Why electrons can have only 2 spin orientations?

[jaumzaum]

I'm studying electron spin/magnetic field and I was taught electrons can have only 2 spin orientations, as seen in the Stern and Gerlach experiment. But this orientations are given in relation to what? Do all electrons in the universe have the only 2 same orientations, or only the electron in the same sample? I mean, let's use the right hand rule to find the spin orientation of the electrons in the silver blade of Stern and Gerlach. Let's say my thumb line passes through Saturn, so it means all electrons in the sample will have the spins pointed to Saturn (in both senses of course). But this means that ALL the electrons in Earth have spins pointed to Saturn too?

 

[tom.stoer]

Theoretically one can use any orientation, there is no preferred direction. So saying that there are two possible values for s^z does not mean that this z direction is fixed globally, it's only a convention w/o a preferred direction; and it's only a convention to call it z.

In a specific experiment the orientation is measured w.r.t. a direction specified by the experimental setup. And by convention we call this the z-direction.

 

[jaumzaum]

Theoretically one can use any orientation, there is no preferred direction. So saying that there are two possible values for s^z does not mean that this z direction is fixed globally, it's only a convention w/o a preferred direction; and it's only a convention to call it z.

In a specific experiment the orientation is measured w.r.t. a direction specified by the experimental setup. And by convention we call this the z-direction.

In the Stern and Gelarch experiment all electrons in the sample seems to spin with the same orientation (that is 10^22 electrons with the same orientation), so I would say of course there is a preferred direction, it's not random (at least in the sample). All z's are aligned. How would you explain that?

 

[tom.stoer]

As I said: it's the experimental setup

In a specific experiment the orientation is measured w.r.t. a direction specified by the experimental setup.

 

[jaumzaum]

As I said: it's the experimental setup

How so? You mean the experimental conditions make the electrons to spin only in that direction?

 

[tom.stoer]

Yes, if you measure spin w.r.t. the orientation of the B-field and call this the z-direction, then you will find exactly these two spin states. If you chose a different direction, you will find again two spin states w.r.t. this new direction. (however in the Stern- Gerlach experiment using a different direction will not be indicated by spatially separation of electrons)

 

[jaumzaum]

Yes, if you measure spin w.r.t. the orientation of the B-field and call this the z-direction, then you will find exactly these two spin states. If you chose a different direction, you will find again two spin states w.r.t. this new direction. (however in the Stern- Gerlach experiment using a different direction will not be indicated by spatially separation of electrons)

Thanks, now I think I've got it. So we scan say the electrons spin in all directions, in the Stern Gerlach experiment they deflect only in the z direction because there was only a field in the z direction. Is it right?

I was thinking the electron rotated only in one direction, like the Earth rotation motion. :/

 

[jaumzaum]

Wait, I still don't get it. In the Stern Gerlach experiment the field in inhomogeneous, it is not only in the z axe. Why the electrons don't form a circumference so? If they spin in all orientations they shouldn't deflect only in the z direction

 

[psmt]

Wait, I still don't get it. In the Stern Gerlach experiment the field in inhomogeneous, it is not only in the z axe. Why the electrons don't form a circumference so? If they spin in all orientations they shouldn't deflect only in the z direction

The field can be inhomogeneous but in the z-direction at all times. The strength of the field depends on z.

 

[vanhees71]

That's not exactly true. Most books do not discuss this issue. I'm not even aware of one, where the SG experiment is completely treated quantum mechanically, although it's not so difficult.

It starts indeed with the choice of the magnetic field. To keep the model as simple as possible a good approximation is
$$\vec{B}=(B_0+B_1 z) \vec{e}_z-B_1 y \vec{e}_y.$$
There must be the second term because of Gauß's Law for the magnetic fields,
$$\vec{\nabla} \cdot \vec{B}=0.$$

As long as $|B_1 z| \ll |B_0$ you can treat the second term as a perturbation, and then you can solve the Schrödinger equation for a wave packet exactly, leading finally to the conclusion that running the particle through a SG apparatus you really get a very accurate spatial separation of particles with spin up and spin down.

You find the calculation here, but it's a manuscript of the QM 2 lecture, written in German:

http://theory.gsi.de/~vanhees/faq/quant/node80.html

 

[Harry Wilson]

The electron can exist in a superposition of states; a mixture of two different states. By measuring the spin in one direction, the electron is forced into a state spinning either up or down in the measured direction. It makes no sense to talk about the x-direction spin if you've just measured the spin in the z-direction.
If you then were to measure the x-spin, you force it to spin (up or down) in the x-direction, and it makes no sense to talk about its z-spin. This is why when you pass it through another z-spin detector, half will be up and half down, despite you having earlier removed all electrons spinning down (say) from the experiment.

electron --> z-detector, remove spin down ---> x-detector, half are up, half down --> z-detector, half are up, half down

 

[psmt]

That's not exactly true. Most books do not discuss this issue. I'm not even aware of one, where the SG experiment is completely treated quantum mechanically, although it's not so difficult.

It starts indeed with the choice of the magnetic field. To keep the model as simple as possible a good approximation is
$$\vec{B}=(B_0+B_1 z) \vec{e}_z-B_1 y \vec{e}_y.$$
There must be the second term because of Gauß's Law for the magnetic fields,
$$\vec{\nabla} \cdot \vec{B}=0.$$

As long as $|B_1 z| \ll |B_0$ you can treat the second term as a perturbation, and then you can solve the Schrödinger equation for a wave packet exactly, leading finally to the conclusion that running the particle through a SG apparatus you really get a very accurate spatial separation of particles with spin up and spin down.

You find the calculation here, but it's a manuscript of the QM 2 lecture, written in German:

http://theory.gsi.de/~vanhees/faq/quant/node80.html

Thanks for the correction! I hadn't noticed that before.

It seems strange, in a way, that treating the y-component of the field as a perturbation is OK, because it would appear that the spin-dependent force on the particle in the y direction is of the same order as that in the z-direction (classically speaking, of course). I'll take your word for it that it works, though.

 

[jaumzaum]

Yep, I still didn't get it.
If there is a y field, why the electrons do not deflect in y direction?

 

[San K]

Yep, I still didn't get it.
If there is a y field, why the electrons do not deflect in y direction?

they do. however it's labelled z - by convention - for convenience/easy of use

 

[tom.stoer]

Perhaps it makes more sense to consider the case w/o B-field b/c then we still do have spin.

Suppose there is an experiment which fixes an axis (I don't tell you how, and I don't tell you which axis). Let's call it the z-axis. Suppose the experiment allows us to measure spin w.r.t. this axis (again I don't explain how this measurement works)

The experimental result for every electron spin is s_z = ± 1/2 w.r.t. this axis.

If we call this axis x-axis then the result is s_x = ± 1/2

If we rotate the setup in some arbitrary way, fix new coordinates and axes, and measure spin w.r.t. these new axes, we get the same results:

For a new z'-axis the result is s_z' = ± 1/2

etc.

 

[vanhees71]

Thanks for the correction! I hadn't noticed that before.

It seems strange, in a way, that treating the y-component of the field as a perturbation is OK, because it would appear that the spin-dependent force on the particle in the y direction is of the same order as that in the z-direction (classically speaking, of course). I'll take your word for it that it works, though.

It's not so strange: What comes out of the analysis is that due to the large constant piece of the magnetic field in $z$-direction, there is a pretty rapid spin precision arount the $z$ axis.

The main point of the Stern-Gerlach experiment is to yield a (sufficiently accurate) entanglement between the spin-z component and the position of the particle after running through the inhomogeneous field, and the rapid precision of the spin around the z-direction leads to an effective time averaging of the xy-components of the spin. That's why the most relevant motion of the particle's position is due to the coupling of the spin-z component to the inhomogeneous part of the magnetic field.

Thus, to a high accuracy a beam of arbitrarily polarized spin-1/2 particles (like neutrons or in the original Frankfurt setup of Stern and Gerlach in 1923 a silver atom) is split into two beams with nearly pure eigenstates of the spin-z component. There is only a little contamination from y-component of the magnetic field.

 

[phyzguy]

Wait, I still don't get it. In the Stern Gerlach experiment the field in inhomogeneous, it is not only in the z axe. Why the electrons don't form a circumference so? If they spin in all orientations they shouldn't deflect only in the z direction

I would say that nobody "gets it". The fact that the result of an electron spin measurement can have only one of two values, regardless of which direction the spin is measured along, is an empirical fact which is not explainable in terms of any simple mechanical model of the electron spin.

 

[vanhees71]

I think to the contrary that the Stern-Gerlach experiment is one of the best understood examples for quantum dynamical behavior and the basic features of quantum theory, including the theory of measurement. So, what do you mean by "nobody gets it"?

Spin can be understood from various points of view. First of all there is the phenomenological point of view: Spin is a feature of some elementary particles associated with a magnetic moment and thus it can be investigated by the movement of these particles in a magnetic field, and that's how it was discovered by Stern and Gerlach. Their ideas about the phenomenon were of course very incomplete and within the then known way to describe quantum phenomena, namely in terms of the Bohr-Sommerfeld model, which already then was known to be incomplete at best, if not inconsistent in itself. As we know today, with the discovery of modern quantum theory in 1925, the latter is indeed the case.

Form the point of view of modern quantum theory, spin is very well understood. The most convincing way to derive and understand it, is to use group theory. Group theory and group representations on the Hilbert space as used in quantum theory are anyway a very important ingredient of modern physics. The idea is to use very basic facts about the mathematical structure of space-time, here the Galilei-Newton space-time model. The space-time can be described by its symmetries (translation invariance in time and space, isotropy of Euclidean space, invariance under Galilei boosts), and in order that the quantum theory of a system like a particle is consistent with this space-time structure, it should admit the representation of the corresponding Galilei group of symmetry transformations of space-time. An "elementary particle" is then defined as being described by an irreducible ray representation of this group, which can be induced from a unitary representation of an appropriate central extension of its covering group. This is quite technical, but the upshot of this mathematical analysis is that an elementary particle is charcterized by its mass (which is also known from macroscopic "point like" objects) and its spin (which is not really known from classical mechanics; the most close analogue is an extended body that may rotate around its center-of-mass point, which itself is at rest; that's where the name spin for this angular-momentum quantum mechanical observable comes from).

Then you can ask, how electromagnetic fields act on such particles. There you have the additional principle of gauge invariance and the most simple realizations to build the corresponding Hamiltonian is, what's called minimal coupling, meaning that you couple the electromagnetic potentials to the electric charge and current provided by the particle. In non-relativistic theory this procedure is unfortunately not a unique, but one of the realizations of minimal coupling leads for spin 1/2 (the most simple case of a particle with spin) to the Pauli equation, which itself can be derived also as the non-relativistic limit of the relativistic Dirac equation for a spin-1/2 particle in an electromagnetic field, and for the relativistic QFTs the principle of minimal coupling is a pretty unique procedure. So there is some logic behind the spin and the implications of its existence to particle dynamics in the electromagnetic field.

The Pauli equation then admits in a not too complicated way to investigate the deflection of a spin-1/2 particle in a inhomogeneous magnetic field, which leads to an accurate description of the Stern-Gerlach experiment. It's particularly intuitive since the semiclassical description turns out to be a very good approximation in the typical setup of the classical experiment.

 

[tom.stoer]

I would say that nobody "gets it". The fact that the result of an electron spin measurement can have only one of two values, regardless of which direction the spin is measured along, is an empirical fact which is not explainable in terms of any simple mechanical model of the electron spin.

It depends what you mean "get it". You can't get it in terms of the formalism of classical mechanics, but you can in terms of quantum mechanics. "Getting it" in terms of quantum mechanics means to get rid of classical explanations.

 

[phyzguy]

It depends what you mean "get it". You can't get it in terms of the formalism of classical mechanics, but you can in terms of quantum mechanics. "Getting it" in terms of quantum mechanics means to get rid of classical explanations.

Agreed, but I had the impression that the OP was trying to form a (classical) mechanical picture of an electron spinning in a certain direction. As you say, these must be discarded in favor of the Hilbert space formalism of quantum mechanics.