## lördag 30 december 2023

### New Year Rubber Model of Real Quantum Mechanics

A popular model of a planetary system is formed by a small ball (Earth) moving on a horizontal elastic rubber sheet suspending a heavy ball (Sun) at the origin $x=0$:

The slope of the rubber sheet gives the force on the little ball balancing centripetal acceleration to make it circle the big ball. The heavy ball suspended by the elastic rubber sheet is a model of gravitational force created by a massive object arising from minimisation of a functional of the form
• $G(\psi ) =\frac{1}{2}\int\vert\nabla\psi (x)\vert^2 dx - \psi (0)$          (1)
over functions $\psi (x)$ vanishing for $x$ far from the origin. The first term of $G(\psi )$ represents elastic energy of compression/expansion $\nabla\psi (x)$ and the -1 factor of $\psi (0)$ is the heavy ball load potential.

A similar model can be used to describe a hydrogen atom with a proton kernel surrounded by an electronic charge density $\psi (x)$ as minimisation of
• $E(\psi ) =\frac{1}{2}\int\vert\nabla\psi (x)\vert^2 dx -\int\frac{\vert\psi (x)\vert^2}{\vert x\vert }dx$     (2)
over functions $\psi (x)$ with $\int\vert\psi (x)\vert^2dx=1$. The first term of $E(\psi )$ is usually named "kinetic energy", which lacks physics since no motion in time is involved. In view of (1) it can better be seen as a form of electronic compression/expansion energy again measured by $\nabla\psi (x)$. The second term involves the kernel potential $\frac{1}{\vert x\vert }$.

We thus see a close similarity of (1) and (2) with both Solar system and Hydrogen atom captured by the same elastic rubber sheet model. The classical Bohr model of an atom as a small planetary system may thus in fact be closer to reality than a probabilistic modern quantum model (with the electron charge density corresponding to the elastic rubber and the proton the central heavy ball).

In any case that is the basic idea of RealQM as a hope for the New Year out of the 100 year mystery of stdQM…

New Years Gift: You can play with this interactive code to explore how the energy of two hydrogen atoms depends on distance to find the molecule H2 as the configuration with minimum energy.