A model is proposed for the hydrogen atom in which the orbiting electron is seen as an objectively real particle. The model is based on the postulate that certain velocity terms can be treated as being affected by relativity. The model provides a mechanism which drives the quantization process and so leads to the discrete energy levels of the atom. The Rydberg formula, being empirically derived, represents the yardstick by which any model for the hydrogen atom must be judged. Rather than develop a model and test it against the Rydberg formula, the approach taken here is to use the Rydberg formula itself as the basis for such a model. The model effectively unifies quantum mechanics with classical mechanics as well as providing a simple mechanical explanation of the Somerfield Fine Structure Constant.
A model is proposed for the hydrogen atom in which the orbiting electron is seen as an objectively real particle. The model is based on the postulate that certain velocity terms can be treated as being affected by relativity. The model provides a mechanism which drives the quantization process and so leads to the discrete energy levels of the atom. The Rydberg formula, being empirically derived, represents the yardstick by which any model for the hydrogen atom must be judged. Rather than develop a model and test it against the Rydberg formula, the approach taken here is to use the Rydberg formula itself as the basis for such a model. The model effectively unifies quantum mechanics with classical mechanics as well as providing a simple mechanical explanation of the Somerfield Fine Structure Constant.
A model is proposed for the hydrogen atom in which the electron is an objectively real particle orbiting at very near to light speed. The model is based on the postulate that certain velocity terms associated with orbiting bodies can be considered as being affected by relativity. This leads to a model for the atom in which the stable electron orbits are associated with orbital velocities where Gamma is n/α, leading to the idea that it is Gamma that is quantized and not angular momentum as in the Bohr and other models. The model provides a mechanism which leads to quantization of energy levels within the atom and also provides a simple mechanical explanation for the Fine Structure Constant. The mechanism is closely associated with the Sampling theorem and the related phenomenon of aliasing developed in the mid-20th century by engineers at Bell labs.
Consideration of the Rydberg Series leads to a model for the hydrogen atom in which the variable of quantisation is the Lorentz factor, Gamma, and not angular momentum as postulated by Niels Bohr and integral to current theories. This is consistent with an atom in which the electron is seen as an objectively real particle having deterministic position and velocity. The electron is seen to orbit at a constant radius irrespective of the energy state of the atom and so there is no change in potential energy between energy states. All of the energy changes are therefore kinetic in nature. The model also provides a simple physical explanation of the hitherto mysterious Fine Structure Constant.
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.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The basis for the hydrogen atom planetary model, which represents the development of the Rutherford-Bohr atom model, is presented. The measurements of the atom polarizability, radius, electric and magnetic moments, as well as the Stark and Zeeman effects are given as an argument. The central paragraph of the Article is a comparison of the binding energy obtained based on wave measurement data (the Rydberg constant) and the binding energy obtained by calculation according to the data of the electron charge and mass fundamental constants and applying the principle of momentum conservation. The substantiation of the atom stability is given proceeding from the atom planetary model. The nature and mechanism of the optical spectrum formation is discussed. The substantiation of the fine structure of the hydrogen atom spectral lines is presented, proceeding from the fundamental association between the electric field strength of the electron charge and the speed of its velocity. The origin of the optical spectrum hyperfine structure is discussed within the planetary model. The difficulty in constructing an acceptable theory of the electronic structure of atoms is primarily due to the complexity of obtaining direct information on the atoms internal structure. The point is that the outermost electron shell of an atom behaves itself like an effective screen for external probing electrical fields. This circumstance sharply limits the possibilities of experimental methods for studying the intra-atomic structure. On the other hand, it is imperative that the research methods being employed be non-destructive ones in order to obtain reliable information concerning the structure of atoms. This means that the action on the atom during measurement process must meet the condition of the electron shell perturbation smallness of the atom or ion being investigated. Concerns associated with the inaccessibility of the electronic structure of atoms served as a pretext for a conclusion of the limited possibilities of the empirical method of knowing the internal structure of atoms at the proper time.
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.