Niels Bohr's development of the quantum theory of the atom and the correspondence principle (his 1912–1923 work in atomic physics and its significance)
Niels Bohr's studies on the quantum theory of the atom over the years 1912–1923 are discussed. The sources of his work on these questions are analyzed. The beginning of his quantum studies in 1912 is described. His famous paper on the theory of the hydrogen atom and on the origin of spectra is analyzed in detail. The subsequent development of his ideas regarding atomic structure is discussed, with an emphasis on his postulates regarding stationary states and radiative transitions and on the development of the correspondence principle. How well he understood the difficulties of the model theory and how he strived for a profound understanding of quantum phenomena are shown.
Niels Bohr's studies on the quantum theory of the atom over the years 1912–1923 are discussed. The sources of his work on these questions are analyzed. The beginning of his quantum studies in 1912 is described. His famous paper on the theory of the hydrogen atom and on the origin of spectra is analyzed in detail. The subsequent development of his ideas regarding atomic structure is discussed, with an emphasis on his postulates regarding stationary states and radiative transitions and on the development of the correspondence principle. How well he understood the difficulties of the model theory and how he strived for a profound understanding of quantum phenomena are shown.
Bohr's 1913 trilogy marks a turning point in the history of quantum physics for its introduction of the planetary model of the atom and its derivation of the Balmer formula for the hydrogen spectrum. It has been studied in the seminal study 1 and is discussed more recently by 2-4. Famously, Bohr's classical paper made two utterly non-classical assumptions, according to which electrons orbited the nucleus in stationary states without losing energy while they emitted radiation only in transitions between these states. As I will argue in this paper, Bohr's 1913 trilogy made a third crucial major assumption, which entailed key elements of the correspondence principle formulated five years later. This third assumption postulated a relation between the mechanical frequency of a stationary state and the radiation frequency in analogy with classical radiation theory. Though this relation was in direct conflict with the key assumptions of stationary states and transitions, Bohr's reasoning—as I want to show—relied heavily on it, and a major part of his initial work was devoted to clarifying how the connection of radiation and motion, the stationary states and the transitions were interrelated. The main aim of Bohr's 1913 trilogy was to derive the Balmer formula from the newly-proposed atomic model. After the introduction of the stationary states and the frequency condition, the derivation follows a three-step recipe consisting of the calculation of the energy from classical mechanics, the quantization of the energy and finally the calculation of the spectral frequencies via the Planck Relation . Given the simplicity of the system considered and the assumption of the stationary states and the frequency condition, the first and last step did not call for an extended argument and Bohr simply gave the relevant equations. As Bohr explicitly acknowledged, the classical radiation mechanism was replaced entirely by the fundamental assumptions of his model: radiation is produced through a transition between stationary states. Though 1 was in direct conflict with the physical assumptions of his model, Bohr's thinking nonetheless depended crucially on it. This is emphasized by the fact that Bohr knew that he could quantize the orbits on the assumption that the angular momentum of the atom is equal to .1 While 1 could not be interpreted as a description of a physical process in a classical sense, Bohr kept the equation and explored its conceptual significance within his new quantum theory of the atom. This was a vital concern for Bohr, first, because he needed a justification; second, because he hoped for insights on how to develop the theory by investigating which parts of classical radiation theory could still be used within the new model. The investigation of this question loomed large in Bohr's 1913 paper and continued to play an important role in his early work on the quantum theory of the atom. For the preceding argument Bohr justified the radiation-motion relation by means of a comparison between the radiation process and the motion of the electron before and after the radiation process. This justification itself was not very convincing: it rested solely on its empirical success and the questionable mean-value argument, and Bohr decided to give a second derivation of the Balmer formula to add further plausibility to 1. Already his first hand waving argument, however, points to some of the central aspects of his attempt to connect radiation and electronic motion: In light of the missing radiation mechanism in quantum theory, Bohr tried to recover a relation between radiation and electronic motion within quantum theory analogous to the one in classical radiation theory. He refrained from proposing a new hypothetical radiation mechanism and shied away from such assumptions until 1924. Instead he resorted to experimental, mathematical and intertheoretical arguments to construct a formal relation between radiation and matter and aimed to get insights into the properties of a radiation mechanism yet to be found. In order to be equivalent to the empirical Balmer formula this function has to be , as Bohr showed by comparing the frequency of a quantum transition and the orbiting frequencies of the stationary states in the case of high quantum numbers, in which classical electrodynamics and quantum theory predict the same numerical value for the radiation frequency. This result implied the consistency of his major assumptions, the radiation-motion connection and the Balmer formula, which entered as an empirical constraint. For the argument the high quantum number limit plays an important role as it renders the condition for the consistency precisely and shows that the assumption of a relation between radiation and motion within quantum theory leads to the agreement of quantum theory and classical theory in this limit. The possibility of an emission of a radiation of such a frequency may also be interpreted from analogy with ordinary electrodynamics, as an electron rotating round a nucleus in an elliptical orbit will emit a radiation which according to Fourier's theorem can be resolved into homogeneous components, the frequencies of which are [5, 14]. This recapturing of the relation between orbital frequency and the radiation frequency is part of Bohr's conclusion: While classical electrodynamics in general is in contradiction with the basic assumptions of the Bohr model, the classical relation between the frequency of radiation and the mechanical frequency can be established at least in the classical limit as a formal relation. To show this is as important to Bohr as to show that his new model predicts the correct Balmer formula. As such the classical relation between motion and radiation is indispensable for Bohr's early work as a touchstone for his two derivations and it is of vital concern in the attempt to explore the break and the continuity with classical radiation theory. The radiation-motion relation continued to play an important role in Bohr's works following the 1913 trilogy; after he had finished his initial papers Bohr adapted his model to account for the Stark and Zeeman effects. The relation between radiation and motion played a major role in the initial work establishing the Bohr model and its extension to the Stark and Zeeman effect. Stemming from classical radiation theory this relation was in contradiction with the major assumptions of the new model of the atom; nonetheless, Bohr's thinking relied heavily on the radiation-motion connection both in establishing his results and in clarifying the foundations of the new model. Compared with his later statements on the correspondence principle there is a remarkable resemblance between the statements in 1913/1914 and the correspondence idea between 1918 and 1923: In both cases the relation between the orbiting frequency and the frequency of radiation and the asymptotic limit between quantum theory and classical electrodynamics were at the core of Bohr's reasoning. Also the idea of the Fourier series is mentioned, which would become ever more important in the development of the correspondence idea.5 While the core elements of the later principle are already apparent in the early phase, they are used very differently in both phases. This shift in the usage of the basic correspondence relation was due to the development of the quantum theory of the atom as a whole. The physicists who developed Bohr's initial model mainly extended the description of the stationary states to more complicated systems with the help of celestial mechanics. In this context the relation between the radiation frequency and the orbiting frequency was replaced by Sommerfeld's quantum condition: . The new relation did not involve any assumptions on the radiation process or the interaction between radiation and matter, it was a statement about the action of a mechanical system and its quantization. The fruitfulness of this new quantization condition led physicists including Bohr to abandon the early quantization procedure. Through this abandonment, the frequency condition became the sole relation of radiation theory, which was needed to determine the series formulae describing atomic spectra. Whereas the loss of the classical radiation mechanism had been a vital concern for Bohr in his early works, this was no longer the case in the development of the quantum theory of the atom. The relation should, however, resurface in Bohr's thinking about the transitions in quantum theory and in this form led to the formulation of the correspondence principle.
It is widely known that Paul Ehrenfest formulated and applied his adiabatic hypothesis in the early 1910s. Niels Bohr, in his first attempt to construct a quantum theory in 1916, used it for fundamental purposes in a paper which eventually did not reach the press. He decided not to publish it after having received the new results by Sommerfeld in Munich. Two years later, Bohr published "On the quantum theory of line-spectra." There, the adiabatic hypothesis played an important role, although it appeared with another name: the principle of mechanical transformability. In the subsequent variations of his theory, Bohr never suppressed this principle completely. We discuss the role of Ehrenfest's principle in the works of Bohr, paying special attention to its relation to the correspondence principle. We will also consider how Ehrenfest faced Bohr's uses of his more celebrated contribution to quantum theory, as well as his own participation in the spreading of Bohr's ideas.
This book gives a detailed study of the development and the interpretation given to Niels Bohr's Principle of Correspondence. It also describes the role that this principle played in guiding Bohr's research over the critical period from 1920 to 1927. Quantum mechanics, developed in the 1920s and 1930s by Bohr, Heisenberg, Born, Schrödinger and Dirac, represents one of the most profound turning points in science. This theory required a wholly new kind of physics in which many of the principles, concepts and models representing reality, that had formed the basis of classical physics since Galileo and Newton, had to be abandoned. This book re-examines the birth of quantum mechanics, in particular examining the development of crucial and original insights of Niels Bohr.
Many teachers like to introduce the Bohr atom toward the end of an introductory physics course. This is an excellent idea, given the historic importance of Bohr’s 1913 work, which provided the bridge from Planck’s quantized interaction of matter and radiation (1900) to the full theory of quantum mechanics (1925-28). Unfortunately, the version of the Bohr atom that appears in many textbooks and is no doubt often presented to students is more wrong than right and may leave both teachers and students wondering why, more than a hundred years later, it is still being taught. This “pedagogic” version postulates that an electron in a stationary state moves in a circular orbit with an angular momentum that is an integral multiple of h/2π (L = nh/2π = nħ)—ħ for the lowest-energy state, 2ħ for the next state, and so on. This picture of the hydrogen atom is wrong in two senses. First it doesn’t conform to our present understanding of the hydrogen atom. This, in itself, is not a reason to scrap it, for the historical development of quantum physics is certainly of interest. But second, it doesn’t conform to the essence of what Bohr actually did. That is a reason not to teach about circular orbits and L = nħ.
In a major work of 1918 (‘On the quantum theory of line-spectra’) Bohr reconsidered the general principles of quantum and atomic theory. Apart from reinterpreting his original theory, he introduced the correspondence principle as a crucial connection between quantum and classical physics. What was the nature of this principle and what role did it play in the understanding of atomic structure? In applying the new principle, Bohr and his assistant H. A. Kramers relied on Einstein’s radiation theory. Bohr used the correspondence principle as a selection principle, and Kramers showed how it could account quantitatively for the Stark effect. However, not all physicists appreciated the Copenhagen ‘magic wand’ and Bohr’s use of it. Sommerfeld was the most prominent of the sceptics, but not the only one. The chapter also briefly considers the role of the correspondence principle in the early phase of the new quantum mechanics.