I present an alternative and rather direct way to derive the well known Schrödinger equation for a quantum wavefunction, by starting with the Klein Gordon equation and applying a directional factorization scheme. And since if you have a directionally factorizing hammer, everything looks like a factorizable nail, I also derive an alternative wavefunction propagation equation in the momentum-dominated limit. This new Schrödinger
p-equation therefore provides a potentially useful complement to the traditional Schrödinger
m-equation's mass-dominated limit.
The need to implement time reversal via complex conjugation in quantum theory has always been a bit of a puzzle. Why should i go to – i under temporal reflection when it has no spatiotemporal dimensions? I’ll provide a new insight into this question by showing how the little-appreciated “quantum-looking” classical Schrödinger equation of Schiller and Rosen faces the exact same problem. Since we know how to escape this problem classically, this observation teaches us one way to solve the problem quantum mechanically too. Big picture: if I’m right, the puzzle over quantum time reversal is connected to the interpretation of quantum theory.
Abstract By the early 1920s physicists had accepted the fact that, depending on the experiment chosen, matter as well as light could be displayed either as compact lumps or as widely spread-out waves. Few pretended to understand this seeming contradiction. The significance of this paradox came a few years later with the Schrödinger equation. But Erwin Schrödinger wasn’t looking for significance. At the time he saw de Broglie’s matter waves only as a way to get rid of Bohr’s “damn quantum jumps.”