1.5Measuring is not reading

When you measure the length of a table, the table had that length before you picked up the ruler. Measuring, in ordinary life and in classical physics, means finding out a value that was already there. This lesson is about an experiment from 1922 in which that cannot be what is going on.

Atoms as compass needles

An atom of silver behaves like a very small bar magnet. Like any magnet it has a strength and a direction, which together make an arrow called its magnetic moment, written $\boldsymbol\mu$. Otto Stern and Walther Gerlach set out to measure it.

Their method uses a magnet with one pole shaped to a sharp edge, so that the field is much stronger near that pole than near the other. In an even field a small magnet only twists. In an uneven one it is also pulled bodily, towards the strong side or away from it, and the size of the pull depends on how the little magnet is pointing. To be exact, the force is proportional to the part of $\boldsymbol\mu$ that lies along the direction of the field. Call that direction $\mathbf{e}_1$ and that part $\mu_1$:

So send a thin beam of silver atoms from an oven through such a magnet and let them land on a plate. An atom whose magnet points along the field is pushed one way, one pointing against it is pushed the other way, and one pointing sideways goes straight on. Where an atom lands tells you its $\mu_1$. The apparatus turns an invisible property of an atom into a visible position on a plate, which is exactly what a measuring instrument is supposed to do.

So the measured value of $\mu_1$ is always $+\mu$ or $-\mu$. That is strange, but one could live with it: perhaps atoms leaving an oven get lined up somehow. What makes it impossible to live with is the second thing the figure shows. Turn the magnet to any other direction and you still get two spots, now along the new direction. Whichever part of the arrow you choose to measure, it comes out as $\pm\mu$.

Three magnets, one arrow

Here is the argument that turns strangeness into contradiction. Pick three directions for the magnet, all in one plane, 120° apart, like the spokes of a three-pointed star. Three arrows of equal length arranged like that cancel:

Now take any arrow $\boldsymbol\mu$ whatever and add up its parts along the three directions. Because shadows add,

This is plain geometry. It is true of every arrow, pointing anywhere. But the experiment says that each of $\mu_1$, $\mu_2$ and $\mu_3$, when measured, is $+\mu$ or $-\mu$. Try to make those two facts agree.

There isn’t one. Three numbers that are each $+1$ or $-1$ add up to $\pm1$ or $\pm3$, never to zero.

Be careful about what this does and does not show. Nobody can measure all three parts on the same atom: the magnet can only point one way at a time, and that is an ordinary practical limit, nothing quantum about it. What we can do is choose freely, for each atom, which one to measure. And whichever we choose, we get $\pm\mu$.

Now suppose that a measurement simply read off a part of an arrow that the atom was carrying all along. Then the atom would have been carrying an arrow whose parts along all three directions were each $\pm\mu$, ready to be read whichever way we turned the magnet. No such arrow exists. So the supposition is wrong: the value that comes out was not sitting in the atom waiting to be read.

Peres puts the general point like this. Once an experiment gives discrete values for the parts of something that can be turned smoothly through any angle, those values cannot be the parts of an arrow that exists independently of the measurement.

The argument talks about the results of measurements that were not made, and it assumes that the atoms leaving the oven are unaffected by which way we later decide to turn the magnet. Both steps look harmless. Reasoning of this kind is put under a much harsher light in chapter 6 of the book, and you should hold it a little warily until then.

What a measurement is, then

If a measurement does not read out a value that was already there, what does it do? Look at what actually happens in the Stern–Gerlach apparatus.

The atom’s magnet interacts with the field, and as a result the atom’s path bends one way or the other. The path is the pointer of this instrument. By the time the atom reaches the plate the two possible paths are far enough apart to be told from each other by ordinary means. The atom then sticks to the plate and leaves a mark, which stays there. Anyone can come and look at it, as often as they like, without changing it, and everyone who looks will agree where it is.

That gives the working definition, in three parts.

• A measurement is an active process. The apparatus and the thing measured act on each other, and the result belongs to the pair of them.
• It ends in a permanent record: a mark, a click, a printed sign. Making a permanent record always involves amplifying something small into something large in a way that cannot be undone.
• The record is objective, in the sense that all observers agree about it. The “quantity” it is said to measure need not be.

One physicist of the 1930s suggested a change of wording that keeps you out of trouble: instead of saying that we measure this or that physical quantity, say that we make a measurement of this or that type. The first phrase smuggles in a value waiting to be found. The second only describes what was done.

In the language of the last lesson, a measurement is simply a test. In lesson 3.3 we shall add one refinement: a measurement is a test whose outcomes have been labelled with numbers.

Which part is quantum?

Notice how the atom was divided up. Its magnetic moment was treated as the mysterious thing under test. Its position, the path through the magnet, was treated as an ordinary classical pointer. That division is ours, not nature’s. We could have treated the whole atom, and the plate as well, as the system under test, and imagined a larger apparatus watching all of it. The theory must then give the same record, and it does. Why it does is a long story that the book leaves to its final chapter.