2.3When paths go dark

Add one more magnet. Three tests in a row, the first and the last identical. Nothing in this lesson needs a new fact about nature: everything follows from two things you already accept, that outcomes are discrete and that a repeated test gives the same answer. Together they force the strangest rule in physics.

Eight clusters

Go back to atoms of spin ½, with two outcomes per test. The first magnet sorts along the vertical; call its outcomes $+$ and $-$. The second sorts along the horizontal, at right angles to the first; call its outcomes $L$ and $R$. The third is a copy of the first.

Every atom now carries a three-part label $(m, \mu, n)$: how it left the first, the second and the third magnet. There are $2\times2\times2 = 8$ labels, and eight clusters of atoms on the plate.

How many atoms are in each? Start with a random mixture from the oven: half leave the first magnet by $+$. With the magnets at right angles the table of the last lesson has every entry equal to $\tfrac12$, so half of those leave the second by $L$, and half of those leave the third by $+$. Each of the eight clusters holds

In particular, of the atoms that left the first magnet by $+$, half leave the third by $-$. The test in the middle has made the third test disagree with the first, half the time. That is the scrambling mentioned in lesson 2.1: testing along the horizontal wipes out what was known along the vertical.

Turn the middle magnet down

Now reduce the field of the middle magnet, gradually. It deflects its two beams less, so they come out closer together. On the plate, the cluster $(m, L, n)$ and the cluster $(m, R, n)$, which differ only in what happened in the middle, slide towards each other. While they can still be told apart, each keeps its eighth.

Keep going until the field is zero and each pair has merged into one.

When two beams that have come by different routes overlap and reinforce or cancel each other, it is called interference. You have met it already: this is the overlapping beams of lesson 1.3 again, with atoms in place of photons.

E. Interference. If a system can go from a given preparation to a given test by several possible paths, the probability of each outcome of that test is not, in general, the sum of the probabilities belonging to the separate paths.

Has probability theory been broken?

It looks as though it has. An atom that started $+$ and ended $-$ must, surely, have gone through the middle by $L$ or by $R$. Those are exclusive alternatives, and for exclusive alternatives probabilities add. An eighth plus an eighth is a quarter. Where did the quarter go?

The rule for adding probabilities is a rule about events: things that happen, and are recorded as having happened. With the middle magnet on, the two routes end at different places on the plate. Each atom leaves a mark that says which way it went. “It went by $L$” is an event, and the probabilities add exactly as they should: each lane holds two clusters of an eighth, a quarter in all.

With the middle magnet off, nothing anywhere records $L$ or $R$. There is no mark, no click, no trace. Then “it went by $L$” is not an event. It is a sentence we would like to be able to say, and the theory of probability has no jurisdiction over it.

This is where quantum physics parts from classical physics for good. In classical physics one can always imagine watching without disturbing, so every route taken counts as an event whether anyone looked or not. Here that is not so. The working rule is:

• if which path was taken could be found out, even in principle, without disturbing the system, the probabilities of the paths add;
• if it could not, they do not.

What does add, in the second case, is the subject of the next lesson.

Where this came from

It is worth noticing how little went into that argument. No equation of motion, no model of an atom. Only this:

A quantum test can depend on a setting that is varied smoothly, such as the strength of a magnet or the angle of a crystal, and yet its outcomes stay fixed and discrete.

The middle field can be dialled smoothly down to nothing. At every strength above nothing, atoms come out in separate beams with fixed shares. At nothing, the first and last tests must agree. The only way to join those two requirements is for the shares to stop adding as the beams merge.

The same experiment can be done on a table-top with three crystals of calcite: horizontal against vertical, then diagonal against anti-diagonal, then horizontal against vertical again. Thin the middle crystal down and the eight beams behave exactly like the eight clusters. Nobody finds this surprising, because for light the wave theory has explained it since 1819: the heights of waves add, their brightnesses do not. The surprise is only that atoms, counted one at a time, do the same.

This lesson follows §2-5. The figure is an interactive version of the book’s Fig. 2.5.