It is shown that the approximation of a strong Coulomb interaction between electrons results in a new model of the atom with a spatial quantization of electrons accompanied by their quantization in energy. This model implies that electrons rotate in circular orbits centered outside the atomic nucleus and only orbit axes pass through it. The Coulomb interaction between electrons leads to a spherically symmetric distribution of their orbits on the surfaces of equipotential spheres of a spherically symmetric electrostatic field of the nucleus. The distribution is similar to “inscribing” electron orbits into faces of regular nucleus-centered polyhedra so each polyhedron corresponds to a certain electron state (s, p, d, f), and a certain set of polyhedra corresponds to a certain period of the Mendeleev Table. It is shown that a spherically symmetric distribution of electron orbits gives rise to the formation of electron pairs in which electron orbits with a common axis are located symmetrically with respect to the nucleus and the orbital magnetic moments of the electrons are oppositely directed. The physical meaning of the electron spin concept becomes clear. The spin turns out to be related to the orbital magnetic moment of an electron and reflects the fact that two electrons of a pair rotate in opposite directions relative to their common axis. So the spin is one of characteristics of the electron state in the atom associated with electron rotation in the orbit centered outside the nucleus. The atomic model gives an insight into the periodicity of changes in the atomic properties with increasing nuclear charge and the reasons for an electron double energy quantization associated with different states and periods. The model shows that the atomic structure and properties can be explained by using concepts of classical mechanics and classical electrodynamics which regard the electron as a particle.
It is shown that the approximation of a strong Coulomb interaction between electrons results in a new model of the atom with a spatial quantization of electrons accompanied by their quantization in energy. This model implies that electrons rotate in circular orbits centered outside the atomic nucleus and only orbit axes pass through it. The Coulomb interaction between electrons leads to a spherically symmetric distribution of their orbits on the surfaces of equipotential spheres of a spherically symmetric electrostatic field of the nucleus. The distribution is similar to “inscribing” electron orbits into faces of regular nucleus-centered polyhedra so each polyhedron corresponds to a certain electron state (s, p, d, f), and a certain set of polyhedra corresponds to a certain period of the Mendeleev Table. It is shown that a spherically symmetric distribution of electron orbits gives rise to the formation of electron pairs in which electron orbits with a common axis are located symmetrically with respect to the nucleus and the orbital magnetic moments of the electrons are oppositely directed. The physical meaning of the electron spin concept becomes clear. The spin turns out to be related to the orbital magnetic moment of an electron and reflects the fact that two electrons of a pair rotate in opposite directions relative to their common axis. So the spin is one of characteristics of the electron state in the atom associated with electron rotation in the orbit centered outside the nucleus. The atomic model gives an insight into the periodicity of changes in the atomic properties with increasing nuclear charge and the reasons for an electron double energy quantization associated with different states and periods. The model shows that the atomic structure and properties can be explained by using concepts of classical mechanics and classical electrodynamics which regard the electron as a particle.
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 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 present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.
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.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.