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# Interfering billiard balls

The miniaturisation of the billiard leads to two remarkable changes in the rules of the game. The first new rule is the uncertainty principle, formulated in 1927 by the physicist Werner Heisenberg. The uncertainty principle states that it is not possible as a matter of principle to fix with complete certainty both the position and the velocity of the electron. To be specific, if the opening through which the electrons are shot into the billiard is narrower than about 0.05 micrometer, then all control over the direction of motion is lost. The direction in which an electron leaves the opening has become completely random. An accurately aimed shot is therefore impossible in principle in the miniaturized billiard.

The second new rule is that of interfering paths. In a usual billiard there might be different ways to shoot the ball into one of the pockets. Suppose that the ball has to pass an obstacle (another ball, for example) in order to reach a pocket. The player can try to pass it from the left or from the right. What he or she will do, is to choose the path which looks most promising and ignore the other one. In the miniaturized billiard the situation is very different. First of all it is not possible to determine a priori which path the electron will follow. The uncertainty principle does not allow that. Only the probability of each path is determined. According to the usual rules of probability one would conclude, that the total probability for a hit is the sum of the two probabilities for a hit via the left and right paths. The rule for adding probabilities in the miniaturized billiard is different. Under certain circumstances the total probability for reaching the pocket can be zero, even though the individual left and right probabilities are non-zero. One speaks of destructive interference. Can you imagine, seeing two paths into the pocket, knowing that the electron has to follow one of these two, and yet finding the pocket empty no matter how often you repeat your shot.

Destructive interference occurs if the two paths differ in length. How much they should differ depends on a property of the electron called its wavelength. In a two-dimensional electron gas the wavelength is about 0.05 micrometer. The condition for destructive interference is that the path length should be an odd multiple of half the wavelength. It is one of the remarkable predictions of quantum mechanics that particles such as an electron sometimes behave as a wave. The miniaturized billiard is called a "quantum billiard" if interference effects govern the motion of the electrons. This is the case if the wavelength of the electrons is not much smaller than the size of the openings. A real billiard ball has a wavelength too, but it is so extremely small that even professional billiard players do not need to worry about interference effects.   Next: Quantum point contacts Up: Chaos in Quantum Billiards Previous: Quantum billiards