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

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.