[Paper Review] Wave mechanics of the hydrogen atom

J. F. Ogilvie|arXiv (Cornell University)|Feb 28, 2016

Atomic and Molecular Physics12 references3 citations

TL;DR

This paper presents the first exact solution of the Schrödinger equation for the hydrogen atom in spheroconical coordinates, extending known solutions in spherical, paraboloidal, and ellipsoidal systems. It demonstrates that electron amplitude functions and quantum numbers are coordinate-dependent artifacts, challenging the general significance of orbital shapes and quantum number assignments in wave mechanics.

ABSTRACT

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

Motivation & Objective

• To extend the exact solutions of the Schrödinger equation for the hydrogen atom beyond the standard coordinate systems.
• To investigate the implications of coordinate system choice on the interpretation of electron amplitude functions and quantum numbers.
• To challenge the assumption that orbital shapes and quantum numbers represent intrinsic atomic properties.
• To provide a comprehensive treatment of wave mechanics in four coordinate systems: spherical polar, paraboloidal, ellipsoidal, and spheroconical.
• To demonstrate that the shape of amplitude functions and quantum number assignments are artifacts of the chosen coordinate representation.

Proposed method

• Solving the time-independent Schrödinger equation for the hydrogen atom using spheroconical coordinates, a system not previously applied to this problem.
• Applying separation of variables to the Schrödinger equation in spheroconical coordinates to derive the radial, angular, and conical wave functions.
• Deriving the energy eigenvalues and quantum number relationships consistent with the new coordinate system.
• Comparing the resulting amplitude functions and quantum number assignments with those in spherical polar coordinates to highlight coordinate dependence.
• Using the solutions to generate accurate plots of amplitude function surfaces and to calculate the absorption spectrum quantitatively.
• Analyzing the implications of coordinate-dependent representations for the interpretation of atomic structure in wave mechanics.

Experimental results

Research questions

• RQ1 How does the wave mechanical solution of the hydrogen atom differ when formulated in spheroconical coordinates compared to other systems?
• RQ2 To what extent do the shapes of electron amplitude functions depend on the choice of coordinate system?
• RQ3 Are quantum numbers assigned to individual wave functions intrinsic properties of the hydrogen atom or artifacts of the coordinate representation?
• RQ4 What are the implications of coordinate-dependent wave functions for the interpretation of atomic orbitals in quantum chemistry?
• RQ5 Can the absorption spectrum of hydrogen be accurately calculated using the wave functions derived in spheroconical coordinates?

Key findings

• The paper presents the first exact solution of the Schrödinger equation for the hydrogen atom in spheroconical coordinates.
• The solutions in spheroconical coordinates yield distinct amplitude functions and quantum number assignments compared to those in spherical polar coordinates.
• The shape of the amplitude function and the quantum numbers used to label it are shown to be artifacts of the chosen coordinate system, not intrinsic properties of the atom.
• The energy eigenvalues and the form of the wave functions are consistent with known results, confirming the validity of the solution.
• The method enables accurate quantitative calculation of the hydrogen absorption spectrum and precise plotting of amplitude function surfaces.
• The findings challenge the general physical significance of orbital shapes and quantum number assignments derived from specific coordinate representations.