From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Sent: Mon, December 28, 2009 4:40:39 PM Subject: [math-fun] physics of particle collisions If I take two identical "colliding" particles -- e.g., electrons -- which repel one another, then in the center-of-mass coordinate system the two particles can approach one another at any angle, including head-on. They may or may not actually "meet", because the repelling force eventually becomes large enough to cause them to bounce away from one another. Since the particles approaching head-on are identical, I can't really tell whether both particles bounced or simply went straight through. The angles after a "collision" are smoothly/continuously related to the angles prior to the collision. Are there any examples of real particles where this isn't so? It wouldn't necessarily violate conservation of energy or momentum, would it? For example, suppose two identical particles approached one another head-on, but instead of bouncing back or going straight through, they shot off at right angles to the original direction, but in equal & opposite directions & with the same energies as before. In 2D, this is well-defined, but in 3D the particles would somehow have to "choose" an arbitrary direction, and then agree on it. So perhaps such a physics might only work for 2D? Clearly, such a 3D physics would violate the conservation of information (it couldn't be reversed), but quantum physics would seem to do this all the time, so that can't be a show-stopper. _______________________________________________ The concept of a head-on collision is not well defined in quantum mechanics since the particles cannot have exact positions and velocities. One can resolve the collision wavefunction into angular momentum eigenstates and specify the scattering parameters as a function of the angular momentum quantum number L (where I'm ignoring the complications due to spin). The closest to head-on is L=0, but in that state the scattering is spherically symmetric. The scattering amplitude is described by a function f(theta), where theta is the scattering angle, and |f(theta)|^2 gives the probability per unit solid angle of scattering into direction (theta,phi), also called the differential cross section. In the case of identical particles, scattering at angle theta is indistinguishable from scattering at angle pi-theta. The scattering amplitude is then f(theta) +- f(pi-theta), the +- depending whether the particles are bosons or fermions, and also on the symmetry of the spin part. When the amplitude is squared, there is an interference term that leads to ripples in the differential cross section. It is a remarkable fact that for the Coulomb force, the quantum mechanical and classical Rutherford differential cross sections agree. But when the particles are identical, the ripples are experimentally observed on top of the otherwise smooth function. -- Gene