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Deconstruction of Quantum Wave Mechanics

Daniel Crespin

Published Nov 14, 2021• 0 citations
OpenAlex

Abstract

The model of the hydrogen atom originally introduced by Erwin Schrödinger in 1926 isa two-body proton -electron system. It relies on the Schrödinger operator H and on thelinear Schr¨odinger time dependent equation. Then Niels Bohr discovered that the modelcontradicts the basic dynamics of the atom and quantum wave mechanics was invented,with incidences here told from a deconstructive standpoint. The final section of this papersketches a new, alternative, continuous, causal and fully deterministic wave mechanics—already mentioned in [1], [2], [3] and elsewhere— crucially relying on H and solvablein terms of its eigenvalues and eigenfunctions, but definitely non-linear. An importantimplication is that the deterministic hydrogen atom must be conceived as a three-bodysystem consisting of proton, electron and photon.

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The model of the hydrogen atom originally introduced by Erwin Schrödinger in 1926 isa two-body proton -electron system. It relies on the Schrödinger operator H and on thelinear Schr¨odinger time dependent equation. Then Niels Bohr discovered that the modelcontradicts the basic dynamics of the atom and quantum wave mechanics was invented,with incidences here told from a deconstructive standpoint. The final section of this papersketches a new, alternative, continuous, causal and fully deterministic wave mechanics—already mentioned in [1], [2], [3] and elsewhere— crucially relying on H and solvablein terms of its eigenvalues and eigenfunctions, but definitely non-linear. An importantimplication is that the deterministic hydrogen atom must be conceived as a three-bodysystem consisting of proton, electron and photon.
The quantum wave mechanics proposed in 1926 by Erwin Schrödinger for the hydrogen atom was a mixture of success and failure. Astonishing and extraordinary success resulted from Schrödinger self adjoint Hamiltonian energy linear operator H. This mathematical object allows the calculation of stationary energies and stationary electron configurations. On the other hand the Schrödinger time dependent equation was such a dramatic failure that it led to the establishment of the current Quantism, or Quantum doctrine, or Copenhagen interpretation of quantum mechanics based on the postulates listed below, or variations of these. This paper discusses the implications that the deterministic time dependent equation has for Quantism.
This chapter considers Schrodinger's model of hydrogen atom and analyzes in details the quantum wave mechanics approach to the motion of the electron in the hydrogen atom. The stationary Schrodinger equation is not easy to solve as it stands, and the whole problem looks rather complicated when expressed in terms of Cartesian coordinates. An essential difference between Bohr's and Schrodinger's models is that there are three quantum numbers, n , l , m , rather than the one number n describing a state of the electron. Moreover, in Bohr's model the atom is, at each instant, in one and only one energy state, that is., at each instant, the state of the atom is definite, but according to Schrodinger's model, the atom can exist in a superposition of states. In other words, the atom can be simultaneously in more than one energy state at a given instant. This is clearly incompatible with Bohr's model.
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.
Through his experiments, the physicist Niels Bohr improved upon Rutherford's model of the atom and developed his model of the atomic structure in 1913 that succeeded in explaining the spectral features of the hydrogen atom. A simple means for extending the conventional non-relativistic Bohr model of the atom to include the wave nature of electrons is presented. As the derivation requires basic knowledge of classical and wave mechanics, it can be taught in standard courses in modern physics and introductory quantum mechanics.
Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.
Abstract In 1926, Erwin Schrödinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrödinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrödinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger’s famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum–mechanical energy eigenfunctions of the hydrogen atom, following Schrödinger’s original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.