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

While instructing an undergraduate physical chemistry course on quantum mechanics, I encountered difficulties in elucidating the concept of the particle-in-a-box thought experiment, initially formulated by Erwin Schrödinger. In particular, the node distribution of the wavefunction. Born's Copenhagen interpretation assigns the modulus of the wavefunction as the probability density distribution of particles. The developers of quantum mechanics, including Schrödinger, de Broglie, and Einstein, have openly expressed reservations about this interpretation. Schrödinger's original conception of the wavefunction is intrinsically linked to the particle's momentum rather than the statistical probabilities of particle locations. Because momentum can be encoded in different forms, it could be useful to create wavefunctions using different encoding equations, which is the major attempt of this manuscript. In addition to quantizing energy levels properly for several example systems, the proposed new equations have yielded a striking and alternative interpretation of Schrödinger's wavefunction that aligns well with classical mechanics. This proposed modification enhanced the simplicity of math and strengthened Schrödinger’s original interpretation. These equations could be baby steps to a brand-new quantum calculation algorithm that is simpler and faster than the established ones.