The world should have waited until 1913, when a Danish physicist N. Bohr proposed an entirely new theory for a description and partial explanation of the spectral properties of the hydrogen atom. His theory was based on two postulates, the so-called Bohr postulates, and both of them were impossible to be understood from the point of view of classical physics. Being based on these postulates and employing classical physics, N. Bohr was able to derive explicit expressions for the energy levels of the hydrogen atom and the wavelengths of possible transitions. The hydrogen-like atom has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.
The world should have waited until 1913, when a Danish physicist N. Bohr proposed an entirely new theory for a description and partial explanation of the spectral properties of the hydrogen atom. His theory was based on two postulates, the so-called Bohr postulates, and both of them were impossible to be understood from the point of view of classical physics. Being based on these postulates and employing classical physics, N. Bohr was able to derive explicit expressions for the energy levels of the hydrogen atom and the wavelengths of possible transitions. The hydrogen-like atom has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
This chapter has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. By limiting the orbiting electrons to a series of circular orbits having discrete radii, Bohr could account for the series of discrete wavelengths in the emission spectrum of hydrogen. Light, he proposed, radiated from hydrogen atoms only when an electron made a transition from an outer orbit to one closer to the nucleus. These physical properties (Ve = 174090.938ms-1& Pe = 1.585860888 × 10-25kgms-1) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength (λ0 = 91.12nm) and the Balmer’s experimental observation (λ0 = 91.18nm) for the emission spectrum of hydrogen atom. There are two other constants δ = 7.199822 × 10-10 eVm and ζ = 1503796277m-1 that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants can be applied to multi-electron systems because they are inherent characteristics of an electron. They can also be derived from the steady state hydrogen atom Schrodinger's equation. For a deeper understanding, these constants may be scrutinised for their determination. The study recommends that current quantum atomic models and equations governing quantum physics should be revisited for better applications and understanding of the physical world.
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.
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.
In this chapter we are going to make a very detailed study of the one-electron atom using relativistic quantum mechanics. The one-electron atom is the simplest bound system that occurs in nature, and plays a central role in both classical and quantum theory. There are two reasons for this. Firstly, it is a model that can be solved exactly at many levels of theory — the classical Bohr atom, the non-relativistic quantum mechanical one-electron atom, and the relativistic one-electron atom with spin zero and spin 1/2. Secondly, most of our understanding of multielectron atoms, molecules and solids is based on this model. Of course, a good description of the hydrogen atom is provided by non-relativistic Schrödinger theory. However, once the relativistic quantum theory had been discovered it was of great importance to validate the theory by describing hydrogen at least as well as Schrödinger theory. As we shall see, the Dirac theory was exceedingly successful in this, and furthermore, at several points in the theory the non-relativistic limit can be taken and the effect of relativity in one-electron atoms can be seen explicitly. This connection with the simpler theory will be made here wherever insight may be gained from doing so. One of the key advances brought about by the advent of quantum theory was that it defined eigenfunctions as well as the eigenvalues predicted by the Bohr theory.
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.