Quantum Mechanical Particles

According to quantum mechanics, a "particle" is not a hard little lump. It is elusive, and usually the only thing you can know about its location is the probability that it can be found within a given spatial region (whatever "it" is if you did find it). For a handful of simple cases the probability can be determined from a solution of the Schrödinger equation. I was not satisfied by staring at the mathematical equations for the solutions, so I created a few animations to get a better feel for this odd creature. The animations are not cartoons; they are derived from exact solutions of the Schrödinger equation. I am not going to describe the solution conditions in great detail. I recommend a course in quantum mechanics for anyone who wants to know more. Note that the animations work with my MS and Firefox browsers, but not with my Netscape browser.

The height of the moving blue contour at any point is proportional to the probability of finding the particle at that point. The first animation shows a particle colliding with an obstacle. The obstacle occupies the region between the vertical red lines. The Newtonian solution would show a billiard ball (a moving vertical blue line of probability) simply bouncing off the wall. The quantum mechanics solution shows a blob of probability approaching the obstacle, sort of fracturing on impact, and mostly bouncing back. A little bit of probability sneaks through the obstacle. In other words, the particle could be on the far side of the obstacle.This phenomenon, called "tunneling," is totally impossible according to classical physics. It is also the basis of some electronic devices, so it really does happen.

A rigid billiard ball bouncing off of a rigid wall creates an impulsive force of essentially zero time duration (a delta function). The quantum mechanical collision force is exerted over roughly the time it takes the particle to travel a distance equal to its diameter (meaning the size of the region where most of the probability is concentrated). The bandwidth of the noise created by a random series of the billiard ball collisions is infinite. The bandwidth for quantum mechanical collisions is large, but finite (see Molecular Noise and Blackbody Radiation).

The second animation is for an obstacle which is sort of like a speed bump, where the classical solution is that the particle always passes over the obstacle. For the quantum mechanics solution there is some probability that it bounces back.

The third animation is sort of like a ditch in the middle of the road. I originally stated that classically the particle would fall in and stay at the bottom of the ditch. But I realized later that the particle would gain kinetic energy as it fell, and if it bounced off the bottom in a lossless collision it could get back out. A quantum mechanical particle just hates to be pinned down within a fixed location, and it does manage to squiggle out.

The final animation represents a wonderful gizmo called a "harmonic oscillator." One example (more or less) of a harmonic oscillator is a child on a swing. The Newtonian form of the case shown here is the red particle attached to a point by the black spring (no gravity, no friction, and the particle does not collide with the attachment point). The blue quantum mechanical blob oscillates with exactly the same frequency as the Newtonian solution, and the most probable location is close to the Newtonian solution. However there is some probability that the particle moves a bit past the limits of the Newtonian motion, which is again classically impossible.

If anyone would like a copy of the Matlab programs used to generate these animations just send me an e-mail.

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