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. 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 wave functions for the hydrogen atom are then guessed based on the historical solutions. 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?