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)].
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)].
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.
This paper re-opens the debate on the failure of quantum mechanics to provide an understandable view of micro-reality. A critique is offered of the commonly accepted ‘Copenhagen Interpretation’ of a theory that is only a mathematical approach to the level of reality characterized by atoms and electrons. This critique is based on the oldest approach to thinking about nature for over 2500 years, known as Natural Philosophy. Quantum mechanics (QM) was developed over the first quarter of the 20th Century, when scientists were enthralled by a new philosophy known as Positivism, whose foundations were based on the assumption that material objects exist only when measured by humans – this central assumption conflates epistemology (knowledge) with ontology (existence). The present critique rejects this human-centered view of reality by assuming material reality has existed long before (and will persist long after) human beings (“Realism”). The defensive view that the micro-world is too different to understand using regular thinking (and only a mathematical approach is possible) is rejected totally. At least 12 earlier QM interpretations are critically analyzed, indicating the broad interest in “what does QM mean?” The standard theory of quantum mechanics is thus constructed on only how the micro-world appears to macro measurements - as such, it cannot offer any view of how the foundations of the world are acting when humans are not observing it - this has generated almost 100 years of confusion and contradiction at the very heart of physics. Significantly, we live in a world that is not being measured by scientists but is interacting with itself and with us. QM has failed to provide explanations: only recipes (meaningless equations), not insights. Physics has returned to the pre-Newtonian world of Ptolemaic phenomenology: only verifiable numbers without real understanding. The focus needs to be on an explicit linkage between the micro-world, when left to itself, and our mental models of this sphere of material reality, via the mechanism of measurement. This limits the role of measurement to confirming our mental models of reality but never confusing these with a direct image of ‘the thing in itself’. This implies a deep divide between reality and appearances, as Kant suggested. This paper includes an original analysis of several major assumptions that have been implicit in Classical Mechanics (CM) that were acceptable in the macroscopic domain of reality, demonstrated by its proven successes. Unfortunately, only a few of these assumptions were challenged by the developers of QM. We now show that these other assumptions are still generating confusions in the interpretation of QM and blocking further progress in the understanding of the microscopic domain. Several of these flawed assumptions were introduced by Newton to support the use of continuum mathematics as a model of nature. This paper proposes that it is the attempt to preserve continuum mathematics (especially calculus), which drives much of the mystery and confusion behind all attempts at understanding quantum mechanics. The introduction of discrete mathematics is proposed to help analyze the discrete interactions between the quintessential quantum objects: the electrons and their novel properties. A related paper demonstrates that it is possible to create a point-particle theory of electrons that explains all their peculiar (and ‘paradoxical’) behavior using only physical hypotheses and discrete mathematics without introducing the continuum mathematical ideas of fields or waves. Another (related) paper proves that all the known results for the hydrogen atom can also be exactly calculated from this new perspective with the discrete mathematics. * Surrey, B.C. Canada (604) 542-2299 spsi99@telus.net Version 2.015 08-05-2017 Begun 23-06-2008 {pp. 82, 70.2 Kw; 800 KB}
The widespread quest to quantize gravity mirrors humanity’s enduring drive to unify quantum and macroscopic physical frameworks. This research initiates its inquiry by integrating quantum mechanics with classical mechanics, yielding novel analytical perspectives, and introduces the Schrödinger-Tu equation—a wave function that incorporates gravitational potential energy and is capable of describing macroscopic objects. Validating the practical viability of merging quantum and classical approaches, several computational cases involving atoms and molecules yield successful outcomes. Additionally, this study derives a wave-mechanical formulation of classical mechanical laws, furnishing direct empirical evidence that quantum mechanics and classical mechanics are not mutually exclusive but rather compatible and complementary. Further substantiation of their compatibility encompasses the following. key points: • The mass term m in the Schrödinger equation (SE) can be scaled to sufficiently large values to characterize macroscopic entities • The steady-state SE (Tψ +Vψ=Eψ) inherently integrates the wave function ψ with the classical energy conservation relation T+V=E; • Potential energy functions within the SE can be derived from macroscopic force fields; • Newton’s second law F=ma and the SE are mutually derivable, establishing a fundamental theoretical linkage; 5.Electron diffraction experiments concurrently demonstrate wave-particle duality, a phenomenon that bridges both mechanical frameworks.This paradigm shift—recasting the relationship between quantum and classical mechanics from one of incompatibility to ompatibility—represents a pivotal theoretical advancement, with the potential to catalyze the development of a generalized wave echanics.
WHAT TO LOOK FOR Although the classical theory developed in Chapter 2 can be used to describe a great many phenomena in the real world, there is little doubt that the proper description of the world is quantum. To a large measure we perceive the world classically and we must find ways to uncover the underlying quantum theory. The process of “quantization” normally consists of turning a classical theory into a corresponding quantum theory. In turn, the “classical limit” is the process by which a quantum theory is brought to its associated classical theory. The parameter ħ = 1.0545 × 10 −27 (≃ 10 −27 ) erg-seconds (cgs units), referred to as Planck's constant, sets the scale of quantum phenomena. In many applications it will prove useful to use “natural” units in which ħ = 1 just so that formulas are less cumbersome. For most typical formulations of quantum mechanics the classical limit means the limit that ħ → 0, and in this text we shall have many occasions to use this definition as well. In addition, we shall also learn to speak about the classical and quantum theories simultaneously coexisting – as they do in the real world – without the need to take the limit ħ → 0. There are three generally accepted “royal” routes to quantization, due principally to Heisenberg, Schrödinger, and Feynman, respectively, and we shall exploit them all. In each case we begin with a single degree of freedom ( N = 1).
Quantum mechanics (QM) and quantum electrodynamics (QED) are often touted as the most successful theories ever. In this paper, this claim is critically evaluated by a test of internal consistency for the ability to calculate the conjugate observables of the nature of the free electron, ionization energy, elastic electron scattering, and the excited states of the helium atom using the same solution for each of the separate experimental measurements. It is found that in some cases quantum gives good numbers, but the solutions are meaningless numbers since each has no relationship to providing an accurate physical model. Rather, the goal is to mathematically reproduce an experimental or prior theoretical number using adjustable parameters including arbitrary wave functions in computer algorithms with precision that is often much greater (e.g. 8 significant figures greater) than possible based on the propagation of errors in the measured fundamental constants implicit in the physical problem. Given the constraints of adherence to physical laws and internal consistency, an extensive literature search indicates that quantum mechanics has never solved a single physical problem correctly including the hydrogen atom and the next member of the periodic chart, the helium atom. Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to the same problem. 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 after Schrodinger, a classical approach is explored which yields a model which is remarkably accurate 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 is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems. Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [1-7] that successfully applied physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrodinger boundary condition: “ 0 Ψ→ 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 1 = n -state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution, modeled as a superposition of accelerating charges, does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math as in the case of the most famous equations of Newton, Maxwell, Einstein, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used. The accurate solution of the helium atom is confirmed by the agreement of predicted and observed conjugate parameters using the same unique physical model in all cases.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]