Coming Back Down Towards the ‘Ground’
The main contention of the previous post can be summed up succinctly as: the Schrödinger equation is to quantum mechanics what the Hamilton-Jacobi equation is to classical mechanics. This is because it was – in a sense – ‘derived’ (really inferred) from it, via the optical-mechanical analogy between idealised particle paths and idealised geometrical light rays, first pointed out by William Rowan Hamilton in the early 1830s (and see, e.g., Masoliver and Ros 2010 for a detailed mathematical exposition). The main postulate of Schrödinger’s wave mechanics was that the action $S$ from Hamilton-Jacobi mechanics becomes the phase of the complex wavefunction $\psi\sim e^{iS/\hbar}$. This meant that we therefore found ourselves three levels of abstraction away from, and ‘floating’ above (so to speak), the ‘ground’ that classical mechanics was founded upon, namely Newtonian mechanics in 3D space. This post now begins the process of thinking about how to come down again to seek a more solid footing, if indeed there is even one to find…
Continue reading “Reverse-Engineering Quantum Mechanics, II.”



