From Particle-in-a-Box Thought Experiment to a Complete Quantum Theory?

09 June 2023, Version 14


The Schrödinger equation provides wavefunctions that describe the behavior of atoms, molecules, and materials, but constructing these wavefunctions by basis sets from fitting experimental data has rendered quantum chemistry predominantly empirical. The computational intensity of this process arises partly from the requirement to integrate wavefunctions across space to calculate energy levels. In this manuscript, a novel equation is proposed that significantly accelerates the computation of energy levels compared to classical quantum mechanics. Leveraging the first law of thermodynamics, which dictates that an equilibrium system maintains its total energy, an energy density function is derived that exhibits spatial uniformity and eliminates the need for normalization and renormalization. Consequently, the energy calculation no longer necessitates the integration of the entire wavefunction and transforms the second-order differential equation into a first-order counterpart. This new equation draws inspiration from the energy conservation method employed in solving classical mechanics problems like the pendulum. Remarkably, it successfully quantizes the energy levels of classical quantum systems, including the particle-in-a-box thought experiment, harmonic oscillator, and free spin, relying on simple rules such as momentum conservation. Therefore, it holds the potential to fit experimental data and reduce computational costs in modern quantum chemistry calculations.


Supplementary materials

MATLAB codes and movies
V4 rotation Y10Y11Y1n1.gif
Rotation wavefunction.
V1 PIAB movie gif
Particle in a box solution.
V3 ClassicalRotationY10Y11Y1n1
Classical rotation wavefunction.


Comments are not moderated before they are posted, but they can be removed by the site moderators if they are found to be in contravention of our Commenting Policy [opens in a new tab] - please read this policy before you post. Comments should be used for scholarly discussion of the content in question. You can find more information about how to use the commenting feature here [opens in a new tab] .
This site is protected by reCAPTCHA and the Google Privacy Policy [opens in a new tab] and Terms of Service [opens in a new tab] apply.