Related Experiment Video
Updated: Jul 7, 2025

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
Published on: May 27, 2020
Revisiting Schrödinger's fourth-order, real-valued wave equation and the implication from the resulting energy levels
1Department of Civil and Environmental Engineering, Southern Methodist University, Dallas, TX 75276, USA.
Abstract:
In his seminal part IV, Annalen der Physik vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger's original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.
Related Concept Videos
The Quantum-Mechanical Model of an Atom
The de Broglie Wavelength
The Energies of Atomic Orbitals
Electron Orbital Model
The first shell is closest to the nucleus, and it has only one subshell with a single spherical orbital called the...
The Bohr Model
Kinetic and Potential Energy of a Wave
In mechanical waves, the amount of energy is related to their amplitude and frequency. In the context of the above examples, large-amplitude earthquakes produce large...

