection rule. I conclude by showing how this new interpretation of the correspondence principle readily makes sense of Bohr’s three puzzling claims. (shrink)

In early 1927, Pascual Jordan published his version of what came to be known as the Dirac-Jordan statistical transformation theory. Later that year and partly in response to Jordan, John von Neumann published the modern Hilbert space formalism of quantum mechanics. Central to both formalisms are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems in the new (...) theory. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canonically conjugate variables, i.e., p’s and q’s. Jordan ran into serious difficulties when he tried to extend his approach from quantities with fully continuous spectra to those with wholly or partly discrete spectra. For von Neumann, not constrained by the analogy to classical physics and aware of the daunting mathematical difficulties facing the approach of Jordan ), the solution of a problem in the new quantum mechanics required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Related to their disagreement about the appropriate general formalism for the new theory, Jordan and von Neumann stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan was the first to state those rules in full generality, von Neumann rephrased them and then sought to derive them from more basic considerations. In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and Nordheim. We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics. (shrink)