A Real-Valued Description of Quantum Mechanics with Schrodinger's 4th-order Matter-Wave Equation

TL;DR

This paper introduces a real-valued, 4th-order Schrödinger equation that accurately reproduces eigenvalues of the traditional 2nd-order equation and reveals the existence of negative energy levels, offering a new perspective on quantum mechanics.

Contribution

It presents a variational formulation of a 4th-order real-valued Schrödinger equation that captures all eigenvalues, including negative ones, expanding the understanding of quantum energy spectra.

Findings

The 4th-order equation produces the same eigenvalues as the 2nd-order Schrödinger equation.

Negative energy levels are identified as a natural feature of the real-valued formulation.

The classical 2nd-order equation is a simplified version that omits negative energy states.

Abstract

Using a variational formulation, we show that Schrodinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V(r), produces the precise eigenvalues of Schrodinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative (repelling) energy levels. Schrodinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V(r), at the expense of missing the negative (repelling) energy levels.