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 a black box of mathematical model. 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 speculated 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 a Hamiltonian slightly different than the classical Schrödinger equation. 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 an ideal harmonic oscillator are then derived. Can this assumption on energy distribution and the pre-mature math in this manuscript lead to a complete quantum theory? Can the idea of energy conservation unify special relativity and quantum?