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International Journal of New Technology and Research

Time-Independent Schrödinger equation as a straight line equation in Cartesian coordinates

Wojciech Szczepankiewicz

Published Jun 20, 2018• 0 citations
Publisher

Abstract

A graphical interpretation of the time-independent Schroumldinger equation in the Cartesian coordinate system for the explicitly set wave function of the ground state and selected excited states of the hydrogen has been presented. It was shown that the equation left-sidedly multiplied by the function psi allows to find its didactic connection with the course of the beam of light emitted by the hydrogen atom in the measuring system of the spectroscope equipped with a prism.

DOI
10.31871/ijntr.4.6.62

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A graphical interpretation of the time-independent Schroumldinger equation in the Cartesian coordinate system for the explicitly set wave function of the ground state and selected excited states of the hydrogen has been presented. It was shown that the equation left-sidedly multiplied by the function psi allows to find its didactic connection with the course of the beam of light emitted by the hydrogen atom in the measuring system of the spectroscope equipped with a prism.
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Based on an exposition of the underlying physics and applied mathematics in arXiv:1603.00899, this paper in five separate parts presents a description of the properties of the amplitude functions of the hydrogen atom according to wave mechanics in the coordinate representation in all four systems of coordinates in which Schroedinger's equations are separable, explicitly I spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal coordinates and IV spheroconical coordinates; part V discusses the implications of these multiple systems in which these atomic orbitals can be directly calculated. Most published plots of orbitals are inaccurate. In particular, parts I - IV contain many accurate plots of the surfaces of amplitude functions in the various systems with a common criterion for the amplitude at which the surface is constructed. Although the plots are generated from amplitude functions in the specified systems of coordinates, the plots are translated to cartesian coordinates that the eye can recognise. The surfaces of the amplitude functions in ellipsoidal and spheroconical functions are previously unreported, with the accurate surfaces in spherical polar and paraboloidal coordinates for a direct comparison.
The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
Abstract The energies, kets and wave functions obtained from the Schrödinger equation for the hydrogen atom are examined in Chapter 9. Three quantum numbers are identified. The energies turn out to be the same as in the Bohr model, and an energy-level diagram appropriate to the quantum description is constructed. Graphs of the probability distributions are interpreted as the electron being in a “cloud” around the proton, rather than at a fixed position: the atom is fuzzy, not sharp-edged. The wavelengths of the five photons of the Balmer series are shown to be in the visible range. These photons are emitted when electrons transition from higher-excited states to the second lowest one, which means that electronic-type transitions underlie the presence of colors in our visible environment. The non-collapse of the atom, required by classical physics, is shown to arise from the structure of Schrödinger’s equation.