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Gravitation and Cosmology

Quantum-Mechanical Problems Geometrically Represented within Virtual Ring Theory

Alexander P. Yefremov

Published Jul 1, 2018• 0 citations
Publisher

Abstract

Virtual Ring Theory implies replacement of a particle with a ghost ring endowed with certain standards that are used to measure distances and mechanical characteristics in integer numbers. This model is applied to three “classical” quantum-mechanical (QM) problems, 1D-box system, H-atom, and linear oscillator. It is shown that the relevant virtual geometric models entail the same energy levels as follow from the respective exact solutions of the Schrödinger equation.

DOI
10.1134/s0202289318030131

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Virtual Ring Theory implies replacement of a particle with a ghost ring endowed with certain standards that are used to measure distances and mechanical characteristics in integer numbers. This model is applied to three “classical” quantum-mechanical (QM) problems, 1D-box system, H-atom, and linear oscillator. It is shown that the relevant virtual geometric models entail the same energy levels as follow from the respective exact solutions of the Schrödinger equation.
The fractal equations of mechanics (quantum and classical) are clearly demonstrated to be definitions of an arbitrary potential on a fractal complex number valued surface. The developed approach helps us to show that a translational motion of any rotating compact object (point-like particle) can be equivalently represented by a specific rotation of a virtual ring described in terms of a fractal “wave function”, the model endowing the particle with a set of quantum characteristics including quantization of the ring’s space translation.
This chapter exclusively deals with the application of Schrödinger equation on various systems such as the particle in a box; the particle in a rectangular three-dimensional box; energy levels for a cubic potential box; the tunnel effect; the final equation for tunnel effect, that is, transparency factor which gives a clue to understand that the probability of transmission decreases rapidly as the width of the barrier increases; quantum mechanical explanation of emission of α -particle; the particle on a ring considering spherical polar co-ordinates; and the particle on sphere. This chapter also comprises the Legendre polynomials, associated Legendre equation, associated Legendre function and spherical harmonics, Hermite polynomials and its applications, and orthogonal properties of Hermite polynomials. It also provides the standard topics like simple linear harmonic oscillator, and its classical and quantum mechanical treatment which also includes the asymptotic and series solutions and energy levels of a linear harmonic oscillator. The Schrödinger equation also plays with rigid rotors comprising its rigorous treatment and also the F and T equations which are very important. The chapter ends with the references, standard several computational problems, questions on concepts, and also standard numerical problems for the benefits of the readers.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
We present a revolutionary theoretical framework that interprets atoms as vibrating ring systems, where quantum numbers correspond directly to classical vibration parameters. This geometric approach provides an intuitive visualization of quantum mechanics while maintaining predictive power comparable to established models. The theory successfully explains hydrogen atom energy levels (0.04% error), predicts atomic radii (9.8% average error), and reproduces periodic trends through effective nuclear charge. By mapping principal quantum number n to harmonic number, angular quantum number l to angular nodes, and magnetic quantum number m to phase patterns, we establish a direct correspondence between quantum formalism and classical vibration modes. The model demonstrates excellent agreement with experimental data while offering unprecedented visual intuition for atomic structure and chemical bonding.Keywords: Quantum mechanics, Atomic structure, Vibration theory, Geometric interpretation, Periodic trends, Quantum numbers visualization
The Geometric Algebra formalism allows for the development of a theory to replace conventional quantum mechanics. Generalizations resulting from the replacement of complex numbers with geometrically feasible three-dimensional objects, followed by unambiguous definitions of states, observables, and measurements, bring into reality clear explanations of strange quantum mechanical features, such as primitively considering atoms as a type of solar system. The three-sphere S\mathbb{S}S3 is transformed into a playground for torsion kind states that do not require abstract Hilbert space vectors. The points S\mathbb{S}S3 evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.