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.

Small Number

My professor of mathematics recently emailed out an interesting problem to his students:

Problem 2006-4. Small Number. What is the smallest number that cannot be described by fewer than thirteen words?

My first response to this question which I thought was close to being right was: "The smallest number that can not be described by fewer than thirteen words." This sounded really nice to me as i counted up the words and saw there were thirteen. However very quickly it becomes apparent that this same number is described by the sentence "The smallest number that can't be described by fewer than thirteen words." (It is also ruined when I spell "can not" correctly as "cannot") Which contains twelve words and therefore ruins my first solution.

My next train of thought was to describe as concisely as possible a number which was less than any number that can be described by 13 words or less. Results in this category had rather bad grammar and were probably too vague to describe one number. I was trying to come up with sentences like "Number less than number less than smallest number minimally described in thirteen words." And this last attempt seems pretty good because by its own definition it is smaller than any other solution that might be described minimally by thirteen words.
However at this point i started thinking about other languages. Other languages definitely still use words and the problem mentions nothing about English only solutions. Also in other languages different amounts of words can be used to describe the same thing described in English. German is one language that is somewhat known for compound words. So maybe if i find some language where the translation of "less than" only takes up one word i could describe the same number but with less than thirteen words.

So at this point the problem takes a quick turn into linguistics where the goal is to find the language which can mostly concisely describe numbers and relationships between them. This would probably lead to the language of mathematics where mathematical characters might be considered words. However mathematics is not a natural language. That is to say it isn't tied to any specific cultural group, instead it is a man made language. If we take the step to allow words from synthesized languages any sentence of words in a given language can be translated into a newly forged language in which that sentence is translated as "figomstormo" and now describes the same number is described in one word which is fewer than thirteen. This reasoning applies to any solution to the problem which uses words at all. The solution "one to the negative nine to the negative nine to the negative nine" Doesn't work as a solution because in my new language that quantity is given the name poiusltur.

At this point it might seem as though the puzzle is unsolvable. However another possibility exists. In order to avoid the translation problem the sentence itself must reference itself and its form so that if it is translated to another language it is no longer the same sentence. For example "The number described in this sentence" would not describe the same number as it would if it were translated into french because it would then be a sentence in french and would be a different sentence. So somehow the sentence must be untranslatable and it seams as though being self referential might somehow accomplish this. Of course my self-referential sentence above doesn't describe a number with a specific value, and also it has fewer than thirteen words. So since I am trying to find a sentence that meets these requirements it is untranslatable (or at least it can't lose words in correct translation), it has thirteen words and is is the smallest number that cannot be described by fewer than thirteen words?

"This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words" The previous sentence describes the number required in the problem. The sentence cannot be changed or translated to contain fewer than thirteen words. It could be translated but any correct translation would have to contain thirteen words or more and it would describe the same number not a smaller one. The only part of it that seems incorrect is that it doesn't describe an exact number, however if it did describe an exact number then by previous arguments it would not be a solution.

"This sentence contains thirteen words or more and describes a number smaller than the smallest number that cannot be described by fewer than thirteen words" Hmm.. of course this self referential thing could get out of hand. Perhaps an infinite loop could be used here to create a infinitely long sentence with nested "a number smaller than a number smaller than..." But really as long as these sentences contain a bit of self reference with respect to the number of words in them (which they must because other wise they could be translated down to one word) then they cannot be described by fewer than thirteen words and therefore the original sentence describes the same number.

Of course the really troubling thing about "This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words", is that despite the sentence's inability to lose numbers it still only describes "The smallest number that cannot be described by fewer than thirteen words." which has twelve words and therefore fails as a solution. Of course it is now on this wonderful trip that I finally see that the question itself asks for "The smallest number that cannot be described by fewer than thirteen words?" The nature of questions and answers indicate that the best solution will be the one that is described by "the smallest number that cannot be described by fewer than thirteen words?" and if it is described by this then it is described by fewer than thirteen words and therefore is not a solution. So there can be no solution.