This Quantum World

Two simple rules

The bare facts

The mathematical formalism of quantum mechanics is (first of all, whatever else it may be) a probability algorithm. It provides us with tools for calculating the probabilities of measurement outcomes. These tools are like machines with an input and an output. Insert the actual outcome of a measurement, and out pop the probabilities of the possible outcomes of other measurements. At the heart of this algorithm are two simple rules.

RFA

Transport VII. The digital art by Eric J. Heller is inspired by the quantum realm of electrons, atoms, and molecules. Here we see the flow pattern for electrons riding over a two dimensional bumpy surface. The whole picture occupies a hundredth of the width of a human hair.

Two definitions to begin with:

Alternatives are possible sequences of measurement outcomes.
With each alternative is associated a complex number called amplitude.

Suppose that you want to calculate the probability of a possible outcome of a measurement M₂ given the actual outcome of an earlier measurement M₁. Here is what you have to do:

Choose a sequence of measurements that may be made in the meantime.
Assign an amplitude to each alternative.
Apply either of the following Rules:

Rule A: If the intermediate measurements are made (or if it is merely possible to infer from other measurements what their outcomes would have been if they had been made), first square the absolute values of the amplitudes of the alternatives and then add the results.

Rule B: If the intermediate measurements are not made (and if it is not possible to infer from other measurements what their outcomes would have been), first add the amplitudes of the alternatives and then square the absolute value of the result.

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