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.
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.
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.
Abstract This paper intends to provide instruction for a single lesson in an introductory course on quantum mechanics by tying together probabilistic methods from statistical mechanics, the use of complex numbers, and a more founded explanation for the derivation of Schrödinger’s time-independent equation. Here presented derivation strongly supports the Copenhagen interpretation of quantum mechanics. Niels Bohr suggested a hydrogen atom model (similar to Kepler’s sun-planet model), where the negatively-charged electron moves in orbit around the heavy positively charged nucleus, this electron having some kinetic and potential energy. Starting from Hamilton’s differential equation for Kepler like atomic model, Schrödinger introduced a new unknown real function ψ in his derivation of time-independent equation (initially, he did not recognize it as the wave function). We show that hypothetical statistical spread of action on many (indefinite) levels is mandatory for the derivation of Schrödinger’s time-independent equation. The result of this hypothesis, the statistical probability constraint formula, is used in this derivation as we look for a minimum of probability for the electron energy distribution E ψ in the whole space. The Euler–Lagrange equation, with normalization of ψ as a constraint limitation, then gives Schrödinger’s time-independent equation for real numbers. A closer inspection of the normalized function ψ shows that it could be complex and is related to the de Broglie–Bohr model of the hydrogen atom. It can be postulated that the classical solution of Schrödinger’s time-independent equation should be a normalized complex function ψ .
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
On 13 March 1926, the first of six papers on wave mechanics by Erwin Schrödinger was published in the Annalen der Physik with the title Quantisation as an eigenvalue problem (Part 1) (Schrödinger, 1926b). The startling first paragraph reads: ‘In this paper, I wish to consider, first, the simple case of the hydrogen atom (nonrelativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the concept of “whole numbers”, merely as such, is not introduced. Rather, when integralness does appear, it arises in the same natural way as it does in the case of node-numbers of a vibrating string. The new conception is capable of generalisation, and strikes, I believe, very deeply at the true nature of quantum rules.’ These papers were the fruits of an extraordinary burst of creativity on Schrödinger's part which resulted from his interactions with Einstein in the latter part of 1925. Central to these exchanges were de Broglie's remarkable researches which culminated in his famous PhD dissertation and published papers of 1924. These events have already be recounted in Chap. 9. The subsequent developments which led to Schrödinger's discovery of the equation which bears his name will be taken up in Sect. 14.2, but let us first understand more about Schrödinger's background. Schrödinger's background in physics and mathematics Education and career up to 1925 Unlike Heisenberg, Jordan, Pauli and Dirac, Erwin Schrödinger was not one of the young Turks who developed Knabenphysik , young man's physics.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.