Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels

TL;DR

This paper revisits Schrödinger's original 4th-order real-valued wave equation, demonstrating it predicts higher energy levels and is too stiff for atomic spectra, thus supporting the use of the 2nd-order complex wave equation in quantum mechanics.

Contribution

It shows that Schrödinger's original 4th-order real wave equation predicts higher energy levels and is less suitable for atomic spectra than the 2nd-order complex wave equation.

Findings

4th-order real wave equation predicts higher energy levels

2nd-order complex wave equation aligns with observed spectra

4th-order equation is too stiff for accurate energy predictions

Abstract

In his seminal part IV, Ann. der Phys. Vol 81, 1926 paper, Schrodinger 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 4th-order in space and 2nd-order in time. In view of the mathematical difficulties associated with the eigenvalue analysis of a 4th-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrodinger splits the 4th-order real operator into the product of two, 2nd-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic 2nd-order, complex-valued wave equation. In this paper we show that Schrodinger's original 4th-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his 2nd-order, complex-valued wave equation that predicted with remarkable success the visible energy levels observed in the visible atomic line-spectra of the chemical elements. Accordingly, the 4th-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, 2nd-order complex-valued wave equation; unless in addition to the emitted visible energy there is also dark energy emitted.