Quantum Pong

I have recently been pondering an idea that I had for adding a twist to the classic arcade game Pong. The game would essentially be described as "Mr. Tompkins plays Pong" The basic game structure of Pong would remain unchanged, with two players controlling paddles on either side of the playing field, trying to reflect the ball past the opponent's paddle. The catch would be that the ball would be treated using quantum mechanics.

The first thing that this means is that according to Heisenberg's uncertainty principle the exact position and the exact momentum of the ball cannot be known at the same time. Heisenberg's uncertainty principle gives an approximate relationship between minimum uncertainty in position and momentum. That is that the product of the two uncertainties are at least equal to the reduced Planck's constant. In this new Pong game Planck's constant (h) would be adjustable so that at very low values (e.g the normal value of h = 6.626068 × 10^-34 m² kg / s) the game would play exactly like normal pong. However at much higher values of h the players would not know the exact position and momentum of the ball, thus there would only be a nebulous cloud where the ball probably is.

Fortunately for Quantum Pong this nebulous cloud can be described very precisely by using the Schrodinger equation. Specifically we would need the two-dimensional, time-dependent, non-relativistic version of this equation Wikipedia gives the following as the general Schroedinger equation.

This represents quite a bit of math but some of the more important elements can be lifted out to gain some understanding of what is going on. H(t) is the Hamiltonian and is essentially the total energy. This is generally given by the sum of the kinetic energy and the potential energy. Because for now we don't need to consider relativity the kinetic energy would be given by this equation from Wikipedia.

The paddles and walls would be the only regions of non-zero potential energy and we would simply consider them to be regions of infinite potential. That means simply that it is impossible for the ball to exist in these regions and will therefore reflect completely off of walls and paddles.

The ψ(t) in the equation is the Greek letter psi and represents the wave function. This by itself is a rather esoteric concept but it is tied closely the nebulous cloud of probability mentioned above. This cloud can be given in more mathematical terms as the probability density function. This function gives the infinitesimal probability that the ball will be found at any given position. Obviously if one looks at the probability of finding the ball at a very specific location the probability will be basically zero. If you look at the probability of finding the ball in a specific region (e.g. the region beyond one players paddle) this probability could be a decent sized probability between 0 and 1. This probability is given mathematically by the integral of the probability density function over the entire region in question. It is important to note that the integral of this function over all the space being considered is one. Using other words this becomes more obvious, because the probability that the ball is found somewhere in all of space is one hundred percent. The mathematical connection between the wave function and the probability is given in this equation from Wikipedia.

P is the probability that the ball will be found in a region epsilon and the probability density is actually given by the magnitude of the wave function squared.

So essentially quantum pong would consist of two players controlling the paddles and reflecting the cloud of probability back and forth at each other. No matter how solidly the paddle was in the center of the cloud when it hit there would be a possibility that the ball continued past the paddle. This would be represented by small parts of the cloud leaking off both players ends of the field untill eventually the probability that the ball is still between the two paddles is very small compared to the probability that it has bled off into the one of the end regions.

Possibly scoring schemes are probably the most interesting aspect of this game. One possibility is that a player gains score based simply on the possibility that the ball has past into the opponents end region and the ball is simply reset when the probability that the ball is still in the playing field goes below a set amount. However one of the most perplexing aspects of quantum mechanics is looking at when the wave function collapses and the exact position of the ball is known. This event corresponds to observation of the system but what constitutes an observation is very unclear. My physics professor suggested that the game should be resolved when at some random or perhaps predetermined event the entire pong game is observed and then the probabilities are evaluated. The ball would be found either still in the playing field in one of the player's end regions. Points would then be rewarded accordingly and the game would reset. But perhaps it might also be possible to count the paddles as observers and to collapse the wave function whenever there was an interaction with a paddle. In this case determining whether there was an interaction or not could get a little hairy.

Other random additions could include:
adding relativity and having a speed of light knob in addition to the Planck's constant knob.
add structure's such as double slits or diffraction gratings in the middle of the field to observe wave effects.
treating the paddles themselves with quantum mechanics.
adding more interesting potential functions such a gravity well in the middle of the playing field.

Also perhaps this idea could be applied to other classic arcade games. Maybe quantum space invaders.