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The hydrogen atom according to wave mechanics - I. spherical polar coordinates

John F. Ogilvie

Published Jun 29, 2017• 4 citations
OpenAlex

Abstract

In the first of five parts in a series, the Schroedinger equation is solved in spherical polar coordinates to yield wave functions that enable an accurate calculation of the frequencies and intensities of lines in the absorption spectrum of the Lyman series in the vacuum-ultraviolet region. Accurate plots of surfaces of amplitude functions illustrate the variation of shapes and sizes varying with quantum numbers k, l, m for comparison with the corresponding plots of amplitude functions in other systems of coordinates.

Related papers

The hydrogen atom according to wave mechanics, parts I, II, III, IV, V

John F. Ogilvie

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.

arXiv:1612.05098•Nov 29, 2016•0 citations

The hydrogen atom according to wave mechanics - II. paraboloidal coordinates

John F. Ogilvie

In the second of five parts in a series, the Schroedinger equation is solved in paraboloidal coordinates to yield amplitude functions that enable accurate plots of their surfaces to illustrate the variation of shapes and sizes with quantum numbers n1, n2, m, for comparison with the corresponding plots of amplitude functions in coordinates of other systems. A useful property of these functions in paraboloidal coordinates is their application to treat the Stark effect, when a hydrogen atom is placed in an isotropic electric field.

•Jun 29, 2017•4 citations

Wave mechanics of the hydrogen atom

John F. Ogilvie

The hydrogen atom is a system amenable to an exact treatment within Schroedinger's formulation of quantum mechanics according to coordinates in four systems -- spherical polar, paraboloidal, ellipsoidal and spheroconical coordinates; the latter solution is reported for the first time. Applications of these solutions include angular momenta, a quantitative calculation of the absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of an amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on the coordinates in a particular chosen system, and are therefore artefacts of that particular coordinate representation within wave mechanics. All discussion of atomic or molecular properties based on such shapes or quantum numbers therefore lacks general significance

arXiv:1603.00899•Feb 28, 2016•2 citations

The hydrogen atom: wave mechanics beyond Schroedinger; orbitals as algebraic formulae derived in all four coordinate systems

John F. Ogilvie

Chemists are aware of the solution of Schroedinger's equations for the hydrogen atom in only spherical polar coordinates, but the spatial variables are separable also in three other systems - - paraboloidal, ellipsoidal and spheroconical; we report here explicit algebraic solutions directly derived in ellipsoidal and spheroconical coordinates for the first time. Our solutions progress from those previously known in spherical polar but not entirely understood, through those little known in paraboloidal, to those in systems of ellipsoidal and spheroconical coordinates unknown before the present work. Applications of these solutions include angular momenta, a quantitative calculation of the discrete absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of a surface of a particular amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on a particular chosen system of coordinates, and are therefore artefacts of that coordinate representation within wave mechanics; a choice of a coordinate system to discuss atomic or molecular properties based on the shapes of amplitude functions or their respective quantum numbers is hence arbitrary

•Jun 20, 2017•2 citations

The hydrogen atom according to wave mechanics in cartesian coordinates

John F. Ogilvie

A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.

arXiv:1709.04759•Aug 30, 2017•2 citations

Solution of spherical equation in 3 dimensions for hydrogen atom with quantum numbers 4≤ n≤ 5

Bambang Supriadi, Zainur Rasyid Ridlo, F Fuadah +3 more

The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .

Journal of Physics Conference Series•Apr 1, 2019•2 citations
In the first of five parts in a series, the Schroedinger equation is solved in spherical polar coordinates to yield wave functions that enable an accurate calculation of the frequencies and intensities of lines in the absorption spectrum of the Lyman series in the vacuum-ultraviolet region. Accurate plots of surfaces of amplitude functions illustrate the variation of shapes and sizes varying with quantum numbers k, l, m for comparison with the corresponding plots of amplitude functions in other systems of coordinates.
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.
In the second of five parts in a series, the Schroedinger equation is solved in paraboloidal coordinates to yield amplitude functions that enable accurate plots of their surfaces to illustrate the variation of shapes and sizes with quantum numbers n1, n2, m, for comparison with the corresponding plots of amplitude functions in coordinates of other systems. A useful property of these functions in paraboloidal coordinates is their application to treat the Stark effect, when a hydrogen atom is placed in an isotropic electric field.
The hydrogen atom is a system amenable to an exact treatment within Schroedinger's formulation of quantum mechanics according to coordinates in four systems -- spherical polar, paraboloidal, ellipsoidal and spheroconical coordinates; the latter solution is reported for the first time. Applications of these solutions include angular momenta, a quantitative calculation of the absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of an amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on the coordinates in a particular chosen system, and are therefore artefacts of that particular coordinate representation within wave mechanics. All discussion of atomic or molecular properties based on such shapes or quantum numbers therefore lacks general significance
Chemists are aware of the solution of Schroedinger's equations for the hydrogen atom in only spherical polar coordinates, but the spatial variables are separable also in three other systems - - paraboloidal, ellipsoidal and spheroconical; we report here explicit algebraic solutions directly derived in ellipsoidal and spheroconical coordinates for the first time. Our solutions progress from those previously known in spherical polar but not entirely understood, through those little known in paraboloidal, to those in systems of ellipsoidal and spheroconical coordinates unknown before the present work. Applications of these solutions include angular momenta, a quantitative calculation of the discrete absorption spectrum and accurate plots of surfaces of amplitude functions. The shape of a surface of a particular amplitude function, and even the quantum numbers in a particular set to specify such an individual function, depend on a particular chosen system of coordinates, and are therefore artefacts of that coordinate representation within wave mechanics; a choice of a coordinate system to discuss atomic or molecular properties based on the shapes of amplitude functions or their respective quantum numbers is hence arbitrary
A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.
The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .