Understanding Measurement Probability in Electron Spin: Explained by Susskind

[rasp]

TL;DR Summary

I’m listening to the Leonard Susskind lectures from Stanford on QM and have a question on probability of finding spin on any axis.

Susskind explains how if you prepare an electron along any axis n (with an electromagnet) and then measure it along any other axis m, the probability of finding the electron with spin up or spin down is given by the angle between the axis. I have left out the linear algebra, because my question is more basic.
Is the probability measured that of finding the electron on axis m, or is the probability measured also whether the spin is up or down?

 

[PeterDonis]

Is the probability measured that of finding the electron on axis m, or is the probability measured also whether the spin is up or down?

Think of it this way:

We prepare an electron in a state of spin up along axis n.

If we measure its spin along axis m where m is the same as n, we have 100% probability of getting spin up and 0% probability of getting spin down.

If we gradually rotate the axis m that we measure along so it makes an increasing angle with axis n, the probability of getting spin up goes down and the probability of getting spin down goes up as the angle increases.

 

[PeroK]

Is the probability measured that of finding the electron on axis m, or is the probability measured also whether the spin is up or down?

There is no sense of "finding an electron on axis m". Whatever axis you choose about which to make a measure of spin you will always get $\pm \frac \hbar 2$, which correspond to spin-up and spin-down.

 

[Nugatory]

Is the probability measured that of finding the electron on axis m, or is the probability measured also whether the spin is up or down?

The probability is something that we calculate, not that we measure. We prepare the particle in a state such that it is spin-up (or down) on a given axis. We are calculating the probability that we will get spin-up (or down) on some other axis if we choose to make that measurement.

 

[PeterDonis]

The probability is something that we calculate, not that we measure.

Actually, we can do both. We can calculate a probability using QM, and we can measure a probability by running the same experiment many times on identically prepared systems and looking at the relative frequencies of outcomes.

 

[rasp]

Thanks. I’m losing this on the concepts, more than the math. I’m thinking that in a 3 dimensional space, there are many axis that are each theta degrees from an existing axis such as to produce a cone. How then do we calculate which one of the many lines to measure?
And a second question which I guess is probably dumber yet, is after we turn off the magnet which initially prepared the electron do we reorient the magnet along a new axis to check for spin. I guess it is correct to say we aren’t measuring the position of the electron but only the spin, which I believe is measured by whether a photon detector along the axis records receiving a photon or not based on the electron moving higher or lower in energy state from the magnetic field. But if we don’t get a photon emitted how do we know the electron wasn’t even on the axis versus being on the axis and pointed down or 180 in the opposite direction?
thanks.

 

[PeroK]

Thanks. I’m losing this on the concepts,

Yes, I think you've misunderstood fundamentally what's going on here.

Is there a particular experiment set-up that Susskind is describing? Sounds a bit like Stern-Gerlach.

 

[PeterDonis]

I’m thinking that in a 3 dimensional space, there are many axis that are each theta degrees from an existing axis

That's correct. The possible directions in which spin can be measured form a 2-sphere (you will see the term "Bloch sphere" used in the literature to refer to this), so to fully specify how we are measuring spin about the axis m, we need to give not only the angle through we we rotate m relative to the axis n about which we prepared the electron to be spin up, but also in which direction we do the rotation--it takes a second angle to describe that. (If you think of the axis n as pointing towards the North Pole of a sphere, then the angle between m and n is the colatitude--the angle down from the North Pole--and the second angle is the longitude--which meridian we choose to rotate the axis m along.)

In QM terms, the relative probabilities for spin up vs. spin down are the same no matter which direction we do the rotation (which meridian we choose); they depend only on the angle of rotation of m relative to n (the colatitude). But there are other more complicated measurements we could make that could give different results depending on which direction we do the rotation (which meridian). In terms of the quantum mechanical wave function of the electron, the relative probabilities for spin up vs. spin down depend only on the relative magnitudes of the two terms in the wave function (the "spin up" and "spin down" terms), which in turn depends only on the angle of rotation (the colatitude); the "which meridian" part corresponds to the relative phase of the two terms.

 

[rasp]

Thanks. I think I understand better how the calculation works but how does the many repeatable experiment to find probabilities work?
what I’m trying to say is after we have aligned the electron spin along axis n, what do we then do experimentally to measure along axis m?

 

[PeroK]

what I’m trying to say is after we have aligned the electron spin along axis n, what do we then do experimentally to measure along axis m?

Electron spin is never aligned along any axis. This is one fundamental point you have failed to grasp.

That's one way it is fundamentally different from classical spin angular momentum.

 

[PeterDonis]

after we have aligned the electron spin along axis n, what do we then do experimentally to measure along axis m?

For electron spin, the preparation process--aligning the spin along axis n--uses the same kind of apparatus as the measurement process: a magnetic field that has a gradient along a specific direction, usually called a Stern-Gerlach magnet (or SG magnet for short). Electrons passing through the field get split into two beams, an "up" beam and a "down" beam.

To prepare electrons in the state of spin up along axis n, you pass them through a SG magnet oriented along axis n and take the electrons that come out in the "up" beam. (The electrons that come out in the "down" beam just get thrown away for this experiment.) To measure the spin of those electrons about axis m, you then pass them through a second SG magnet oriented along axis m and look at which way they come out--in the "up" beam or the "down" beam of the second magnet. If you do this for a large enough number of electrons, the relative numbers of electrons in each output beam of the second SG magnet give a measurement of the relative probabilities of spin "up" and spin "down" along axis m.

 

[PeterDonis]

Electron spin is never aligned along any axis.

Actually, this isn't strictly true for spin-1/2 particles like electrons. Every possible spin state for such particles corresponds to a definite point on the Bloch sphere, and every point on the Bloch sphere corresponds to a definite direction of a spin axis along which the spin of the electron in that state is "up". (The "down" state for that axis corresponds to the antipodal point on the sphere--i.e., it's the "up" state about an axis pointing in the opposite direction.) So every spin-1/2 state can be thought of as the spin being aligned along a definite axis.

The real issues come with states involving the spins of 2 or more electrons, or spins of higher magnitude than spin-1/2, or measurements of the spin along an axis that is not aligned with the axis of the electron's state.

 

[Nugatory]

Thanks. I think I understand better how the calculation works but how does the many repeatable experiment to find probabilities work?
what I’m trying to say is after we have aligned the electron spin along axis n, what do we then do experimentally to measure along axis m?

In principle (and Google for “Stern-Gerlach single electron” to see just how much experimental difficulty is hiding behind those words) we pass a beam of electrons through an inhomogeneous magnetic field aligned on axis $n$. The magnetic field splits the incoming beam into two output beams, one of spin-up and one of spin-down particles. The spin-up beam gives us our initial state, a bunch of electrons prepared in the state “spin-up on axis $n$”. Now we feed that beam into another inhomogeneous magnetic field, this one aligned along axis $m$ and see what fraction comes out in the spin-up beam from the second device.

 

[PeroK]

Actually, this isn't strictly true for spin-1/2 particles like electrons. Every possible spin state for such particles corresponds to a definite point on the Bloch sphere, and every point on the Bloch sphere corresponds to a definite direction of a spin axis along which the spin of the electron in that state is "up". (The "down" state for that axis corresponds to the antipodal point on the sphere--i.e., it's the "up" state about an axis pointing in the opposite direction.) So every spin-1/2 state can be thought of as the spin being aligned along a definite axis.

The real issues come with states involving the spins of 2 or more electrons, or spins of higher magnitude than spin-1/2, or measurements of the spin along an axis that is not aligned with the axis of the electron's state.

I'd be surprised if the OP is talking about the Bloch sphere.

 

[Nugatory]

Electron spin is never aligned along any axis.

Actually, this isn't strictly true for spin-1/2 particles like electrons.

The two of you may be talking at cross-purposes here. I’m don’t think that PeroK was saying that the states are not one-to-one with points on the Bloch sphere, but rather that unlike classical spin the spin axis is not a classical (counterfactually definite) dynamical attribute of the particle.

 

[rasp]

Electron spin is never aligned along any axis. This is one fundamental point you have failed to grasp.

That's one way it is fundamentally different from classical spin angular momentum.

 

[rasp]

Perhaps aligned is a wrong word. What wording should we say when we prepare the electron in a magnetic field and then measure its spin along the same axis of that field? My understanding is that we will find that spin up or down to be either 100% or 0%. I hope I’m correct about that.
My most troubling questions revolve around the experimental procedures to verify the probabilities of finding the spin along any other axis.

 

[PeroK]

Perhaps aligned is a wrong word. What wording should we say when we prepare the electron in a magnetic field and then measure its spin along the same axis of that field? My understanding is that we will find that spin up or down to be either 100% or 0%. I hope I’m correct about that.
My most troubling questions revolve around the experimental procedures to verify the probabilities of finding the spin along any other axis.

Are you talking practically or theoretically? Practically, the SG experiment doesn't work with an electron in any case, as the Lorentz force dominates. It can be done with silver atoms. I'm sure you can find a description of that online.

For practical reasons, a lot of quantum experiments are done with photon polarization instead of electron spin.

 

[PeterDonis]

What wording should we say when we prepare the electron in a magnetic field and then measure its spin along the same axis of that field?

We could say that the electron is in an eigenstate of the spin measurement operator that is being used, but that quite likely doesn't help very much.

More important, though, the point @PeroK was making was not about the words being used. It was just to caution you that, while in some ways it's OK to think about an electron whose spin about a particular axis you have just measured to be "up" as having its spin aligned along that axis, in other ways it isn't. That will be true no matter what ordinary language words you use to describe what's going on. That's why physicists don't use ordinary language when they really want to be precise in describing a particular bit of physics; they use math.

 

[PeterDonis]

My most troubling questions revolve around the experimental procedures to verify the probabilities of finding the spin along any other axis.

What do you find troubling about the procedure I described in post #11? There are practical issues with actually realizing it literally with electrons, as @PeroK has pointed out, but it can certainly be realized with objects like silver atoms (that's what Stern and Gerlach originally used) whose spins can be treated like the spins of electrons for purposes of the experiment.

 

[andrew s 1905]

If it's the lectures I have watched Susskind is not describing a Stern-Gerlach experiment. He considers an isolated electron (some how fixed, in an unspecified way, in space) in a suitable magnetic field that can be aligned in different orientations.

Regards Andrew

 

[PeterDonis]

Susskind explains how if you prepare an electron along any axis n (with an electromagnet) and then measure it along any other axis m, the probability of finding the electron with spin up or spin down is given by the angle between the axis.

If it's the lectures I have watched Susskind is not describing a Stern-Gerlach experiment.

The description given in the OP certainly sounds to me like a S-G experiment, but perhaps not.

@rasp, can you give a link to the actual video? That way we can take a direct look at what Susskind is saying.

 

[Nugatory]

Perhaps aligned is a wrong word. What wording should we say when we prepare the electron in a magnetic field and then measure its spin along the same axis of that field? My understanding is that we will find that spin up or down to be either 100% or 0%. I hope I’m correct about that.

Any measurement of the spin of an electron, no matter how we've prepared the electron and no matter which axis we choose, will always give us either spin-up or spin down. There are no in-between results, and this is different from how classical spin works.

If we measure the spin on a given axis, and then measure the spin again on the same axis, the two results will teh same with 100% probability. This is just the $\theta=0$ case of preparing the electron to be spin-up (or spin-down) along axis $n$ (that's the first measurement) and then measuring its spin on the axis $m=n+\theta$ (the second measurement).

 

[Nugatory]

If it's the lectures I have watched Susskind is not describing a Stern-Gerlach experiment. He considers an isolated electron (some how fixed, in an unspecified way, in space) in a suitable magnetic field that can be aligned in different orientations.

That's an idealization intended to illustrate the underlying principles without being distracted by the experimental difficulties of separating the effects of magnetic moment from the much larger effects of the Lorentz force on an isolated electron.

The Stern-Gerlach experiment as originally performed is an electron-spin measurement; the silver atom is just a convenient carrier for the electron whose spin we're measuring. What's going on here: silver atoms are massive, electrically neutral so unaffected by the Lorentz force, and have only one unpaired outer electron so that the total magnetic moment is essentially equal to the magnetic moment of that one unpaired electron, which is what we're trying to measure.

The experiment has since been done (with much greater difficulty) with hydrogen atoms where there is no question but that we're looking at a single electron.

 

[andrew s 1905]

That's an idealization intended to illustrate the underlying principles without being distracted by the experimental difficulties of separating the effects of magnetic moment from the much larger effects of the Lorentz force on an isolated electron.

Exactly, and the up down spin state is determined by if a photon is emitted or not. A thought experiment. Regards Andrew

 

[vanhees71]

The probability is something that we calculate, not that we measure. We prepare the particle in a state such that it is spin-up (or down) on a given axis. We are calculating the probability that we will get spin-up (or down) on some other axis if we choose to make that measurement.

And we can indeed measure whether the predicted probabilities are correct by repeating the same experiment over and over again with equally prepared and measured electrons. The larger you make this ensemble with the more significance you can verify (or falsify) the predicted probability for the outcome of the polarization measurement. In this sense the probabilities are observable but, of course, only on an ensemble of many equally performed preparations of and measurements on the electrons, not by a measurement on a single electron.

 

[rasp]

And we can indeed measure whether the predicted probabilities are correct by repeating the same experiment over and over again with equally prepared and measured electrons. The larger you make this ensemble with the more significance you can verify (or falsify) the predicted probability for the outcome of the polarization measurement. In this sense the probabilities are observable but, of course, only on an ensemble of many equally performed preparations of and measurements on the electrons, not by a measurement on a single electron.

For electron spin, the preparation process--aligning the spin along axis n--uses the same kind of apparatus as the measurement process: a magnetic field that has a gradient along a specific direction, usually called a Stern-Gerlach magnet (or SG magnet for short). Electrons passing through the field get split into two beams, an "up" beam and a "down" beam.

To prepare electrons in the state of spin up along axis n, you pass them through a SG magnet oriented along axis n and take the electrons that come out in the "up" beam. (The electrons that come out in the "down" beam just get thrown away for this experiment.) To measure the spin of those electrons about axis m, you then pass them through a second SG magnet oriented along axis m and look at which way they come out--in the "up" beam or the "down" beam of the second magnet. If you do this for a large enough number of electrons, the relative numbers of electrons in each output beam of the second SG magnet give a measurement of the relative probabilities of spin "up" and spin "down" along axis m.

thank you Peter. Your answer is most helpful. QM, a fascinating journey I’m just beginning!

 

[PeterDonis]

the up down spin state is determined by if a photon is emitted or not

Huh? What experiment are you talking about? In the SG experiment the up or down result is determined by which beam the electron (or the atom, in the actual realizations that have been mentioned) comes out in. There is no photon emission anywhere.

 

[andrew s 1905]

Huh? What experiment are you talking about? In the SG experiment the up or down result is determined by which beam the electron (or the atom, in the actual realizations that have been mentioned) comes out in. There is no photon emission anywhere.

As explained it is a idealised experimental arrangement described by Susskind in his lectures. I don't know if it has been realized or is just his thought experiment. Most recently I listened to it in his entanglement lectures on YouTube.

As the OP was referring to these lectures I was just pointing out that the references to the SG experiment were not what the OP had been exposed to.

I know what a SG experiment is and how it works he was not describing it.

I would propose you listen to the lectures and then take it up with Susskind if you feel so inclined.

Regards Andrew

 

[PeterDonis]

As explained it is a idealised experimental arrangement described by Susskind in his lectures. I don't know if it has been realized or is just his thought experiment.

I'm not aware of any experiment, either an actual one or a thought experiment, where the up/down spin of an electron is measured by emission or non-emission of a photon. Perhaps there is one that I'm not aware of, or perhaps you have misunderstood something.

I would propose you listen to the lectures

Nobody has provided any links so I don't know what lectures are being referred to. Nor do any of us know whether the lecture you listened to is the same one that the OP listened to.

 

[andrew s 1905]

I'm not aware of any experiment, either an actual one or a thought experiment, where the up/down spin of an electron is measured by emission or non-emission of a photon. Perhaps there is one that I'm not aware of, or perhaps you have misunderstood something.
Nobody has provided any links so I don't know what lectures are being referred to. Nor do any of us know whether the lecture you listened to is the same one that the OP listened to.

Perhaps someone should have asked the OP for the link before assuming it was a SG arrangement.

I listened to the entanglement lectures in this set .

Hope this helps although I don't recall in which lecture it first turns up.

Regards Andrew

 

[PeterDonis]

Perhaps someone should have asked the OP for the link before assuming it was a SG arrangement.

Someone (namely me) has asked the OP for a link, because of the question you raised. We'll see if one is provided.

 

[PeterDonis]

Hope this helps although I don't recall in which lecture it first turns up.

Then I'm afraid it won't help. I'm not going to watch multiple lectures of close to 2 hours each just to try and spot what you think you saw.

 

[andrew s 1905]

Then I'm afraid it won't help. I'm not going to watch multiple lectures of close to 2 hours each just to try and spot what you think you saw.

If you don't believe me that's your problem not mine.

Regards Andrew

 

[PeterDonis]

If you don't believe me

It's not a question of "belief". It's a question of not having enough information to even make an evaluation. All you've given me is a vague reference to something said in one of a series of long lectures, and you can't point me to the specific lecture and the specific thing that was said so I can watch it for myself. Nor can you give me a reference to a textbook or paper where the thing you are talking about is described. What am I supposed to do?

Moreover, the real question is not whether you've convinced me of anything, but whether you've given any information that will help the OP. I don't see how you have, since the same issues I raised above also apply for the OP. What is the OP supposed to make of your information?