The path-integral formulation of quantum mechanics has promoted the study of classical periodic trajectories. In this spirit, the three-body Coulomb problem with a nucleus of infinite mass is considered in the special case of a rigid rotation. Using only Newton's fundamental law for a classical atomic system, one can show that both electrons must be equidistant from the nucleus, and one obtains the solutions previously derived by Klar [Phys. Rev. Lett. 57, 66 (1986); Z. Phys. D 3, 353 (1986); J. Opt. Soc. Am. B 4, 788 (1987)]. Nevertheless, if one allows the charge at rest to be arbitrary, one finds that for Z less than unity unsymmetrical solutions may exist. The stability of each of these configurations is considered. In each case an exponentially increasing solution is present, but the unstable character is less marked in the unsymmetrical configuration. Possible consequences for atomic systems are discussed.
The path-integral formulation of quantum mechanics has promoted the study of classical periodic trajectories. In this spirit, the three-body Coulomb problem with a nucleus of infinite mass is considered in the special case of a rigid rotation. Using only Newton's fundamental law for a classical atomic system, one can show that both electrons must be equidistant from the nucleus, and one obtains the solutions previously derived by Klar [Phys. Rev. Lett. 57, 66 (1986); Z. Phys. D 3, 353 (1986); J. Opt. Soc. Am. B 4, 788 (1987)]. Nevertheless, if one allows the charge at rest to be arbitrary, one finds that for Z less than unity unsymmetrical solutions may exist. The stability of each of these configurations is considered. In each case an exponentially increasing solution is present, but the unstable character is less marked in the unsymmetrical configuration. Possible consequences for atomic systems are discussed.
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.
In this work, the quantum-mechanical problem of the motion of two material points of different masses on a three-dimensional sphere with a non-fixed position of the center of mass of the system is formulated on the basis of the previously solved classical problem. It is shown that the established Schrödinger equation includes two different reduced masses, depending on the distance between the points. For the case of the interaction potential of points, depending only on the distance between them, this equation allows the separation of variables into a radial, depending on the relative distance and both the reduced masses and the spherical part. The equation for the spherical part depends only on one of the above reduced mass and allows one to formulate and solve the problem of a rigid rotator - the distance between the points is fixed. The solution and spectrum of the problem of a rigid rotator are found. It is shown that the spectrum of the system has an upper limit that does not depend on the distance between points, in contrast to the spectrum in a flat space.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
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.
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 fact that an orbiting electron does not collapse into the nucleus of its parent atom has thus far been considered a major obstacle to a classical interpretation of the stable states of orbiting electrons. Quantum mechanics avoids the very problem by discussing the probability of finding it, a method that confessedly has been very fruitful in the exploring the behaviour of elementary particles. Nonetheless, why should necessarily those two approaches be regarded as each others' enemies? In this paper it is shown that the classical mechanistic approach is still capable of explaining the eternal, circular movement of an electron around a nucleus. It is possible if reformulating the laws of action involved, returning to the simple electrostatic model, based upon Coulomb's law (1785, 1771). Further, in this paper it is also discussed, how electromagnetic radiation due to the de-excitation of excited electrons, orbiting around a positive nucleus, can be explained classically, as a sudden peak in the otherwise zero electric field, due to the inwards spiralling movement connected to the de-excitation. The concept of a distinct particle, the 'photon' is thereby rejected.