A Particle in Mexican Hat Potential

The potential here is a standard harmonic oscillator parabola with a guassian "bump" in the middle so it looks like an upside down Mexican hat. This potential is believed to be the "symmerty breaking" feature that mediates the Higgs boson in the Standard Model of quantum field theory. Of course the potential in quantum field theory is 3 dimensional but three dimensions are hard to depict so I have chosen 1 dimension, x, here. As labeled, the black trace is the potential that the particle is in, the red trace is the force it feels at any given location, the blue trace is the signed velocity of the particle, and the green trace is the time spent per unit distance moved by the particle. The green trace value should be proportional to the envelope of the square of the quantum mechanical wave function, psi, for the quantum mode which corresponds to the initial potential energy of the particle. But in quantum field theory, the particle will sometimes tunnel through the center hill of the potential, so a simple relation to the classical value if 1/|vx| cannot be expected. For more information see Quantum Mechanical Modes