4.3: Repeated Measurements of Spin

If you have a beam of electrons with randomly oriented spins, when you measure the z spin of the beam you get half of the electrons showing a spin of \(\ \hbar / 2\) and half showing a spin of \(\ -\hbar / 2\).

Suppose that you block off the beam with negative z spin. Send the beam with positive z spin into a second Stern-Gerlach machine. What do you get?

Unsurprisingly, the second SGz machine shows that every electron that comes into it has +z spin. You wouldn’t expect anything else. After all, we divided up the electrons that went into the first SGz machine based on their z spin, and threw out the ones that didn’t have +z spin.

What if you put the +1/2 out put of the SGz machine into an SGx machine?

When the x component of the spin angular momentum of an electron is measured, just as with the z component it only takes on values of \(\ +\hbar / 2\) or \(\ -\hbar / 2\). Because the x axis is perpendicular to the z axis, you wouldn’t expect knowing whether the z spin of the electron was along the +z or −z direction to tell you anything about the whether the x spin was positive or negative. And, indeed, that’s what’s observed. For each electron with spin +z that goes into an SGx machine, there’s a 50% chance it will come out the +x output, and a 50% chance it will come out the −x output.

Things get interesting when you add one more SG machine to the mix. Take the electrons that were first measured to have +z spin, and were then measured to have +x spin. That is, at the first SG machine (an SGz machine), we’re throwing out the electrons with −z spin, and at the second SG machine (an SGx machine), we’re throwing out the electrons with −x spin. What happens if you send these electrons through another SGz machine? You might expect all of them to come out through the +z output; after all, we already know from a previous measurement that all of these electrons have a positive z component of spin angular momentum. In fact, however, this is not what’s observed! If you construct this experiment, you find that the final SG machine, an SGz machine, puts out electrons through either output with a 50% chance for each!

The fact that angular momenta were quantized was the first thing about quantum mechanics that was completely at odds with our intuition and our experience with classical physics. This is the second thing. It seems that, somehow, by measuring the x spin of the electrons, we lost information about the z spin of the electrons. To explain this and similar experiments, the theory of quantum mechanics includes formalism that shows that it is impossible to know certain pairs of observables at the same time. This is related to the famous Heisenberg Uncertain Principle, about which we will say more in a later chapter. If you know the z spin of an electron, you know nothing about its x spin; were you to measure the x spin, you have a 50% chance of measuring either +1/2 or −1/2. Likewise, if you know the x spin, you know nothing about its z spin.

The same result is observed if, instead of the +x output, we take the −x output of the second machine. We have a beam of electrons who all were first measured to have positive z spin, and were then measured to have negative x spin. As before, if we measure the z spin again, we find that we have a 50% chance of measuring +z, and a 50% chance of measuring −z.

The quantum weirdness goes deeper than that. It turns out that it’s not just that you don’t know. The particles themselves do not have a definite state! If you’ve measured the z spin of an electron, the electron does not have a definite x spin! The jargon we use to describe this is to say that the electron is in an “indefinite state”, or that it is in a “mixture of states”. In this case, the x spin state of the electron is a mixture of the +1/2 and −1/2 states.

In Quantum Mechanics, certain observables may not be known— do not even take on definite values— at the same time as certain other observables.

At this point, you might object, reasonably so, that we could have neglected an effect of our measuring devices. Charged particles with angular momenta interact with magnetic fields. Could it not be that our devices aren’t only deflecting the electrons’ paths, but also rotating those electrons? That is, after the first SGz machine, the electrons coming out of the +z output do have z spin of +1/2. But when they go through the SGx machine, perhaps it’s not just measuring the x spin, but also rotating the electrons so that their angular momenta no longer as up along the z axis as they were before. Indeed, it’s clear that the state of the system is changed when the x angular momentum is measured. Must it really be something particular to quantum mechanics?

To answer that question, suppose that after we’ve sent the beam through the SGx machine, dividing it into a beam of electrons with positive x spin and a second beam with negative x spin, we recombine those two beams. Take the recombined beam and put that into the third SGz machine. What do we observe?

The beam did go through the SGx machine, so any effect it has on the beam has happened. Remember that the beam coming out of the +1/2 output of the SGx machine had an indeterminate z spin; likewise for the beam coming out of the −1/2 output of the SGx machine. Yet, somehow, by recombining the beams, we are able to restore the information about the z spin of the electrons! Again, if we make it so that the beam has a very low intensity, and only one electron is going through the apparatus at a time, exactly the same result is observed. In a sense, by recombining the beams, we never really did measure the x spin of the electron. Sure, the SGx machine measured it. . . but we never let the measurement go beyond that, we never let it go into any other experimental apparatus, we didn’t let any physicists know about it, we didn’t record the spin of any given electron.

There is something peculiar about measurement that changes the state of a system. Yet, exactly what is a measurement is not entirely clear. Indeed, the “measurement problem” in quantum mechanics has troubled physicists for nearly a century, and remains a point of active debate today. We will discuss this in greater detail in a later chapter.

For the time being, however, review the results of the various experiments combining SG machines together. The set of observations that we see can not be explained by pure classical physics. The fact that particles have quantized values is already unfamiliar enough. Add to that the fact that for some pairs of observables, such as the z and x components of angular momentum, you can’t know both observables at the same time. Finally, on top of all of that, you can destroy, but then somehow reconstruct, information about the state of a given observable based on whether there are multiple paths a particle could have followed, and how those paths are put together.

In future chapters, we will explore the mathematical formalism that physicists have developed to model this behavior.