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

It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry empirical for almost all elements and molecules. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I proposed an equation that quantizes the energies orders of magnitude faster than classical quantum mechanics. I assumed that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with an equation significantly different than the classical Schrödinger equation. This new equation is inspired by the energy conservation method used in solving classical mechanics problems such as the pendulum problem. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and harmonic oscillator are derived. We can see that the new equation is much simpler than Schrödinger equation in obtaining consistent quantized rotational and vibrational energy levels. Thus, it has the potential to be widely used to free up computational pressure in modern quantum chemistry calculations.