You say: "the magnetic field is known not to do work", this applies to the (magnetic part of the [generalized, depending on teaching tradition(*)]) Lorentz force, which is the force exerced by (electro-) magnetic fields on charges.

It is not so appropriate to say that there is a rotation of the spin (by 90°, in your example, which, by the way is not the largest possible change).

(a) When a charged object with angular momentum (spin in case of a free electron) is subject to a homogeneous field, then the result is a precession of the spin about the field direction. The scalar product of magnetic moment and magnetic field remains unchanged (and so is therefore the magnetic potential energy. The Larmor precession represents an extra (kinetic) energy, though, if orbital motion was involved).

(b) there is no net force exerced by a homogeneous field onto the spin.

All this, however, is not the situation of the Stern Gerlach experiment and for two reasons: the net charge of the electron leads to cyclotron motion and the beam is therefore bent such as to make the experiment impossible (this is actually, why it was [cleverly] done with neutral silver atoms). Second the deflection is induced by a field gradient. The field gradient is what exerces a force onto the moment and makes a difference between spin up and down.

This deflection, which is necessary to now be able to "measure" the spin component along the field gradient (split beam) of course represents some work being done. In case of an electromagnet kept at constant field (and gradient) this work is effectively done by the power supply.

(*) In some texts the electric component q*\vec{E} is included in what is called "Lorentz force", in others not.

Kai, thank you for your considerations. I owe  a clarification.

1. I meant that the velocity of the electron is parallel to magnetic induction B and hence there is no magnetic Lorentz force evB. This is easy to insure by using a coil for creation of the field (and other ways).

 2, Of course if one has a small magnet (or a current loop) it will rotate and the magnetic field will do some work leading to rotational kinetic energy shown in this rotation. But the electron can not rotate. If the spin is -1/2 this gives a term which must add kinetic energy (increase the velocity). But if it is +1/2 the velocity must decrease. But this is not always possible (if v is very low or 0). I would be very thankful to understand what do you think about this?

3. I am not aware how the magnet looses or gains energy (what is the  mechanism if you know). This is curious from the point of view that magnetic field is carried by virtual photons. If there is a small rotating magnet the lead magnet must loose energy at all cases. But with the nonrotating electron it must also take energy.

4. I know about the precession but I akso know that  in magnetic field sz must take -1/2 or 1/2. How do this synchronize? I think that precession is a classical physics scope event where the other is pure quantum effect.

Dear Ilian,

"unfortunately", both classical and quantum physics are involved and this occurs in ways that may seem nontrivial to you. And for now I will not find the time to go through all of this. I will just give a few hints.

Quantum mechanical spin (of an electron) is not a classical degree of freedom with which you could associate kinetic energy. Its properties are those of a quantum mechanical angular momentum though.

When pondering about classical and quantum behavior of angular momentum it might be a good idea (in case you have some familiarity with it) to consider the behavior of protons in NMR. Here, alternating fields induce a precession. When one does it right, the spin can be oriented to be precessing with the arrow of momentum lying in the xy-plane (field along z). It then takes time to come back to the equilibrium situation (spins in stationary (= "time independent") |+1/2> or |-1/2> states). The process of coming back to equilibrium is called relaxation and ultimately has to do with dissipation (which destroys q.m. phases. The "rotating" state induced by the 90° pulse is also just a q.m. superposition of the |+1/2> and |-1/2> states. This must be so, because for a spin of 1/2, there exist only two linearly independent states).

Concerning the work done by the power supply of an electromagnet: if an object in the magnetic field changes (the z-component of) its moment in the magnet, then so does the (z-component of the) flux density associated with that object. This change of Bz creates an electromotoric force ( rot(\vec{E} \propto dB/dt ), which in turn creates an additional current in the coil (law of induction). If the power supply "wants" to manage keeping the current constant, it will have to react accordingly corresponding to a power increase or decrease (and hence work done or "received", depending on the sign of dB/dt).

Coming back to the spin states and your point 4: you say "sz must take -1/2 or 1/2", which is not entirely true: these states are the stationary states. Any other states, including the precessing ones are also possible, but not stationary [which actually is what makes them useful in NMR diagnostics].

Edit: note added. When the electron enters the magnet with finite momentum along the magnet axis, then also there is a change of flux in the magnet and associated electromotoric force. Again the opportunity for the magnet power supply to provide the work required to make the physics/energetics right.

Dear Kai,

after paying attention to your considerations I realized that the electron can not change its velocity as a result of the spin projection (the superposition of spin up and down and following precession of spin is very unstable - in fact this is like Schroedinger cat with an open door of the box). The change of velocity would in fact be against Maxwell equations - the magnetic field can not accelerate a charge.

So the electron after taking spin up or down - can not accelerate and can not rotate but it would have been imparted with caring a rotational hidden momentum. In fact this is close to an Einstein-de Haas  experiment but with lack of the body - just an electron(s) stripped away from it . When there  is a body (in E de H exp) it will rotate. Now without a body  the rotation (angular momentum) stays now in the electron. But any  electron would carry angular momentum anyway (it has spin). So I think an electron gone thru a magnetic field will not take any energy or momentum. Which mean there is no energy accepted from the magnet. Than nothing in the magnet changes . This is what I called nothing in the question. But the direction of the spin of the e- has changed. This is something.

So there is a change induced by a body that doesn't change. This seems odd but maybe does not contradict to laws.

Sorry the notices maybe quite clumsy for reading.

I am sorry but I don't agree. If the moment of the electron (or whatever else) changes, then this provokes  reaction in the magnet coils.

The non-stationatity of the precessing state is not an instability, at constant field it is a periodic motin (a mechanical top with its rotation axis inclined with respect to the gravitational field does this, too).

A homogeneous magnetic field does not accelerate electrons ( in the sense of changing the absolute value of velocity ) but an inhomogeneous field does exerce a force onto the spin. Inevitably, this will happen upon entering and exiting the magnet.

What you are trying to confider is not an easy piece of cake. You have to calculate and argue very carefully.

I  perfectly understand in Einstein de Haas experiment that the magnet (or coil) must experience either a decrease in energy and (or) a rotation counter that of the body. Of course this are the conservation laws. In this case the electrons with their spin serve as a medium that transfer energy or momentum from the magnet to the rotating body.

If this process is detached than the rotation is somehow freezed in the electron. If these electrons are brought later in contact with a body they will transmit the rotation to the body. But until then there must be some change in the electron? (Beside of the change of the direction of the spin). If the electrons were a body (I easily would accept this body is rotating) but they are not.

I have actually lost track on what your sentence So there is a change induced by a body that doesn't change is meant to say. Sorry. 

So I try to go back to your original post where you wrote:

• you have since then stated that the trajectory should exactly be along the axis of the magnet. ( (a) This is perpendicular to the S-G geometry. (b) For all practical purposes this would represent an extremely hard requirement, since the phase space for this would be zero? But we can put this aside for the moment).
• If the magnet is of finite length, the field at both ends is necessarily inhomogeneous.
• Unless you have prior knowledge on the spin state before "it enters the magnet / magnetic field" you have no means to discuss whether a change occurs to the electron. You would probably have to describe the spin state as (1/sqrt (2))(|1/2> +|-1/2>) and then design your setup such as to allow you to check for diltoeues to that after passage of the magnet. Is that correct?
• To see whether something is strange you would have to write down the evolution of that initial state under the Hamiltonian considered. This is potentially nontrivial, since the magnet is not just imposing a constraint (the field) but also sensing / reacting to the electron. (the energetics involved is totally irrelevant on the magnet's macroscopic scale,  but your interest lies in the microscopic scale!!)
• etc etc??

So, with all these connotations, would you still say your introductory statement is a valid one? What would be the experiment setup allowing you to decide on such a statement? I think this should be clarified before going any further. 

Initially I envisaged a SG sort appliance but realised that it will mask the problem (because of Lorentz magnetic force). 

Note that is not something I am sure about but I just want to clear it for myself. I am sorry I also totally forgot about Zeeman splitting but I also has a problem with it, which will be clarified below.

So it goes like this:

1. The spin of the electron (e-) is measured and is taken to be perpendicular to its velocity.

2. The electron enters a coil (toroid) which delivers a magnetic field B, so v is linear to B (and there is no rotating Lorentz force).

Now there two variants:(as I believe)

A) as you say: the e- is a superposition (precessing) around B. After leaving the magnet(toroid) it is exactly in the same state as when it enters. The energy from Zeeman splitting is returned to the field. (When it is inside there is Zeeman splitting - when it is outside there is not - what is the mechanism of this is a charade to me)

So then there is nothing from nothing? No change in B, no change in e-. This is an example (I think) of the measurement problem. Can the interaction with the field measure the spin or not? Maybe this a resolution to the problem I posted?

B) the e-  is forced to take (is measured by the field) spin up or down. When it comes to Zeeman energy, it is restored and the only thing that happened is the change of spin. This is because there is no body to which e- can deliver angular momentum. If there was a body as in Einstein de Haas it will rotate (and B will change). But the problem is that there is no a body and the electron can not rotate as it is pointlike. Maybe it (e-) is not pointlike and can rotate it some way. By the way string theory insists that it is not and I think they say it can rotate (not in a classical mean).

Now this can also be a resolution.

Another issue:

I said earlier that I think that e- can not gain energy (kinetic) in B field. So e- in the described above situation can not accelerate. But then again after considering more carefully I am not sure. The total energy of e- must conserve? E=Ek+Ep where Ep comes from Zeeman energy and can be positive or negative depending on spin projection. Then according Ek must decrease (increase) after all in B area?

On magnetic postential energy and Zeeman splitting:

Rather you may want to see it from another perspective: If "an object" or "a system" (QM: in some state) changes its energy when being subjected to a magnetic field, then it is said to possess a magnetic moment. The magnitude of that moment is given by the derivative of the energy with respect to the field. (If to you "the magetic field" is H, as given by SI units then multiply by µ0-1, if the field is B= µ0H) then just take the derivative with respect to B. (either the gradient or component wise)

Objects/systems with (e.g. spin) angular momentum and total angular momentum quantum number J actually come with 2J+1 degenerate states belonging to J (unless that degeneracy is lifted, as e.g. for electron states in so-called crystal or ligand fields) . Each of these states responds with a different change of energy when exposed to a magnetic field and therefore exhibit different energies in nonzero field. The corresponing pattern of non-degenerate (and thus 'split') energies is referred to as Zeeman splitting.

OK Ilian, thanks for your clarification. There may be practical limitations to actually compose an experimental setup which does what you'd like it to do. A toroid magnet e.g. possesses a far field that decays with d-3, where d is the distance from its center. So, strictly speaking we never have a zero field....

But I think it's not necessary and we can try to go along your idea. So let's say that at some point we are able (by whatever means) to prepare an electron in a state of linear motion, (essentially some kind of plane wave) and with spin known to be in one of the eigenstates of orientation along x, e.g. \sigmax=+\hbar/2.

As the field (at whatever strength) is along z we know that the state we have prepared is not a spin eigenstate, these would be given by \sigmaz=\pm\hbar/2. So that state is going to be non-stationary, i.e. it has a "non-trivial" time evolution. How to determine it?

We can represent the state by a superposition of eigenstates, the time evolution of which is known (loosely: \psi(t) = psi(t=0)*exp(i E t / \hbar)). Now, we have some Zeeman splitting, therefore the energy of the two eigenstates is not the same, and therefore, you will observe some time dependence. Maybe think of the magnetic field being constant for the moment. Are you able to write down the linear combination needed and to evaluate the corresponding time dependence? Can you also compute some "interesting" quantities such as the time dependent expectation values of energy and the three components of the spin / magnetic moment, i.e. (t), (t) and (t) ?

Notice that I wrote expectation values. Only in eigenstates is the spectrum of an operator single valued. (Here is indeed a connection to the 'measurement problem', as you suspected).

Edit: you will probably have noted a second problem: if we were to define the electron to be in a state of well defined momentum along z, then it would be represented by a plane wave, spatially totally unconstrained along z. Then it would make no sense to talk about the location of the electron.

We would have to prepare a wave packet to overcome this. This packet will inevitably spread out with time...

Well I will think about this. But would probably take a week.

Well Kai, I consider this but it would take a week or so. I'll get in contact with a letter on researchgate's post.

no problem with me - take your time.

I think at least a part of problem lies in fact that what we call 'spin' is in reality only a projection of momentum on the selected direction in space.  Specifically, for electron, its total momentum is \sqrt(s(s+1)) (=\sqrt(3)/2) NOT s, and its projection can only take two possible values, either +s (=1/2, 'spin up') or -s (=-1/2, 'spin down').  At the same moment nothing can be said about two other projections separately.  Thus even if sx is fixed then sy and sz are not.

That's true, Marek, of course. Nevertheless, we can write down an equation of motion in the presence of external fields and look for the time evolution. Then we can tell what we can say and what we cannot. (Spin resonance is a good example I think (as I wrote before), be it ESR or NMR)

Kai, the purpose of my short message was that it is incorrect to speak about orthogonality (or parallelism) between spin and external field.  Thus the classical view is inherently flawed.  Accordingly, there is not "nothing" and "something" in the mentioned experiments, but rather "something" and "something else".

Hi Kai.

I have sent a letter in your box about the case.

Marek, agreed in case of (mis-) taking spin as a classical vector. Nevertheless one can do the proposed calculation (Ilian actually did) and then obtains a time dependent state the vector components of which vary like (,,) \propto (cos(wt),sin(wt),0) with w \propto B the precession frequency.

In absence of other interactions (dissipation) the spin will not convert to an eigenstate. Only if I ask (by some experiment) "what's the spin projection along z" will I get either "+1/2" or "-1/2" as the answer... and zero on average. This is where my thoughts went when answering. I think that the latter notion oftentimes gets overlooked when e.g. paramagnetic response is being taught (or learnt): thermodynamic equilibrium is assumed to be present, but this actually means that interactions must be existent (even if we assume to be discussing the paramagnetic response of non-interacting angular momenta).

Just my $0.02.

Something is wrong here.  If indeed state vector components "vary like (,,) \propto (cos(wt),sin(wt),0)" and the notation means the time average of a, then not only the last component (sz) is equal to zero, but the same goes for sx and sy - isn't it?  If this is the case then spin becomes totally irrelevant.  On the other hand, when solving equations of motion, we need momentary values of time-varying parameters, not their time-averaged values - at least from classical perspective.  I agree that some interactions are necessary to understand the paramagnetic response of non-interacting angular momenta.  Those interactions (with phonons, for example) are the reason for Prob(-1/2 --> +1/2) being equal to Prob(+1/2 --> -1/2) in absence of external field, and no net magnetization.  When the field is present, then those transitions occur with different probabilities and the time-averaged value (observed quantity) becomes non-zero, and is not quantized! Nevertheless, the transitions -1/2 +1/2 occur all the time, in unpredictable moments, without any periodicity.  Without extra interactions some magnetic moments would be aligned along the field, while others - in opposite direction, forever.

Back to the original question.  In Stern-Gerlach-type experiments a single electron need not to flip its spin.  The magnetic field only 'sorts' "free" (i.e. not interacting with anything else) electrons, accordingly to the projections of their momenta in respect to the field direction.  No need for energy flow.

Marek,  in "vary like (,,) \propto (cos(wt),sin(wt),0)", does not mean "time average" but "time dependent expectation value". The phenomenon which is described then is precession. The underlying assumption is that at t=0 a state was prepared in which is +1/2, i.e. the spinor state \chi = 1/sqrt(2)*(|up>z + |down>z).

(Edit: in spin resonance experiments, such a preparation might e.g. be achieved with a suitable 90° pulse.)

Let us think of a classical experiment. We have N small magnets in a liquid oriented with north pole up.  We passe the system through a permanent magnet with north pole down. All these small magnets change direction. Outside the magnetic field remain all down. What is energy difference of the system of the magnets before and after the interaction (ignoring friction effects) with the magnetic filed? I think the same. Would not be the same situation for the spins? 

No. The liquid would provide a thermal bath (= relaxation). For a single spin, this is exactly what does not happen. There is no mechanism (no interactions) for generating thermal equilibrium. Again: think of NMR, the (nuclear) spin precession is exactly what provides the signal. The interaction with external degrees of freedom is weak enough so that the time scale for returning to equilibrium (T1) is macroscopically observable. (For some isotopes of rare gases e.g. this can extend to hours, so that the gas can be transported in bottles with the nuclear spin polarization being preserved! This has been successfully used to do NMR of the lung!)

Dear Spyros Messoloras, some more side notes:

A single spin is no classical object. there is no classical experiment for this to do then. And, as far as spin is concerned, actually no spin is classical.

Since, however, spin is related to angular momentum we can think of a classical angular momentum: the mechanical analog would be a spinning top. Now put a supported, frictionless spinning top in the gravitational field with some inclination with respect to the vertical: the angle between the angular momentum vector and the gravitational force will not change - instead the top precesses.

My question is simple. Does the spin up (before the interaction) and spin down (after the interaction) has the same energy or different? No need to complicate the matter with gravitational interactions. So state up and state down in free space with no interactions have the same energy or not?

In a classical experiment we can avoid complications of thermodynamics and stay only in classical electromagnetism. Two magnets interact and the question is if the exchange energy during their interaction. Off course in a classical system the final state of a magnet even if it is initial state was up does not have to be down as in a quantum spin.

Why I talk of both quantum and classical situations. Because the question is doing so.  

Dear Spyros, I actually do not understand you latest post, sorry.

Both a quantum spin as well as a "classical" angular momentum start to precess when subject to a torque. The potential energy does not change. This means that the position of minimum potential energy will not be reached. This only possible through relaxation mechanisms. This means, that e.g. something like paramagnetic response (Curie law) requires the notion of thermal equilibrium to materialize

From my perspective and understanding these aspects have been thouroughly discussed in this thread.

Sorry if I got you wrong (I fear so). Gravitation was not mentioned as a complication but to create the analog between a magnetic moment (or spin) in a magnetic field and a mechanical spinning top.