Principles and Concepts
of Quantum Mechanics
Implications of
Quantum Mechanics

10. The Probability Law of Quantum Mechanics.
Randomness.

Summary
The single quantum mechanical version of reality we perceive is chosen at random, with a probability that depends on the ‘size’ of the wave function.

One of the most puzzling things about quantum mechanics is that, when there are many versions, the version we perceive is not predictable; it is apparently chosen at random (by what means we do not know)! But if an experiment is done many times, there is a certain probability that we will perceive outcome i. We will illustrate using a modified Schrödinger’s cat experiment.

A. Randomness in a Modified Schrödinger’s cat experiment. We wish to run this experiment many times, so, to avoid killing so many cats, we will modify it. Instead of the detector being hooked up to the box with the cat inside, it is connected to a memory device. Each time the experiment is run (the detector is turned on for 5 seconds and then turned off), the device records a 1 if a nuclear decay is detected and a 0 if no decay is detected. On a given run, quantum mechanics does not tell us whether a 0 or a 1 will appear.

B. Probability. Suppose we run the same experiment 10,000 times and we get 6,500 ones (6,500 decays) and 3,500 zeros (3,500 no decays). Now we run the experiment 10,000 times again. Then, from having done similar sets of experiments many times, we know that we will get close to 6,500 ones and 3,500 zeros. That is, after many runs of the experiment, the fraction of zeros and ones is not just any number between 0 and 1. Instead it will be near .65 for ones and .35 for zeros. That is, there is a probability of (near) .65 of obtaining a one and a probability of (near) .35 of obtaining a zero.

C. Probability and the wave function. There is a formula linking a property of the wave function and probability. Suppose we focus on a single nucleus. At time 0, say, the nucleus is not decayed, so its wave function is [not decayed]. But after 5 seconds, its wave function is a combination of [not decayed] and [decayed]. This is written as a sum in quantum mechanics, so we have

[wave function after 5 sec]=a(0)[not decayed]+a(1)[decayed]

The a(0) and a(1) are ‘coefficients;’ their values—actually their values squared—tell ‘how much’ of the wave function corresponds to not decayed and how much to decayed. Pictorially, |a(0)|2 and |a(1)| 2 tell ‘how much’ of the ‘mist’ making up the wave function is in the not-decayed aspect and how much is in the decayed aspect.
The link to probability is the following: Suppose

|a(0)|2=.35, |a(1)|2 =.65.

Then the probability of perceiving [not decayed] is .35 and the probability of perceiving [decayed] is .65. (Note that in accord with principle [P1]sec 1, we are not saying that the probability of the outcome being not decayed is .35; we are only saying the outcome of perceiving not decayed is .35.)

D. The |a(i)|2 probability law. As before, we start out at time 0 in the [not decayed] state. As time progresses, the state will be some combination of [not decayed] and [decayed].

[wave function at time t]=a(0,t)[not decayed]+a(1,t)[decayed]
|a(0,t)| 2 + |a(1,t)| 2=1

The probabilities for the two states at time t are then

probability of not decayed at time t=|a(0,t)| 2
probability of decayed at time t=|a(1,t)| 2

This statement generalizes. Suppose at time t, the quantum state of a system is the sum of several possibilities, , with the ith possible ‘state,’ i , having coefficient a(i). Then
[P9] The |a(i)|2 probability law. If an experiment is run many times, a physical reality with characteristics corresponding to version i will be perceived a fraction |a(i)|2 of the time.
Exactly the same law, but in a different guise, is called the probability law, or the Born rule (Born first proposed the idea of probability), where is the wave function at position x. (In this form, it is often interpreted as implying that the probability of finding the particle at x is . But this is an unwarranted interpretation; see No Evidence for Particles.)

E. Conservation of probability.
There are two important points about principle 3-1. The first is that coefficients a(i) are determined by the Schrödinger equation. The second is that, even though the coefficients may change in time, because of the special nature of the Schrödinger equation, the sum of the coefficients squared is always 1. And so, from principle [P9], the sum of the probabilities is always 1, exactly as it should be. This is called conservation of probability.

F. Perception vs. ‘actuality.’ Because our world appears to be objective, it is most natural to assume there is a single, actual, objective outcome to an experiment. But one is not allowed, a priori, to make that assumption when dealing with the interpretation of quantum mechanics. In fact, as we indicated in section 2, the perceptions predicted by quantum mechanics agree quite well with what we actually perceive without assuming there is just a single version of reality.

G. Consistency of quantum mechanics proper and the probability law. Principle [P9] cannot be deduced solely from the Schrödinger equation plus the properties of the wave functions. However, suppose we agree that the probability of perceiving outcome i is a function of |a(i)|2 so that p(i)= f ( |a(i)|2). Then there are several ways to show that the only functional form consistent with the mathematics of quantum mechanics is the conventional f ( |a(i)|2) = |a(i)|2. The law f ( |a(i)|2) = (|a(i)|2)2, for example, would give inconsistent results. I think this observation, that

[P10] The |a(i)|2 probability law is the only functional form consistent with the rest of conventional quantum mechanics.

strongly suggests that the origin of the law must, to a large extent, somehow be within conventional quantum mechanics, even though we haven’t yet figured out how (see, however, the section on Probability in the Mind-MIND Interpretation). That is, it doesn’t make sense for there to be a probabilistic ‘mechanism’ that is entirely independent of the laws of quantum mechanics, but just happens to be consistent with those laws.  