In Part I we constructed the Quantum Mechanics of a charged unitary entity and prescribed the form in which such a particle interacts with other charged particles and matter in general. In this second part we extend the description to the hydrogen atom testing the correctness and accuracy of the general description. The relation between electron and proton in the atom is described systematically in a construction that is free from analogies or ad-hoc derivations and it supersedes conventional Quantum Mechanics (whose equations linked to measurements can be recovered). We briefly discuss why the concept of isolation built in Schrödinger's time evolution is not acceptable and how it immediately results in the well known measurement paradoxes of quantum mechanics. We also discuss the epistemic grounds of the development as well as those of conventional Quantum Mechanics.
In Part I we constructed the Quantum Mechanics of a charged unitary entity and prescribed the form in which such a particle interacts with other charged particles and matter in general. In this second part we extend the description to the hydrogen atom testing the correctness and accuracy of the general description. The relation between electron and proton in the atom is described systematically in a construction that is free from analogies or ad-hoc derivations and it supersedes conventional Quantum Mechanics (whose equations linked to measurements can be recovered). We briefly discuss why the concept of isolation built in Schrödinger's time evolution is not acceptable and how it immediately results in the well known measurement paradoxes of quantum mechanics. We also discuss the epistemic grounds of the development as well as those of conventional Quantum Mechanics.
We reconstruct Quantum Mechanics in a way that harmonises with classical mechanics and electromagnetism, free from mysteries or paradoxes as the \emph{collapse of the wave-function} or \emph{Schrödinger's cat.} The construction is inspired by de Broglie's and Schrödinger's wave mechanics while the unifying principle is Hamilton's principle of least action, which separates natural laws from particular circumstances such as initial conditions and leads to the conservation of energy for isolated systems. In Part I we construct the Quantum Mechanics of a charged unitary entity and prescribe the form in which the entity interacts with other charged entities and matter in general. In Part II we address the quantum mechanics of the hydrogen atom, testing the correctness and accuracy of the general description. The relation between electron and proton in the atom is described systematically in a construction that is free from analogies or ad-hoc derivations and it supersedes conventional Quantum Mechanics (whose equations linked to measurements can be recovered). We briefly discuss why the concept of isolation built in Schrödinger's time evolution is not acceptable and how it immediately results in the well known measurement paradoxes of quantum mechanics. We also discuss the epistemic grounds of the development and provide a criticism of instrumentalism, the leading philosophical perspective behind conventional Quantum Mechanics.
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.
Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
The atom model of Quantum Mechanics (QM) was conceived from an unsolved paradox. Indeed, Schrödinger’s equation has been deducted by considering a free electron, but it is applied for the atom, where the electron is inside a potential. In order to eliminate the nonsense, quantum theorists proposed a ridiculous postulate: they claim it makes sense to use the equation because it gives results in agreement to experimental data. The unsolved paradox evidences that Schrödinger’s equation cannot be applied to the physical conditions considered in the QM atom model, and that his equation actually requires some special conditions not considered in the theory (for instance, the electron helical trajectory, rejected by Heisenberg). The banishment of the aether has introduced several paradoxes in the development of Theoretical Physics. And because the theorists have neglected other paradox (from the mathematical probability the spectacular successes of Bohr’s hydrogen atom cannot be accidental), these two unsolved paradoxes introduced dramatic consequences in the development of Nuclear Physics.