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If you're not completely confused by quantum mechanics, you do not understand it. (1911-) First, let's consider how are ordinarily computed: When an event consists of two mutually exclusive events, its probability is the sum of the probabilities of those two events. Similarly, when an event is the conjunction of two statistically independent events, its probability is the product of the probabilities of those two events. For example, if you roll a, the probability of obtaining a multiple of 3 is 1/3 = 1/6+1/6; it's the sum of the probabilities (1/6 each) of the two mutually exclusive events '3' and '6'.

You add probabilities when the component events can't happen together (the outcome of the roll cannot be both '3' and '6'). On the other hand, the probability of rolling two fair dice without obtaining a 6 is 25/36 = (5/6)(5/6); it's the product of the probabilities (5/6 each) of two independent events, each consisting of not rolling a 6 with each throw. Quantum Logic and [Complex] Probability Amplitudes: In the quantum realm, as long as two logical possibilities are not actually observed, they can be neither exclusive nor independent and the above does not apply. Instead, quantum mechanical probability amplitudes are defined as whose absolute values squared correspond to ordinary probabilities.

The phases (the angular directions) of such complex numbers have no classical equivalents (although they happen to provide a for the existence of the conserved classical quantity known as electric charge). To obtain the amplitude of an event with two unobserved logical components: • For EITHER-OR (exclusive) components, the amplitudes are added. • For AND (independent) components, the amplitudes are multiplied. In practice, ' AND components' are successive steps that could logically lead to the desired outcome, forming what's called an acceptable history for that outcome. The ' EITHER-OR components', whose amplitudes are to be added, are thus all the possible histories logically leading up to the same outcome. Following, the whole thing is therefore called a 'sum over histories'.

These algebraic manipulations are a mind-boggling substitute for statistical logic, but that's the way the physical universe appears to work. The above quantum logic normally applies only at the microscopic level, where 'observation' of individual components is either impossible or would introduce an unacceptable disturbance. At the macroscopic level, the observation of a combined outcome usually implies that all relevant components are somehow 'observed' as well (and the ordinary algebra of probabilities applies). For example, in our examples involving dice, you cannot tell if the outcome of a throw is a multiple of 3 unless you actually observe the precise outcome and will thus know if it's a '3' or a '6', or something else.

Similarly, to know that you haven't obtained a '6' in a double throw, you must observe separately the outcome of each throw. Surprisingly enough, when the logical components of an event are only imperfectly observed (with some remaining uncertainty), the probability of the outcome is somewhere between what the quantum rules say and what the classical rules would predict. (2007-07-19) On the 'Statistics' of Elementary Particles A direct consequence of quantum logic: Pauli's Exclusion Principle In very general terms, you may call 'particle' some part of a quantum system. Swapping (or switching) a pair of particles is making one particle take the place of the other and vice versa, while leaving everything else unchanged. Although swapping particles may deeply affect a quantum system, swapping twice will certainly not change anything since, by definition, this is like doing nothing at all. 'Swapping' can be defined as something that does nothing if you do it twice.