Agemoz's Physics Blog

I’ve done quite a bit of testing on my differential equation solver, including verifying the quantized states of the hydrogen atom. It seems to be working delightfully well, identifying non-quantized solutions as unbounded such that Psi doesn’t meet at least one of the boundary conditions.

So, I was ready to try the axial solution for a three charge configuration represented by the u,d,u proton arrangement of charged particles. I had to make a number of assumptions: that the quark charges have to be spaced in the statically stable arrangement with the two +1/3 up quarks opposite each other and the down -2/3 quark in the center. I assumed a momentum term (1/r^2 in the potential), since I did the simulation on one of the up quarks. Symmetry implies no momentum term for the down quark. I did not assume any specific separation distance, but included that as a configurable setting in the simulation. I was able to vary E (up quark energy level), separation distance, and other settings in real time to see how that impacted the Psi solutions. I only solved the case along the axis of the three particles (a 1 dimensional Schroedinger equation) since symmetry will dictate what happens off axis due to no off-axis contributions to the wave function second derivative.

I had postulated (about three posts ago) that there could not be multiple eigenstates for solutions, at most only one because of the system boundary conditions. Sure enough, running this simulation appears to show there are no time independent eigenstates. The Schroedinger equation is over-constrained by the boundary conditions.

This doesn’t mean there are no solutions, but it almost certainly means there’s no time independent solutions for this charge configuration. It’s also possible that using some other potential quantization numbers (n,l) may yield a time independent solution, but I really doubt it–now that I’ve run a number of scans on quite a few quantum well types, I’m learning how to recognize a converging solution and I am not seeing it here.

Agemoz

Tags: physics, quantum, quantum theory, special relativity

This entry was posted on July 27, 2022 at 9:34 pm and is filed under Physics. You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.