| About transitioning from Classical Mechanics to QM, guided by observations. There is a very interesting approach in the quantum physics book by Eisberg and Resnick, section 5.2 To arrive at the Schrödinger equation Eisberg and Resnick construct what they refer to as a plausibility argument. The goal: to arrive at a wave equation that when solved for the Hydrogen atom will have the electron orbitals as set of solutions. Eisberg and Resnick state 4 demands: -1. Must be consistent with the de Broglie/Einstein postulates. wavelength=h/p, frequency=E/h -2. Must be such that for a quantum entity followed over time the sum of potential energy and kinetic energy is a conserved quantity. -3. Must be such that the equation is linear in \Psi(x,t): any linear combination of two solutions \Psi_1 and \Psi_2 must also be a solution of the equation. (Motivation: in experiments electron diffraction effects are observed. Interference effects can occur only if wave functions can be _added_.) -4. In the absence of a potential gradient the equation must have as a solution a propagating sinusoidal wave of constant wavelength and frequency. Eisberg and Resnick proceed to show that the above 4 demands narrow down the possibilities such that arriving at the Schrödinger equation is made inevitable. To me the second demand is particularly interesting. The second demand is equivalent to demanding that the work-energy theorem holds good. The recurring theme: the work-energy theorem. I have a (html)-transcript of the Eisberg & Resnick treatment that I can make available to you. There is a youtube video with a presentation that is based on the Eisberg & Resnick plausibility argument. https://youtu.be/2WPA1L9uJqo In that video the presentation of the plausibility argument is in the first 18 minutes, the rest of the video is about application of the Schrödinger equation. |