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Physical Review

An Extension of Bohr's Correspondence Principle to Apply to Small Quantum Numbers

Pierce Waddell Ketchum

Published Nov 1, 1924• 2 citations
Publisher

Abstract

Generalized correspondence principle for the Bohr atom.---It is shown that when an electron spirals in from an orbit of quantum number ${n}_{2}$ to an orbit of quantum number ${n}_{1}$, the frequency of the radiation emitted is equal to an integer (${n}_{2}\ensuremath{-}{n}_{1}$) times the mean frequency of rotations in its orbit during the spiraling, assuming that the number of rotations made for a change of quantum level $\mathrm{dn}$ is proportional to $\mathrm{dn}$. That is, the mean is defined as the ratio to (${n}_{2}\ensuremath{-}{n}_{1}$) of the integral from ${n}_{1}$ to ${n}_{2}$ of ${\ensuremath{\nu}}_{\mathrm{c}} \mathrm{dn}$ where ${\ensuremath{\nu}}_{c}$ is the classical frequency of rotation corresponding to any value of $n$. No explanation is offered as to why radiation begins and stops at integral values of $n$, nor why the radiation emitted in the manner assumed is monochromatic.

DOI
10.1103/physrev.24.463

Related papers

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•Mar 3, 2017•0 citations
Generalized correspondence principle for the Bohr atom.---It is shown that when an electron spirals in from an orbit of quantum number n2{n}_{2}n2​ to an orbit of quantum number n1{n}_{1}n1​, the frequency of the radiation emitted is equal to an integer (n2\ensuremath−n1{n}_{2}\ensuremath{-}{n}_{1}n2​\ensuremath−n1​) times the mean frequency of rotations in its orbit during the spiraling, assuming that the number of rotations made for a change of quantum level dn\mathrm{dn}dn is proportional to dn\mathrm{dn}dn. That is, the mean is defined as the ratio to (n2\ensuremath−n1{n}_{2}\ensuremath{-}{n}_{1}n2​\ensuremath−n1​) of the integral from n1{n}_{1}n1​ to n2{n}_{2}n2​ of \ensuremathνcdn{\ensuremath{\nu}}_{\mathrm{c}} \mathrm{dn}\ensuremathνc​dn where \ensuremathνc{\ensuremath{\nu}}_{c}\ensuremathνc​ is the classical frequency of rotation corresponding to any value of nnn. No explanation is offered as to why radiation begins and stops at integral values of nnn, nor why the radiation emitted in the manner assumed is monochromatic.
Correspondence theorem for frequencies.---It is well known that the frequency emitted by a hydrogen atom as it falls from one of its quantized states to another may be expressed as a mean value of the frequency of motion of the electron (or overtone thereof) when averaged in the proper manner over orbits intermediate between the initial and final states. In the present paper it is proved that, similarly, the Compton shift is a properly taken mean of the classical Doppler shift. The quantum frequency actually scattered is thus a properly taken average of the frequency which would be scattered on the classical theory as the electron is accelerated from its state of rest to its final recoil condition.Correspondence principle for intensities.---In like manner, the amount of light scattered in various directions may be determined if it is assumed that the intensity in the quantum theory equals a proper average of the intensities scattered according to the classical theory. A comparison is made with observed data on the scattering of $\ensuremath{\gamma}$-rays.The characteristic feature of the present paper is that the corresponding classical electron is assumed to have the same direction of motion as the scattered quantum, whereas an actual classical electron would from symmetry recoil straight forward in the direction of the incident beam, as in Compton's and Woo's theories of intensities. This new point of view eliminates the difficulty of a constant correction-factor which has been encountered by Compton and Woo in their explanation of intensity relations.
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.
Turning back to Niels Bohr's model of the hydrogen atom, the paper is trying to resolve one of the issues associated with the model.While N. Bohr in the framework of his model did not find any relation between electron's motion in the atom and radiation emitted, there is a reason to believe that the relation is bound to exist.Furthermore, it is already given by Rydberg's formula that expresses the frequency of radiation as a difference of two other frequencies associated in some manner with electron's rotation.Based on this approach a new value for the radius of the ground stationary orbit is calculated.It appears to be more than Bohr's radius by a factor of 1.588.The question why both values for the radius of the ground stationary orbit (Bohr's radius and a new one) are compatible with Rydberg's formula is considered.Finally, what exactly Niels Bohr overlooked in his model of the hydrogen atom, is discussed.It is established that Rydberg's formula can be derived from the more general parameterized postulates, when Niels Bohr used, regardless of the value of the parameter.
In his doctoral thesis in 1911, Bohr reached the conclusion that at thermal equilibrium metal atom has no magnetic effect. Therefore, when establishing the hydrogen atom model in 1913, he did not consider the Lorentz force, but only the Coulomb force. The radius of the hydrogen atom he obtained is half smaller, and the orbital rotation frequency is twice larger. It must be multiplied by 1/2 to be consistent with the value of the Rydberg experiment. He proposed to calculate the photon frequency by using the average value of the zero frequency of infinite quantum number and the basic frequency, but it is not applicable to the energy levels for the ordinary quantum numbers, and later he called it the correspondence principle. In our modified Bohr model, we eliminate and solve Bohr's half frequency problem; The photon energy is equal to the difference between the two energy levels, and the photon frequency is equal to the difference between the two rotation frequencies of the electron; All excited states of hydrogen atom are elliptical orbits, and the nucleus is located in the center of the ellipse; The microscopic model of photonic structure is established, and the microscopic mechanism of atomic luminescence and quantum entanglement is given.
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.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.