IIG.  Probability in Quantum Mechanics.

1. Quantum Mechanics Gives Probabilities. 

Quantum mechanics allows more than one possible outcome to an experiment, but only one of them is perceived.  The theory cannot say, in a specific instance, which possible outcome will be perceived.  However, if an experiment, say the half-silvered mirror experiment, is run many times, quantum mechanics can predict, on average, the fraction of experiments that a given outcome will be observed.

2. Illustration of Probabilities in the
Half-Silvered Mirror Experiment. 

To illustrate, we will use the half-silvered mirror experiment of sections IID  and IIE.  The equations of quantum mechanics tell how the wave function proceeds in time.  So when it hits the mirror, the equations (which depend on the thickness of the silver coating) will tell how much of the wave function goes along the vertical path and how much goes along the horizontal path.  (A rough picture of “how much” of the wave function goes along a particular path is that it is the average density of the “cloud” times the volume in which the wave function is nonzero.) Suppose the thickness of the silver is such that 60% of the wave function goes along the vertical path and 40% along the horizontal path.  Then if we have a counter attached to each detector,  and if we run the experiment 1,000 times, the detector on the vertical path will register the passage of the wave function approximately 600 times and the detector on the horizontal path will register the passage of the wave function approximately 400 times.

3. No Explanation of the Probability Law. 

Is there some rationale or conceptual picture that explains why the probability depends on the relative amount of the wave function on each “path?”  There is not.  This is the second mystery of quantum mechanics.  Often, it is simply assumed to be a separate law of nature, a law that connects the other (time evolution) mathematics of quantum mechanics to our perceptions.  (See, however, section IIIE, subsections 6 and 7.)