Question

Use logic and reasoning to contradict Bohr’s correspondence principle

Answer 1

In modern literature the Correspondence principle is often defined as the requirement for the quantum theory to go over to the classical theories in the limit of large quantum numbers, or in other words as . Already in the time of Bohr it was the trend to perceive the principle in this matter, and to use it accordingly. Nobelprize winner Max Born also cites Bohr’s correspondence principle as such. In his lectures of 1933 Born states that; “die neue Mechanik für den Grenzfall großer Massen oder großer Bahndimensionen in die klassische Mechanik übergeht”. In 1925, in the paper The Fundamental Equations of Quantum Mechanics, P. A. M. Dirac named the correspondence principle as the requirement that: “the classical theory gives the right results in the limiting case when the action per cycle of the system is large compared to Planck’s constant , and in certain other special cases.” According to these formulations, the correspondence principle functions as a constraint on theorizing; this means that it only acts in a limit. However this correspondence limit sometimes called ‘Planck’s correspondence principle’ does not capture the full body of the correspondence principle as defined by Niels Bohr. Léon Rosenfeld recalled that he once suggested to Bohr that Planck’s found correspondence between classical theory and quantum theory was the first germ of the correspondence principle, but that Bohr disagreed fervently: ‘He said emphatically that; “it is not the correspondence argument. The requirement that the quantum theory should go over to the classical description for low modes of frequency, is not at all a principle. It is an obvious requirement of the theory”. But if it is not as Bohr himself states a limiting principle, what is the full definition of the correspondence principle?
While doing a survey on the work of Bohr we will find that there is quite some ambiguity surrounding the correspondence principle and that it may not be fully clear what this principle exactly covers. We will see that for this reason the correspondence principle is often ill understood by fellow scientists. It is therefore important to differentiate between Bohr’s own understanding of the correspondence principle and how his peers eventually used it. In the first chapter we shall mainly focus on Bohr’s own writing on the correspondence principle and in the second chapter elaborate on what status it held in the larger physics community. Also I want to give some special attention to the philosophical considerations of Bohr in chapter 3, as I feel that this philosophical side shaped him as a physicist. I shall refer to earlier papers on the correspondence principle and elaborate on what these papers conclude. It is worth mentioning that Bohr’s statements concerning the correspondence principle changed considerably in his later works and therefore it does not reflect well the way he thought about the correspondence principle in the beginning of his career. For this reason we will leave out his later statements concerning the correspondence principle in the first two chapters and focus primarily on the time period between 1913-1925.
A. Bokulich divided the formulations made by Bohr on the correspondence principle into three categories; ‘first we have the frequency theorem interpretation, according to which the correspondence principle is an agreement between the Fourier decomposition of the classically defined frequency and the quantum frequency. This interpretation holds in the limit of large quantum numbers and is therefore only an asymptotic agreement. Secondly, there is the intensity interpretation, which signifies the thought that there is a statistical relation between the classical intensity and the quantum intensity. The quantum intensity is then depending on the number of quantum transitions and therefore the probability for this transition to occur. The classical intensity is equal to the square of one component of the classical described motion (Fourier decomposition). Finally, there is also the selection rule interpretation, which is to be understood as the principle that each allowed quantum transition corresponds to one harmonic component in the classical motion. We will see that in the works of Bohr and his peers we will encounter all three interpretations of which we will try to define ‘the’ correspondence principle. I make use of these three interpretations when analyzing Bohr’s work, as they are generally used in literature surrounding Bohr’s correspondence principle. Although not all Bohr’s remarks on the correspondence principle can be subdivided into these categories, it is good as a basic to think about whilst trying to define the correspondence principle. However the use of these three interpretations will actually also underline the point I will make; that the correspondence principle is very difficult to place under one coherent interpretation. I will also show in the paper that the correspondence principle was in its first formulations more a mathematical principle, although not clearly stated in one version by Bohr, and that it slowly changed. At the end of 1925 we will see that it had become more of a guiding principle, which helped Bohr and his peers to look in the right direction for answers surrounding atomic questions. So the role and the formulation of the correspondence principle underwent changes that can be brought back to the changes in the atomic model on which it was based.

Niels Bohr was a Danish physicist who lived from 1885 until 1962. He was born and died in Copenhagen. In 1922 he won the Nobel Prize in physics and is best known for his work on quantum physics. Besides his scientific work, Bohr spent a great deal of his time on the philosophical interpretation of it; in his later years he even became more known for these philosophical contributions than for his contributions to physics. He was in the beginning of his career first and foremost a physicist. His lasting contributions to the quantum theory included the Copenhagen interpretation, the Bohr atom, the thesis of complementarity, and the correspondence principle. The correspondence principle can be called one of the most fascinating and interesting parts of Bohr’s philosophy. It reflects perfectly his personal way of thinking and it was a prime example of what Einstein called Bohr’s ‘musicality’ and ‘unique instinct and tact’. To understand the basics correspondence principle one first needs to understand the context in which it was formed: the old quantum theory.
The old quantum theory was an “interim” theory, developed on the basis that classical theories could not fully describe atomic systems. Max Jammer, an Israeli physicist and philosopher of physics, aptly describes the pre-1925 'theory' as follows:

“Every single quantum-theoretic problem had to be solved first in terms of classical physics; its classical solution had then to pass through the mysterious sieve of quantum conditions or, as it happened in the majority of cases, the classical solution had to be translated into the language of quanta in conformance with the correspondence principle. Usually, the process of finding "the correct translation" was a matter of skilful guessing and intuition rather than of deductive and systematic reasoning.”

Especially this ‘mysterious sieve of quantum conditions’ was a part of the old quantum theory that was more based on the intuition of a scientist, than on hard mathematical rules. This was also the reason that many scientists first rejected the quantum theory.
The first theory that seemed to solve the problem that classical theories could not describe atomic systems was proposed by Niels Bohr in the Philosophical Magazine of July 1913 in a three-part paper titled ‘On the Constitution of Atoms and Molecules’. It is called an ‘epoch-defining paper’ and caused the first definite split with the classical theories. In the paper Bohr articulated the postulates of the old quantum theory. In a long letter to Ernest Rutherford, a British physicist known for his great experimental contributions to the atomic theory, on March 6, 1913 he sent his paper and wrote:

"Enclosed I send the first chapter of my paper on the constitution of atoms. [...] I hope that you will find that I have taken a reasonable point of view as to the delicate question of the simultaneous use of the old mechanics and of the new assumptions introduced by Planck's theory of radiation. I am very anxious to know what you may think of it all...".

We see that Bohr knew that his blending of quantum and classical ideas was highly ‘delicate’, but he defined the rules and combined the two theories in an ingenious manner. Bohr used in his paper Ernest Rutherford's model of the atom, according to which a relatively high central charge is concentrated into a very small volume in comparison to the rest of the atom (the nucleus) and with this central volume also containing the bulk of the atomic mass of the atom. The electrons orbit the nucleus in planetary trajectories. However, in an attempt to explain some of the properties of matter with this atom-model Bohr encountered a problem; it was unstable according to classical electrodynamics. According to the model, the electron, an accelerated charged body, radiates energy in such a manner that it would eventually collapse into the nucleus. Bohr's solution to this problem was to incorporate Max Planck's theory:

"The result of the discussion of these questions seems to be a general acknowledgment of the inadequacy of the classical electrodynamics in describing the behavior of systems of atomic size. Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant".

Bohr summarized his quantum theory by means of two assumptions or postulates:

“(1) That the dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis.
(2) That the latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory."
So the atomic systems can only exist in a series of individual states, which can be thought of as stable periodic orbits around the nucleus. Bohr also often compared these orbits to the orbits of our planetary system, from which he got his inspiration. These periodic orbits are labeled by the principal quantum number , with the lowest orbit labeled , the next etc. When the electron is in one of these states, its motion can be correctly described by classical mechanics. However, when an electron makes a jump from one state to the next, the theory no longer applies. So we see a mix of quantum and classical theories. Whereas the first postulate still relies on the classical mechanics as defined by Isaac Newton, the second postulate is the official break with the classical theories. We will see that, in later papers, this second postulate will develop in to the frequency interpretation.
Caution is needed when we speak of the quantum jumps made from the stationary states, because it is now generally accepted that a photon is released, however Bohr himself did not embrace this theory up until the mid-1920s. Bohr preferred to think of the emitted radiation as a wave, rather than a particle. He stated his distrust of light quanta clearly in his lecture in 1922 where he said that: “in spite of its heuristic value, however, the hypothesis of light-quanta, which is quite irreconcilable with so-called interference phenomena, is not able to throw light on the nature of radiation.” Bohr’s resistance for accepting the particle theory was also tied to the importance of being able to analyze radiation into its harmonic components, which was in essence tied to the basis of the correspondence principle. As John Stachel notes: “It was indeed, his reliance on the correspondence principle that seems to have been a principal motive for Bohr's distrust of the photon concept”. In the end Bohr would be led to embrace the concept of the photon and recognize its heuristic value, although not before the collapse of his atomic model. we already saw that there was quite some flexibility in the definition of the Correspondence principle and that there was not one coherent interpretation of it. If the principle was however not even definably clear in the work of its founder Bohr, how did his peers perceive it? It is important to understand the distinction between the correspondence principle in Bohr’s own understanding and what it became to mean for the physics community of his time. We can divide the responses to Bohr’s correspondence principle into roughly three categories; ‘those who misunderstood the principle (e.g., Born and Rosenfeld), those who embraced and developed it (e.g., Kramers and van Vleck), and those who seemed to understand it, but nevertheless mistrusted it (e.g., Sommerfeld and Pauli)’. We cannot say who understood the correspondence principle completely and if it was even possible to do so, as it was so flexible in its definition. Even when we look at the supporters of the correspondence principle we can see that there were quite a few whom, according to Bohr, didn’t fully grasp the correspondence principle. For example Léon Rosenfeld, a Belgian physicist and collaborator of Bohr, wrote about Bohr’s annoyance ‘over his failure to have correctly understood the substance of this principle’.
Bohr himself was quite aware of the misinterpretations that his often-difficult written essays on the topic could induce. He realized a lot of physicists regarded his work as being ‘philosophical’ and ambiguous. For example he wrote to Sommerfeld: “I should not like you to get the impression that my inclination to pursue obscure, and undoubtedly often false analogies makes me blind to that part of the formation of our conceptions that lies in unveiling of the systematic of the facts. Even if I were blind, I would only need to glance at your book to be healed”. The book Bohr talks about was Sommerfeld’s famous Atombau that at that time gave one of the most complete overviews of the quantum theory.
There is indeed quite an obscurity surrounding the correspondence principle and its use, which persists even in today’s literature. A possible explanation could be that Bohr’s general writing style was complex and sometimes difficult to read, seeing how Bohr’s process of writing was quite different than most scientists. A common writing process is first thinking a thought through before writing it down, for Bohr writing was an important part of his cognitive process. He tried to explain his thoughts in his papers with as much clarity as possible, which sometimes led to a painful process vividly described by Heisenberg: “Bohr would always change the sentences again and again. He could have filled half a page with a few sentences and then everything was crossed out and changed again. And even when the whole paper was almost finished- say, ten pages or so – the next day everything would be changed over again. So it was a continuous process of improvement, change and discussions with others… The final text of Bohr’s paper was so subtle and he would think about half an hour whether in a certain case he would use indikativ or konjunktiv and so”. We can see that Bohr considered it important to clarify his thoughts as much as possible, however, this sometimes had the opposite effect, making it even more ambiguous. Could it have happened that, because of this ambiguity and the certain flexibility in the interpretation of Bohr’s texts on the correspondence principle, Bohr’s original principle underwent supplementation, modification and metamorphosis? Did it come to have a different function in the physics community than originally intended by Bohr? Multiple physicists, for example Hendrik Kramers and John van Vleck, used and extended the term “correspondence principle” to cover a wide range of correspondence-type arguments that were important in the development of quantum mechanics, but which were not necessarily implied in the formulations of Bohr.
Among the physicists whom ‘correctly’ understood Bohr’s principle there were also a lot of critics. Their main argument against Bohr’s atomic theory, not entirely unjustified, was that the basis of it was a messy, patchy combination of classical and quantum elements. Or as Henry Mergenau later phrased it: “Bohr’s atom sat like a baroque tower upon the Gothic base of classical electrodynamics”. Among skeptics the most noticeable were Arnold Sommerfeld and Wolfgang Pauli.
In this chapter I will try to give some insight into the question “how was the correspondence principle, as formulated in the old quantum theory, received by the physics community?” I will pay attention specifically to the thoughts of the physicists who were closely involved in the development of the quantum theory such as; Sommerfeld, Heisenberg, Pauli, Kramers and others. I believe by looking at the general responses on the correspondence principle of Bohr, we can obtain more insight in the correspondence principle and how it was used and altered.