# B Polarized Electron in a Rotating Reference Frame

#### metastable

I tried asking a similar question in cosmology but got no answer there so here goes...

Suppose I am on a windowless spacecraft in the middle of an intergalactic void. I know that the spacecraft is spinning from measuring the centrifugal forces but have no way of observing the outside universe other than what occurs in my spacecraft. At the center of mass in the spacecraft is an electron trap containing an electron with its spin axis polarized along an axis of the spacecraft in such a way that for each rotation of the spacecraft, the electron spin axis completes one rotation. I now release the electron from the spacecraft in such a way that the electron's spin axis continues to rotate at the same rate that it did when it was inside the craft. Now the spacecraft moves away to a great distance and using thrusters reduces its angular speed to zero. I know that the electron is polarized in a rotating reference frame.. The reference frame of the polarized electron is now rotating relative to what?

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#### Nugatory

Mentor
You're asking about the spin of an electron, and that brings in all the complexities of quantum mechanical half-integral spin. Did you intend that, or would a rotating ball suffice for the question you're trying to ask? I'm assuming a rotating ball in the comments below and in that case...
Now the spacecraft moves away to a great distance and using thrusters reduces its angular speed to zero......The reference frame of the polarized electron is now rotating relative to what?
The motion of the spacecraft is completely irrelevant to the behavior of the electron.

There is a non-inertial reference frame (which you're calling "a rotating reference frame" - this is somewhat sloppy language but everyone does it) in which the ship and the object were both initially at rest; after the ship moves the object is still at rest using this frame while the ship is moving in a giant circle.

There is an inertial reference in which the object and the ship were both initially rotating; after teh ship moves and kills its angular momentum the ship is now at rest using that frame and the object is still rotating.

You are free to describe the entire situation, both before and after the ship moves, using either frame, or any other that you choose. In practice, you will want to choose whichever ones makes it easiest to calculate whatever you want to calculate.

#### Dale

Mentor
The reference frame of the polarized electron is now rotating relative to what?
Relative to any inertial frame, including the one where the spacecraft is now at rest.

#### vanhees71

Gold Member
You're asking about the spin of an electron, and that brings in all the complexities of quantum mechanical half-integral spin. Did you intend that, or would a rotating ball suffice for the question you're trying to ask? I'm assuming a rotating ball in the comments below and in that case...

The motion of the spacecraft is completely irrelevant to the behavior of the electron.

There is a non-inertial reference frame (which you're calling "a rotating reference frame" - this is somewhat sloppy language but everyone does it) in which the ship and the object were both initially at rest; after the ship moves the object is still at rest using this frame while the ship is moving in a giant circle.

There is an inertial reference in which the object and the ship were both initially rotating; after teh ship moves and kills its angular momentum the ship is now at rest using that frame and the object is still rotating.

You are free to describe the entire situation, both before and after the ship moves, using either frame, or any other that you choose. In practice, you will want to choose whichever ones makes it easiest to calculate whatever you want to calculate.
Ironically, $s=1/2$, in quantum mechanics is the most simple case of spin you can think of. It leads to the notion of two-level systems, with which you can explain a lot in introductory quantum mechanics (in fact nearly all conceptual principles can be explained by this simple example and an only somewhat more complicated case of two, three,... spins 1/2).

Ironically, what's really difficult is to describe spin in the macroscopic relativistic realm. It's all pretty simple, if not beautiful, in non-relativistic physics, where you have as an example the rigid body (spinning top). If it comes to both special and general relativistic macroscopic physics, I think it's still not completely solved how to really describe it in all details. There's of course a plethora of papers on this over some decades, but it's still not really understood.

For some recent work, see e.g.,

#### Nugatory

Mentor
Ironically, s=1/2s=1/2s=1/2, in quantum mechanics is the most simple case of spin
I appreciate the irony, but what you're (reasonably) calling "the most simple case of quantum mechanical spin" is something that everyone else is calling "a can of worms that should not be opened in a B-level relativity thread"
what's really difficult is to describe spin in the macroscopic relativistic realm.
That's also true, but this question doesn't appear to involve rotational speeds high enough to introduce the relativistic complications.

#### vanhees71

Gold Member
Hm, well. I thought it's about relatistic dynamics because of the reference to cosmology and setting it up in outer space in a space craft ;-). Unfortunately, I think even the nonrelativistic treatment of quantum mechanical systems in non-inertial frames of reference is way beyond B level.

I think Schmutzer is the standard reference for this problem:

https://onlinelibrary.wiley.com/doi/pdf/10.1002/prop.19770250102

#### metastable

You're asking about the spin of an electron, and that brings in all the complexities of quantum mechanical half-integral spin. Did you intend that, or would a rotating ball suffice for the question you're trying to ask?

So if the elevator spins at 1000rad/sec, and discharges a single electron into the vacuum, will the electron retain any of the elevator’s classical angular speed or will the electron’s spin be the entirely quantum-mechanical concept of spin? If the emitted electron does retain some or all of the elevator’s 1000rad/sec classical spin, what would this additional spin (above and beyond the intrinsic quantum mechanical spin) be relative to?
then I posted my slightly more refined question in the first post in the high energy / particle physics section and it was moved here.

Suppose I am on a windowless spacecraft in the middle of an intergalactic void. I know that the spacecraft is spinning from measuring the centrifugal forces but have no way of observing the outside universe other than what occurs in my spacecraft. At the center of mass in the spacecraft is an electron trap containing an electron with its spin axis polarized along an axis of the spacecraft in such a way that for each rotation of the spacecraft, the electron spin axis completes one rotation. I now release the electron from the spacecraft in such a way that the electron's spin axis continues to rotate at the same rate that it did when it was inside the craft. Now the spacecraft moves away to a great distance and using thrusters reduces its angular speed to zero. I know that the electron is polarized in a rotating reference frame.. The reference frame of the polarized electron is now rotating relative to what?

#### Dale

Mentor
I tried asking a similar question in cosmology but got no answer there
$14\ne 0$

It would be better to say that you didn’t understand or didn’t like the answers than that you didn’t get any. You clearly did, and many of the same people post in both places, so they might be irritated at the dismissal of their posts.

#### metastable

Relative to any inertial frame, including the one where the spacecraft is now at rest.
I appreciate yours and their answers but I don't understand why the lone electron polarized in a "rotating" (or "non-inertial") frame is in fact considered in a non-inertial reference frame. Wouldn't an electron that was polarized in a non-rotating frame take the same path (ie no acceleration) as one that was polarized in a rotating frame (also no acceleration)?

#### Nugatory

Mentor
but I don't understand why the lone electron polarized in a "rotating" (or "non-inertial") frame is in fact considered in a non-inertial reference frame.
Everything is always “in” all reference frames, so the object (not an electron! They don’t rotate!) is certainly “in” the non-inertial frame in which the ship was initially at rest.

Note the scare-quotes around the word “in” above. People often speak of something being “in” a reference frame, but that’s sloppy and inaccurate terminology; some of the difficulty here may be that this sloppiness has misled you. Anytime that someone says “in a reference frame”, they’re really saying something more like “using the coordinates assigned by that reference frame” and clearly I can use any reference frame I please to assign coordinates to points on the surface of the rotating object.

There’s a non-inertial frame in which the spatial coordinates of those points are constant (the first one described in my previous post) and inertial one in which those coordinates vary periodically with time (the second one). The rotating object is no more “in” one than the other, and it would be a good exercise for you to try writing down the transformations between those two frames.

#### metastable

the object (not an electron! They don’t rotate!)
Thank you for your answer but the above quote gets to the heart of what I am trying to understand. You said electrons don't rotate... so does it mean it is not possible to "polarize" one or more electrons in a spinning craft such that changes to the vector of polarization match the rotation rate of the craft? What would that be called if it isn't called "spinning" or "rotating" the electrons? (please excuse my potentially sloppy language-- it's not deliberate).

#### Dale

Mentor
I don't understand why the lone electron polarized in a "rotating" (or "non-inertial") frame is in fact considered in a non-inertial reference frame.
I don’t understand your confusion. You are the one who set up the scenario and you are the one who specified that it was at rest in the non-inertial frame. So how can you possibly be confused about that? You are the one who specified it. You even identified that there were fictitious forces. You were very specific.

#### Nugatory

Mentor
You said electrons don't rotate... so does it mean it is not possible to "polarize" one or more electrons in a spinning craft such that the vector of polarization matches the rotation rate of the craft? What would that be called if it isn't "spinning" the electrons? (please excuse my potentially sloppy language-- it's not deliberate).
The word “spin” is used with electrons for historical reasons, but it means a quantum-mechanical property of point particles that is altogether unrelated to the classical notion of an object spinning around its axis. Electrons do have a magnetic moment so can be aligned with a magnetic field (I think that’s what you mean by “polarized”) but it has absolutely nothing to do with the magnetic moment of a rotating charged object.

#### metastable

I don’t understand your confusion. You are the one who set up the scenario and you are the one who specified that it was at rest in the non-inertial frame. So how can you possibly be confused about that? You are the one who specified it.
Because on wikipedia:

A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame.[1]

^I didn't understand why a lone electron with a changing polarization vector experiences acceleration compared to one with a non changing polarization vector.

#### Nugatory

Mentor
Because on wikipedia:

A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame.[1]
And stuff like that is the reason that Wikipedia is not an acceptable reference under the forum rules. It’s not exactly wrong, but it’s also nowhere near right, and it is unlikely that the anonymous Wikipedian who wrote that understood the subtleties here.

#### Dale

Mentor
A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame.[1]

^I didn't understand why a lone electron with a changing polarization vector experiences acceleration compared to one with a non changing polarization vector.
Hmm, I think that definition is a little confusing. I don’t think I would use it.

Newton’s first law says that a free particle (no interactions with other objects) travels in a straight line at constant speed. This is the principle of inertia. So inertial frames are ones where Newton’s first law holds and non-inertial frames are ones where it does not hold.

The presence of fictitious forces in your frame indicates that it is non inertial because free objects would accelerate due to the fictitious forces.

#### pervect

Staff Emeritus
Thank you for your answer but the above quote gets to the heart of what I am trying to understand. You said electrons don't rotate... so does it mean it is not possible to "polarize" one or more electrons in a spinning craft such that changes to the vector of polarization match the rotation rate of the craft? What would that be called if it isn't called "spinning" or "rotating" the electrons? (please excuse my potentially sloppy language-- it's not deliberate).
The quantum description of spin may be helpful to you. You could perhaps get a better answer in the quantum forum, but I'll give my best shot at it here. What I'm taking away from your posts is that I think you have the idea an electron is literally "spinning". This is not the case. We call the quantum property that the electron has "spin", but it's not actually spinning. What I'll do now is go into a bit what the quantum propety we call "spin" is about, in basic terms.

I could be misunderstanding your point, but this is my interpretation after some thought about what lies behind your question.

Let us start our exploation of "spin" with one of the famous experiments that led us to believe that electrons have spin - the Stern-Gerlach experiment. <<link>>.

In this experiment, neutral silver atoms are shot through a magnetic field. These neutral atoms (not electrons) are deflected either "up" or "down".

This is sometimes described as the beam being split into two "polarized" parts. See for instance the following quote. From another site:

One of the cornerstones of quantum mechanics is the Stern-Gerlach effect. An unpolarized beam of silver atoms is passed through a strong magnetic field gradient and splits into two polarized beams. This effect is one of the main motivations to postulate that electrons have spin, in particular spin-1/2.
I am guessing that this may be what you were thinking of when you wrote the word "polarized", but there is some question in my mind as to what you actually meant by this phrase, it had me scratching my head for a bit. If you meant something else, you might want to clarify.

So, what is going on here? Basically, the silver atoms are acting like little bar magnets. This is called a magnetic dipole moment, or just a magnetic moment.

If the silver atoms acted like classic bar magnets, the beam would not split into two parts. Rather, the beam of atoms would spread out. Some of the little magnets would be oriented one way, others would be oriented in other ways, the orientations would be random. Depending on the orientation of the magnets, they might be attracted in the direction of the magnetic field, repelled and move in the opposite direciton, or be completely unaffected.

What makes the quantum experiment is that the beam does split into two distinct parts, it does not just spread out. One beam is attracted to the magnets, one is repelled. We usually say that the spin is either "up" or "down", (this assumes the magnets are oreinted vertically), and that passing through the Stern-gerlach apparatus "measures" the spin of the silver atom. There is no atom in the beam that is unaffeced passing through the magnetic field - it is deflected one way or the other, there is no "middle ground". This is one of the important differences between the quantum behavior and the classical behavior. And it's not at all inttuitive.

Now - how does this relate to classical spin? If we had a spinning point charge, it would not act like a bar magnet at all. IT would have no magnetic moment. A ball of spinning charge, though, would act like a little bar magnet. So the electron is in some respects a little like a spining ball of charge, in that it has a magnetic moment, but if one tries to ask questions like "how big" the ball of charge is, one does not necessarily get sensible answers. The electron acts like a point particle, not like a little ball of charge, in other experiments.

We can say, unequivocally, that the electron has a magnetic moment, this has been measured experimentally. So it's similar to the spinning ball of charge in that it has a magnetic moment, but it's different than a spinning ball of charge, too.

See for instance the wiki article on the electron magnetic moment <<link>>.

One final point. The Stern Gerlach experiment used silver atoms, and not electrons. This may be a source of puzzlement. Why use silver atoms, why not use electrons? The basic issue is that the electrons are not heavy enough. The wave nature of the electron would make the splitting of the electron beam into two parts not measurable, because of the Heisenberg uncertanity principle. This is from memory - I know I've read this, but alas, I don't recall all the details. And I believe it was rather tehcnical as well, and this is just an overview.

This is a limit of the Stern-Gerlach experiment itself rather than anything really fundamental, more sophisticated experiments can and have detect the magnetic moment of the electron, and it behaves just like the non-classical magnetic moment of the silver atoms in that when we measure it, it's in one of two states, it's either "up" or "down".

Perhaps there is a better way to talk about spin than the Stern-Gerlach experiment, but from what I recall, that's how it's usualy introduced. But I'd certainly encourage you to read more about it spin if you are interested . However, reading popularizations may be of limited use. So I don't have any specific recommendations of what you can read at an introductory level.

The detailed mathematics is quite interesting, there are a pair of complex numbers, the square magnitude of one number gives the probability of finding the electron in the "up"state, the squared magnitude of the other number gives the probability of finding the electron in the "down" state. Electrons can be in what's called a "superposition" of quantum states, as well. It's quite interesting, and very relevant to understanding quantum mechanics, but I think we're drifting away from your quesiton into deeper waters, so I'll stop here.

That's the story in a nutshell.

I've focussed here on the electron, and noted that it's the magnetic moment that's the physical feature associated with 'spin'. I have not discussed the magnetic fields present in a spinning frame of reference containing a classical point charge, but that would be another post - and this one is already rather long.

#### jartsa

I tried asking a similar question in cosmology but got no answer there so here goes...

Suppose I am on a windowless spacecraft in the middle of an intergalactic void. I know that the spacecraft is spinning from measuring the centrifugal forces but have no way of observing the outside universe other than what occurs in my spacecraft. At the center of mass in the spacecraft is an electron trap containing an electron with its spin axis polarized along an axis of the spacecraft in such a way that for each rotation of the spacecraft, the electron spin axis completes one rotation. I now release the electron from the spacecraft in such a way that the electron's spin axis continues to rotate at the same rate that it did when it was inside the craft. Now the spacecraft moves away to a great distance and using thrusters reduces its angular speed to zero. I know that the electron is polarized in a rotating reference frame.. The reference frame of the polarized electron is now rotating relative to what?
Electron does not retain the rotation of its orientation. It's a gyroscope. It retains its orientation really well. The rotation of the orientation is not retained at all, to a very good approximation.

Any other object would be better for this experiment - may I suggest a broom stick.

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#### Dale

Mentor
I don't think that the OP's question is really about QM, I think it is about reference frames. He just chose an unfortunate example. @metastable can you clarify?

#### metastable

What I'm taking away from your posts is that I think you have the idea an electron is literally "spinning". This is not the case.
We call the quantum property that the electron has "spin", but it's not actually spinning.
I am guessing that this may be what you were thinking of when you wrote the word "polarized", but there is some question in my mind as to what you actually meant by this phrase, it had me scratching my head for a bit. If you meant something else, you might want to clarify.
So the electron is in some respects a little like a spining ball of charge, in that it has a magnetic moment, but if one tries to ask questions like "how big" the ball of charge is, one does not necessarily get sensible answers. The electron acts like a point particle, not like a little ball of charge, in other experiments.
We can say, unequivocally, that the electron has a magnetic moment, this has been measured experimentally.
Electron does not retain the rotation of its orientation. It's a gyroscope. It retains its orientation really well. The rotation of the orientation is not retained at all, to a very good approximation.

Any other object would be better for this experiment - may I suggest a broom stick.
I don't think that the OP's question is really about QM, I think it is about reference frames. He just chose an unfortunate example. @metastable can you clarify?
Thanks for all the answers. I will try to summarize my remaining confusion on the matter. It pertains to whether the electron can or cannot physically rotate its orientation.

^I read that the electron has a magnetic moment, and the moment is considered a vector pointing from the south to the north pole of the magnet / electron.

Suppose on the spaceship, I've determined by measuring the centrifugal forces that the spaceship is rotating at 1 rad/sec. Suppose the spaceship has an X, Y, and Z axis which are locked relative to the ship (ie the X, Y, and Z of the ships reference frame is also rotating at 1 rad/sec)

Is it possible to do the following:

Suppose the ship is rotating 1rad/sec about its X axis. While the electron is in the trap at the center of mass of the ship, measurements are taken 1.1 seconds apart to determine the orientation of the elecron's magnetic moment relative to the ships X, Y and Z axis. Suppose that within a couple of percentage points of accuracy I determine in both measurements taken 1.1 seconds apart that vector of the electron's magnetic moment is substantially parallel to the Y axis of the ship (ship rotates about X axis). Since the ship rotates 1 rad/sec about the X axis, the vector of the Y axis changes at 1 rad/sec. Since both measurements of the electron's magnetic moment vector taken 1.1seconds apart show that it is parallel within measurement limits to a rotating Y axis, is it reasonable to conclude that the orientation of the vector of the magnetic moment of the electron is rotating at a rate of at least 1 rad/sec?

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#### Dale

Mentor
The gyromagnetic ratio for an isolated electron is 1.76E11 rad/s/T. So an electron with a precession rate of 1 rad/s would imply a magnetic field of 5.7 pT. Form your description of the geometry it would be 5.7 pT in the x direction.

#### metastable

So an electron with a precession rate of 1 rad/s would imply a magnetic field of 5.7 pT.
Is there a way to release the electron from the craft in such a way that I could expect it to continue precessing unless further acted on?

#### Dale

Mentor
Is there a way to release the electron from the craft in such a way that I could expect it to continue precessing unless further acted on?
As long something with that gyromagnetic ratio remains in a magnetic field of 5.7 pT then it will continue precessing regardless of the presence or absence of the ship. However, being a quantum mechanical particle there is a lot of messy quantum mechanical considerations for a single electron including not being able to measure its state without causing the wavefunction to change. You would do much better to consider a classical object with the same gyromagnetic ratio or to consider a large ensemble of electrons such that you can make measurements on the ensemble and obtain results approximately equal to the expectation value (known as the classical limit).

I think that your insistence on using "an electron" is detracting from the substance of your actual question which I believe is about the reference frames and not the quantum mechanics of electrons.

#### Nugatory

Mentor
Is there a way to release the electron from the craft in such a way that I could expect it to continue precessing unless further acted on?
The electron is precessing because it is in a powerful magnetic field. There are two possibilities:
1) The equipment generating the field is in the ship, so if the ship and the electron are separated the electron will no longer be subject to that field and will no longer precess. Exactly what does happen depends on the details of how you separate the two.
2) The equipment generating the field is not in the ship. In this case, the presence or absence of the ship is altogether irrelevant.

#### metastable

The electron is precessing because it is in a powerful magnetic field. There are two possibilities:
1) The equipment generating the field
So does this mean it would require a constant supply of energy to cause the electron to continuously change the vector of its magnetic moment?

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