We use 3 equations as postulates: Pem referring to electromagnetism, Pgrav referring to gravity, and Pqm referring to quantum mechanic and defining the wave function. Combining Pem with Sommerfeld's quantum rules corresponds to the original quantum theory of Hydrogen, which produces the correct relativistic energy levels of atoms (Sommerfeld's and Dirac's theories of matter produces the same energy levels, and Schrodinger's theory produces the approximation of those energy levels). Pqm implies that the wave function is solution of both Schrodinger's and Klein-Gordon's equations in the non interacting case while, in the interacting case it implies Sommerfeld's quantum rules: Pem and Pqm then produce the correct relativistic energy levels of atoms (the same as Dirac's energy levels). We check that the required degeneracy is justified by pure deduction, without any other assumption (Schrodinger's theory only justifies one half of the degeneracy). We observe the connection between Pqm, Quantum Field Theories and tunnel effect. From Pgrav we deduce an equation of motion very similar to general relativity (with accuracy 10^{-6} at the surface of the Sun), our postulate being explicitly an approximation. First of all, we discuss classical Kepler problems (Newtonian motion of the Earth around the Sun), explain the link between Kelpler's law of periods (1619) and Plank's law (1900) and observe the links between all historical models of atoms (Bohr, Sommerfeld, Pauli, Schrodinger, Dirac, Fock).
We use 3 equations as postulates: Pem referring to electromagnetism, Pgrav referring to gravity, and Pqm referring to quantum mechanic and defining the wave function. Combining Pem with Sommerfeld's quantum rules corresponds to the original quantum theory of Hydrogen, which produces the correct relativistic energy levels of atoms (Sommerfeld's and Dirac's theories of matter produces the same energy levels, and Schrodinger's theory produces the approximation of those energy levels). Pqm implies that the wave function is solution of both Schrodinger's and Klein-Gordon's equations in the non interacting case while, in the interacting case it implies Sommerfeld's quantum rules: Pem and Pqm then produce the correct relativistic energy levels of atoms (the same as Dirac's energy levels). We check that the required degeneracy is justified by pure deduction, without any other assumption (Schrodinger's theory only justifies one half of the degeneracy). We observe the connection between Pqm, Quantum Field Theories and tunnel effect. From Pgrav we deduce an equation of motion very similar to general relativity (with accuracy 10^{-6} at the surface of the Sun), our postulate being explicitly an approximation. First of all, we discuss classical Kepler problems (Newtonian motion of the Earth around the Sun), explain the link between Kelpler's law of periods (1619) and Plank's law (1900) and observe the links between all historical models of atoms (Bohr, Sommerfeld, Pauli, Schrodinger, Dirac, Fock).
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 article is a continuation of Part(I) to demonstrate more aspects for a successful merge of classical mechanics to the quantum theory. The article starts first with the deduction of the general form of the complimentary energy, next a robust firm mathematical proof is given indicates that the complimentary energy has a wave nature and it is a potential energy, then the radiation condition has been determined, a vector formula expresses the corresponding force was deduced and this force was called adhesion force. A mathematical convergence process has proved the duality of the electromagnetic wave parameters and particle parameters, which results in the existence of a “wave-particle” object. While finding the relationship between the electromagnetic wave parameters and particle parameters, a mathematical relationship was found indicating that not every moving charged mass generates a magnetic field. This relationship for the mass was linked to the effective mass tensor, The geometric representation of the constant energy surface and its relationship to the energy levels of both the orbital model (Bohr model) and the cloud model for different geometric 3D shapes were investigated as a sample of the possible geometric interpretations of the energy equation, that is by employing the triangle method, it is possible to create a geometric representation of the constant energy surfaces. The relation between the energy levels, quantum numbers, and band width is represented geometrically. As a result of proving the relationship between the motion of an electromagnetic wave and the motion of a particle associated with the wave, two new, unprecedented mathematical formulas were developed. The first formula specifies a moving charged mass and its relationship to the resulting wavelength. Thus, the radius of the charged mass is obtained in terms of its wavelength and mass density. The second formula specifies the kinetic energy of a photon. The grand potentials are the potential energy of the Newton’s Law of Universal Gravitation, Coulomb’s Law, and Inverse Square Law of Electromagnetic Waves Intensity. From the potential energy of Newton’s law of universal gravitation, the wavelength of the gravitational wave, as well as the wavelength of quantum gravity were deduced. A mathematical relationship was then deduced that determines the lengths of the functions of conic sections, and in particular the length of perimeter of ellipses. From the potential energy of Coulomb’s law of attraction, a mathematical formulation of the energy levels in general for any atom of an element was obtained. This mathematical formulation was compared with the results obtained by Bohr and Schrödinger for the hydrogen atom. As for the inverse square law of electromagnetic radiation intensity, several innovative mathematical equations have been deduced that link different parameters together. Following the contents of this article requires to be quite familiar with the contents of the first part with the same title, as well as a complete understanding of the concepts of intermediate parameters derived and used in the various equations. Otherwise, it will be extremely difficult to follow the contents of this article.
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.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
Quantum physics begun with discretising the energy of resonators (Planck 1900). Quantum systems exhibit a substantially smaller amount of stationary states than classical systems (Einstein 1907). Planck’s and Einstein’s worked within statistical physics and electromagnetism. The first step toward quantum mechanics was, perhaps, Bohr’s 1913 atom model. The task was to explain the stability of the atoms and the frequencies and intensities of their spectral lines. Two of these three tasks concern stationary properties. Heisenberg’s 1925 matrix mechanics mastered them through a radical “reinterpretation of kinematic and mechanical relations”, where that article tackles the harmonic oscillator. The Bohr orbitals result directly from Schrodinger’s 1926 wave mechanics, though the discretisation method is that of classical resonators. The reuse of the classical expressions for the kinetic and potential energies needs justification. Without that, the tunnel effect remains a mystery provided that justification through an axiomatic deduction of the stationary and time-dependent Schrodinger equations from Euler‘s and Helmholtz’s rather than Newton’s and Hamilton’s representations of classical mechanics. Referring to Einstein, the stationary quantum states are selected out of the classical continuum not through the classical eigenvalue method, but using the intrinsic discreteness of the stationary Schrodinger equation and energy conservation. Unbound states are naturally included. The effective potential energy is always smaller than the total energy; a quantum particle does not ‘tunnel’ through a barrier, but jumps over a hill. The smooth transition from classical to quantum mechanics facilitates to teach and understand the latter one. One can give decent details why reality cannot at all be represented by a continuous field.
From the Quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seen to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory of or the description of reality. It leads to the insight that, if gravity is a fundamental interaction and Quantum Mechanics is universally valid, the gravitational field will have to be quantized, not at least because of the inconsistency of semi-classical theories of gravity. The quantization must be adroitly sufficient, which implies specifically that the subsequent quantum hypothesis must be foundation autonomous. This can't be accomplished by methods for quantum field hypothetical procedures. The goal of a hypothesis of Quantum Gravity would then be to recognize the quantum properties and the quantum elements of the gravitational field. If this means to quantize General Relativity, the general-relativistic identification of the gravitational field with the space time metric has to be taken into account. The quantization must be reasonably sufficient, which implies specifically that the subsequent quantum hypothesis must be foundation free. This can't be accomplished by methods for quantum field hypothetical techniques. One of the fundamental prerequisites for such a quantization technique is, that the subsequent quantum hypothesis has a traditional breaking point that is (in any event roughly, and up to the known phenomenology) indistinguishable from General Relativity.
Be that as it may, should gravity not be an essential, yet an incited, lingering, developing connection, it could in all likelihood be an inherently old style marvel. Should Quantum Mechanics be regardless generally substantial, we needed to expect a quantum substrate from which gravity would result as a new traditional marvel. What's more, there would be no contention with the contentions against semi-old style hypotheses, in light of the fact that there would be no gravity at all on the substrate level. The gravitational field would not have any quantum properties to be caught by a hypothesis of Quantum Gravity, and a quantization of General Relativity would not prompt any principal hypothesis. The target of a hypothesis of 'Quantum Gravity' would rather be the ID of the quantum substrate from which gravity results. The requirement that the substrate theory has General Relativity as a classical limit – that it reproduces at least the known phenomenology – would remain. The paper tries to give an overview over the main options for theory construction in the field of Quantum Gravity. Because of the still unclear status of gravity and space time, it pleads for the necessity of a plurality of conceptually different approaches to Quantum Gravity. The most essential motivations for the development of a theory of Quantum Gravity are generally supposed to be based on two (probably interrelated) types of problems(i) the mutual conceptual incompatibility between General Relativity on the one hand and Quantum Mechanics and Quantum Field Theory on the other hand.