# Doubt regarding measurement of spin in quantum mechanics as per 1st chapter of Quantum Mechanics, A Theoretical Minimum by Leonard Susskind

Susskind starts with an experiment in which he measures the spin of a particle , which can either take a value of $$+1$$ or $$-1$$ along any particular axis. He takes a measuring device $$\mathcal A$$ which can be aligned at different angles to measure the spin.

Suppose he measures the particle keeping $$\mathcal A$$ aligned to the $$+z$$ axis. It either returns $$spin =+1$$ or $$spin =-1$$. Say it is $$+1$$. If we do not disturb the particle and measure the spin again, after resetting $$\mathcal A$$, it again shows $$+1$$ and continues to show $$+1$$ on each successive measurement provided we do not disturb the particle and $$\mathcal A$$ is aligned along $$+z$$ axis.

Then he turns the device around $$180^\circ$$ so it faces the $$-z$$ axis as it shows $$-1$$ and continues to do so as many times the experiment is repeated. Now he mentions that if the device is turned $$90^\circ$$ instead of $$180^\circ$$ so it faces the $$+x$$ direction then it either shows $$+1$$ or $$-1$$. If we measure many times with $$\mathcal A$$ it randomly shows $$+1$$ and $$-1$$ so that the average of the outputs is $$0$$.

Now if we measure the spin along $$+z$$ axis by turning the device back to its original position i.e. along $$+z$$ axis we no longer get a value of $$+1$$ over successive measurements as was the case initially, and it randomly shows $$\pm 1$$. He argues that this is because measuring the spin along $$+x$$ direction has changed the particle's spin and it is no longer $$+1$$ always in the $$+z$$ direction. He says it is the property of any quantum mechanical system that it changes when an observation or measurement is made.

What I cannot understand is why measuring along the $$+z$$ axis does not effect the spin along $$+z$$ axis. Should not the $$+1$$ spin that we initially measured repeatedly, change? Am I missing something?

Insightful question! If quantum measurements are invasive, then why don't invasive measurements disturb the very property which is being measured?

First, think about what is meant by a "measurement". Suppose I repeatedly measure some particular quantity, and each time I get a different random number, because each measurement disturbs the value of that very quantity. In what way would this be an "informative measurement"? It would not inform you anything about the next "measurement", or the previous "measurement". It might as well be a random number generator. Even if that number is in fact representative about something at the system at that particular time, it couldn't be verified by experiment.

Because we want to map our theoretical framework to experiments, physics is only interested in (potentially) "informative measurements", those which could be used to verify/predict/explain some other informative measurement. This means we look for ways to measure quantities which appear to be stable, or possibly made stable, even after we measure them.

Susskind is describing one such informative measurement. If one is careful enough, experimentally, one finds that this measurement is indeed stable, just as he describes. That's exactly why he's talking about it; if it wasn't stable experimentally, we wouldn't have to try to come up a framework that explained it. Since it is stable experimentally, these measurements are certainly telling us something about what is really going on.

Now, is it really true that a repeated measurement of spin in the z-direction doesn't change the system at all? To carefully answer that, you need to delve into quantum foundations, down where you are speculating about what is actually happening under the hood of the quantum formalism, and no one really knows what's going on down at that level. But one influential "toy model", by Rob Spekkens, explores the viewpoint where the spin in the z-direction is not even an actual quantity to be measured, but instead is a state of knowledge (knowledge about some deeper level of reality). And in the Spekkens model, such a repeated measurement can change the state of the system, but it changes it in such a way that our knowledge of the system is unchanged. I think you'll easily see that even with this wrinkle, such a measurement is still "informative" in just the right way.

In quantum mechanics, the act of measurement disturbs the system and hence influences its quantum state.

When you measure the spin in $$z$$ direction, the state of the particle is said to collapse to a specific eigenstate $$|z\rangle$$. Subsequent measures in the same direction result in identical results.

But if you instead measure the spin in the $$x$$ direction then the state collapses into the $$|x\rangle$$ state. However, the old state is disturbed and therefore, the old information is lost. So if you now again measure the spin in $$z$$ direction then you can get the spin angular momentum in either $$\pm$$ direction.

• But why does subsequent measures in the same direction give same result. If measurement changes the state of the system should not it give different results on subsequent measurements in the same direction? Commented Sep 9, 2023 at 11:27
• @SuprativMondal It is how "nature works". Or at least, if we do model things as such in our formalism, we make correct predictions. Commented Sep 9, 2023 at 11:29
• @SuprativMondal if you measure spin in a new direction, the spin will be placed in a new eigenstate on the new basis. Measuring on that same basis will of course produce the same result precisely because it is in a known eigenstate. Although it is stated in the answer that a measurement disturbs the system, obviously that is not exactly true in all cases as we see here with repeated measurements on the same basis. That statement is commonly used, but should not be taken overly literally. Commented Sep 9, 2023 at 14:24

Away from the theoretical descriptions, one can measure the spin of an electron as follows. At the output of the Stern-Gerlach experiment, thanks to the inhomogeneous magnetic field, one obtains separated electrons whose magnetic north pole is oriented downward in one partial beam and upward in the other. If I now use partial beam 1 and let it run through another identical magnetic setup, I get the same deflection of the electrons, i.e. only one spot instead of the two.

If I now arrange a magnetic field rotated by 90°, I get two spots at the output again. This must be the case, because I use dipoles, which are aligned exactly 90° to the external field.

What I cannot understand is why measuring along the +z axis does not effect the spin along +z axis. Should not the +1 spin that we initially measured repeatedly, change?

Stern and Gerlach performed an experiment using an inhomogeneous magnetic field. By observing the deflection of the electron beam in two directions, they came across the interplay between external magnetic field and electron spin. From the uniform distribution of the magnetic dipoles of the electrons from a thermal source, the particles are deflected in two directions in an inhomogeneous magnetic field.

Not as sharp spots, but because of the equal distribution of the dipole directions into two extended spots. In short, their deflection is different because of the initial position of the spins. Their polarization on the other hand is perfect, the spins point either down or up.

Its long outdated interpretation.

In the old times measurements like Stern-Gerlach were made on beams and a beam passing a spin polarisation splitter was considered as a preparation of a statistical ensemble $$\psi$$ with sure expectation $$E(s_z,psi)=1/2$$.

Passing the prepared beam through an identical splitter bedind the first, yields the results you describe.

But never in such an experiment the same electron is measured twice. The indicator electrons are absorbed and cannot be followed anymore.

So the conclusion is, its the statistical state, that changes.

This so called statistical interpretation (see eg. Ballentine as standard) has been verified 100% by the single photon experiments of Aspect (Nobel price).

The spin rotation of photons by two polarisators even happens between preparator and second preparator as Poisson statistics of completely independent single photon experiments, if the measurements is made only after the second polarisator.

Conclusion as always in quantum mechanics: time evolution is hard coded as the boundary condition (lattices eg) and the potential active in the solution process of the Schrödinger equation.

Single independent measurements on a single particle system yield constant values for eigenstates and Poisson statistics for everything else described by the expectations over the states involved.