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Physics Essays

Modification of the Schrödinger equation from Bohr’s theory

Noboru Kohiyama

Published Sep 11, 2021• 0 citations
Publisher

Abstract

The mass-energy equation ε = mc2 is obtained from Bohr’s theory. The Schrödinger equation (SE) is modified using this equation. In the hydrogenlike atom, the modified SE is solved for consistency with the eigenvalues of electron’s energy derived from the Dirac equation. Based on the consistency of their eigenvalues, the microwave emission (e.g., 21 cm radio wave) by the hydrogen atom is expressed.

DOI
10.4006/0836-1398-34.3.261

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The mass-energy equation ε = mc2 is obtained from Bohr’s theory. The Schrödinger equation (SE) is modified using this equation. In the hydrogenlike atom, the modified SE is solved for consistency with the eigenvalues of electron’s energy derived from the Dirac equation. Based on the consistency of their eigenvalues, the microwave emission (e.g., 21 cm radio wave) by the hydrogen atom is expressed.
Bohr theory was not satisfactory as it failed to provide an interpretation of relative line intensities and splitting of spectral lines in the presence of electric field and magnetic field in the hydrogen spectrum. This theory also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this chapter, we consider the application of Schrödinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrödinger equation can be exactly solved. Other one-electron system like He + , Li 2+ etc. can also be solved exactly. The chemical system with more than one-electron cannot be solved exactly due to electron-electron repulsion, which is the most problematic term.
Bohr theory was not satisfactory as it failed to provide an interpretation of relative line intensities and splitting of spectral lines in the presence of electric field and magnetic field in the hydrogen spectrum. This theory also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this chapter, we consider the application of Schrodinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrodinger equation can be exactly solved. Other one-electron system like He + , Li 2+ etc. can also be solved exactly. The chemical system with more than one-electron cannot be solved exactly due to electron-electron repulsion, which is the most problematic term.
The quantum mechanical problem of the hydrogen atom is treated by use of a finite difference equation in place of Schrödinger's differential equation. The exact solution leads to a wave vector energy expression that is readily converted to the Bohr-Rydberg formula. (The calculations here reported are limited to spherically symmetric states.) The wave vectors reduce to the familiar solutions of Schrödinger's equation as c --> infinity. The internal consistency and limiting behavior provide support for the view that the equations employed could well constitute an approach to a relativistic formulation of wave mechanics.
The Bohr radius was derived using the non-relativistic de Broglie approximation, not the relativistic de Broglie equation. In this work, we will derive a relativistic Bohr radius from both the relativistic de Broglie approximation and relativistic kinetic energy. The Bohr model of the hydrogen atom has since been replaced by modern quantum mechanics and the Schrodinger equation. It is well known that the Schrodinger equation confirms the first Bohr radius as the most probable radius, but the Schrodinger equation itself is non-relativistic. The Dirac equation can likely be used, but there seem to be very few publications on the relativistically corrected Bohr radius. This is something we will discuss.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.