Can Stern-Gerlach spin alignment be seen as a result of EM radiation of precessing magnetic dipole?

[Duda Jarek]

Stern-Gerlach experiment is often seen as idealization of measurement. Using strong magnetic field, it makes magnetic dipoles (of e.g. atoms) align in parallel or anti-parallel way. Additionally, gradient of magnetic field bends trajectories depending on this choice.

Magnetic dipoles in magnetic field undergo e.g. Larmor precession due to τ=μ×B torque, unless μ×B=0 what means parallel or anti-parallel alignment.

Precession means magnetic dipole becomes kind of antenna, should radiate this additional kinetic energy. Thanks to duality between electric and magnetic field, we can use formula for precessing electric dipole, e.g. from this article:

Using which I get power like 10^−3 W, suggesting radiation of atomic scale energies (∼10^−18 J) in e.g. femtoseconds (to μ×B=0 parallel or anti-parallel).

So can we see spin alignment in Stern-Gerlach as a result of EM radiation of precessing magnetic dipole?

Beside photons, can we interpret other spin measurement experiments this way?

[exchemist]

5 hours ago, Duda Jarek said:

Stern-Gerlach experiment is often seen as idealization of measurement. Using strong magnetic field, it makes magnetic dipoles (of e.g. atoms) align in parallel or anti-parallel way. Additionally, gradient of magnetic field bends trajectories depending on this choice.

Magnetic dipoles in magnetic field undergo e.g. Larmor precession due to τ=μ×B torque, unless μ×B=0 what means parallel or anti-parallel alignment.

Precession means magnetic dipole becomes kind of antenna, should radiate this additional kinetic energy. Thanks to duality between electric and magnetic field, we can use formula for precessing electric dipole, e.g. from this article:

Using which I get power like 10^−3 W, suggesting radiation of atomic scale energies (∼10^−18 J) in e.g. femtoseconds (to μ×B=0 parallel or anti-parallel).

So can we see spin alignment in Stern-Gerlach as a result of EM radiation of precessing magnetic dipole?

Beside photons, can we interpret other spin measurement experiments this way?

According to my (admittedly rusty) understanding, the only way an atom can radiate in this situation would be by transitioning between the different allowed space-quantised energy levels, i.e. for an atom with angular momentum J, between the 2J+1 values of M, the quantum number of the projection of the angular momentum vector along the field direction. Absorption of energy to bump up electrons from lower energy to higher levels is the basis of EPR (known as ESR in my day). The splitting of the energy levels is modest and therefore these energy transitions absorb (and emit) photons in the microwave region of the spectrum. 

I'm not sure I follow what you mean by spin alignment being the result of radiation. The alignment simply occurs as an external field is applied. But, to pursue your line of thought, in the absence of a field there will be equal numbers of atoms in each of the M states, because they are degenerate. When the field is applied, this is no longer so and the atoms will adopt a Maxwell-Boltzmann distribution among the energy levels of M, with more in the lower (more aligned) energy levels and fewer in the upper (more anti-aligned) ones. This, I suppose, must involve either emission of microwave radiation or non-radiative relaxation processes. I have never seen this described and don't know which is dominant. However as the Einstein transition probability for spontaneous emission goes up with the cube of frequency, it may be that spontaneous emission processes in the microwave region are so infrequent as to be negligible. But maybe someone else here, ideally with experience of EPR, will know.   

According to my old Herzberg, the energy, W, of the states in an applied magnetic field is W(0) + hoM, where W(0) is the energy in the field-free case and o is the Larmor precession frequency. That implies that for transitions between adjacent energy states, (i.e. for which ΔM=1,), ΔW = ho = hν, i.e. the microwave frequency of the emitted or absorbed photon is the Larmor precession frequency.

I'm not sure if this deals with your query, but maybe it's a start.    

 

 

Edited January 23, 2023 by exchemist

[Duda Jarek]

For atom the dominating magnetic dipole moment can come e.g. from angular orbital momentum - in which case shouldn't Larmor precession be of angular momentum direction?

For any (also macroscopic) magnet in external magnetic field there is τ=μ×B torque leading to precession, what means oscillating dipole - type of antenna, radiating energy with power as the above formula ... until reaching the lowest energy state: μ×B=0 having minimal kinetic terms.

[exchemist]

Just now, Duda Jarek said:

For atom the dominating magnetic dipole moment can come e.g. from angular orbital momentum - in which case shouldn't Larmor precession be of angular momentum direction?

For any (also macroscopic) magnet in external magnetic field there is τ=μ×B torque leading to precession, what means oscillating dipole - type of antenna, radiating energy with power as the above formula ... until reaching the lowest energy state: μ×B=0 having minimal kinetic terms.

Yes, that's why the angular momentum quantum number is called J rather than S. The values of J can be any integer between L+S and I L-S I. The total angular momentum of the atom is √(J(J+1)).h/2π.

But you need to be careful with applying classical radiation concepts in QM systems. Recall that the Bohr model of the atom failed because classically an electron in orbit around the nucleus, being an accelerating electric charge, should radiate, lose energy and fall into the nucleus. Which it doesn't. Hence the move to Schrödinger's quantised model of orbitals, in which the electron can only radiate by making transitions between a small number of discrete, allowed energy levels, determined by solutions to his equation.

Exactly the same is true in the case of the space quantisation we are discussing here. The atom can only move between discrete allowed energy levels, denoted by different values of M, the quantum number for the allowed projections of the angular momentum along the field direction.  

[Duda Jarek]

Precession is not just "a classical concept" - it applies also to electron, even for much more complex acrobatics in spin echo: https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance#Pulsed_electron_paramagnetic_resonance

 

I think you agree that macroscopic magnet in external magnetic field would get torque, precession ... undergo "classical EM radiation" down to minimal energy μ×B=0 ... as also observed in Sterin-Gerlach.

So how small could such magnet be? What happens when this magnet becomes extremely tiny: of size of atom ... electron?

[exchemist]

2 minutes ago, Duda Jarek said:

Precession is not just "a classical concept" - it applies also to electron, even for much more complex acrobatics in spin echo: https://en.wikipedia.org/wiki/Electron_paramagnetic_resonance#Pulsed_electron_paramagnetic_resonance

 

I think you agree that macroscopic magnet in external magnetic field would get torque, precession ... undergo "classical EM radiation" down to minimal energy μ×B=0 ... as also observed in Sterin-Gerlach.

So how small could such magnet be? What happens when this magnet becomes extremely tiny: of size of atom ... electron?

Nope, you emphatically do not get classical EM radiation. This is a quantised system. Please read my 2 previous posts. 

We can get onto spin-echo EPR later, once we have agreed on the basics.  

[Duda Jarek]

So how do you understand/explain Stern-Gerlach experiment: why these atoms align in parallel or anti-parallel way (as "classical magnets" would through radiation of abundant kinetic energy)?

What happens with the difference of energy and angular momentum (between initial random and final aligned spins) if it is not radiated as EM wave?

[exchemist]

16 minutes ago, Duda Jarek said:

So how do you understand/explain Stern-Gerlach experiment: why these atoms align in parallel or anti-parallel way (as "classical magnets" would through radiation of abundant kinetic energy)?

What happens with the difference of energy and angular momentum (between initial random and final aligned spins) if it is not radiated as EM wave?

When a compass needle aligns with the Earth's magnetic field, are you telling me it emits radiation? 

[Duda Jarek]

Conservation laws e.g. Noether theorem say that change of energy, momentum, angular momentum - has to be compensated with opposite change e.g. in EM field, for example:

excited atom <-> deexcited atom + EM wave carrying difference of energy, momentum and angular momentum

From the other side, accelerating charges/dipoles leads to radiation of some energy as EM waves ... and e.g. spin echo in pulsed EPR shows everything works down to electron scale.

I think you agree that classical magnet in magnetic field should EM radiate energy and finally align (to zero torque μ×B=0) ... they also see it for atoms in Stern-Gerlach ... if you claim these are different mechanisms, please elaborate on the mechanism seen in Stern-Gerlach?

Cannot we see it as

unaligned "random" spin <-> aligned spin + EM wave carrying difference of energy, momentum and angular momentum?

Edited January 23, 2023 by Duda Jarek

[exchemist]

1 hour ago, Duda Jarek said:

Conservation laws e.g. Noether theorem say that change of energy, momentum, angular momentum - has to be compensated with opposite change e.g. in EM field, for example:

excited atom <-> deexcited atom + EM wave carrying difference of energy, momentum and angular momentum

From the other side, accelerating charges/dipoles leads to radiation of some energy as EM waves ... and e.g. spin echo in pulsed EPR shows everything works down to electron scale.

I think you agree that classical magnet in magnetic field should EM radiate energy and finally align (to zero torque μ×B=0) ... they also see it for atoms in Stern-Gerlach ... if you claim these are different mechanisms, please elaborate on the mechanism seen in Stern-Gerlach?

Cannot we see it as

unaligned "random" spin <-> aligned spin + EM wave carrying difference of energy, momentum and angular momentum?

The mechanism in the Stern Gerlach experiment is as written up in the textbooks on QM, surely? 

My understanding of this, admittedly from my 1st year as an undergraduate, 50 years ago, is that the atom precesses at the appropriate Larmor frequency for the M state it occupies, and carries on doing so indefinitely, unless it encounters radiation of the right (i.e. Larmor) frequency to stimulate  a transition to a different M state. Regarding energy conservation, what I think happens is that those states with an antiparallel component (of magnetic moment: due to the -ve charge on electrons the magnetic moment points in the opposite direction to the angular momentum itself)  -  acquire energy from the field, while those with a parallel component give energy to the field, the net effect being zero, in line with conservation of energy. 

However, once the states are split, statistical thermodynamics enters the picture, causing the ones in the higher states to tend to lose energy and drop down to lower states. As I say above, I think this relaxation process will be primarily non-radiative, because of the improbability of spontaneous emission at such low frequencies. Non-radiative relaxation will manifest itself as heat in the material.

So no, I do not think there is significant radiation emitted in the Stern-Gerlach experiment. But I'll be interested to see if anyone else has a different view.

P.S. Be aware, too, that the notion of precession is a semi-classical one. In QM, only the compment of the magnetic moment or angular momentum along the field  z-direction is defined. The orientation of the component in the x,y plane , i.e. the component that "rotates" in the precession model, is undefined. This is a consequence of the uncertainty principle (non-commuting operators)   

Edited January 23, 2023 by exchemist

[Duda Jarek]

Textbook explanation of this kind of quantum effects is usually "unitary evolution ... magic happens ... unitary evolution" ... I would gladly see a better one, getting inside this magic, and experimentally it slowly becomes accessible - e.g. they measured ~20 attosecond delay of photoemission: https://www.science.org/doi/10.1126/science.1189401 ... what is happening during such 20as?

If atom has magnetic dipole moment e.g. due to angular momentum, then in external magnetic field it gets torque, hence its axis should precess - what means additional kinetic energy. There is tendency to release abundant energy, and mechanism for oscillating dipole - due to acceleration of charge/dipole, as e.g. for electron in bremsstrahlung.

However, there appears question, issue of quantization of such released EM energy - I completely agree it can directly manifest as heat ... but what if they are single atoms in vacuum?

Quote

Non-radiative relaxation will manifest itself as heat in the material.

The belief that everything EM related is through quantized photons probably comes from pertubative approximation of QFT e.g. seeing Coulomb interaction through photon exchange ... but this is our approximation - fundamental question should be for non-perturbative.

A common alternative is quantization through emitter/absorber - usually being atoms of quantized energy states. E.g. cosmic microwave background radiation seems just a thermal noise of EM degrees of freedom - quantized when absorbing its energy by atoms.

The problem starts with antennas e.g. linear - producing cylindrically-symmetric EM waves. Assuming such wave consists of a finite number of photons, going with distance to infinity such discrete photons would become infinitely diluted, large ...

[exchemist]

2 minutes ago, Duda Jarek said:

Textbook explanation of this kind of quantum effects is usually "unitary evolution ... magic happens ... unitary evolution" ... I would gladly see a better one, getting inside this magic, and experimentally it slowly becomes accessible - e.g. they measured ~20 attosecond delay of photoemission: https://www.science.org/doi/10.1126/science.1189401 ... what is happening during such 20as?

If atom has magnetic dipole moment e.g. due to angular momentum, then in external magnetic field it gets torque, hence its axis should precess - what means additional kinetic energy. There is tendency to release abundant energy, and mechanism for oscillating dipole - due to acceleration of charge/dipole, as e.g. for electron in bremsstrahlung.

However, there appears question, issue of quantization of such released EM energy - I completely agree it can directly manifest as heat ... but what if they are single atoms in vacuum?

The belief that everything EM related is through quantized photons probably comes from pertubative approximation of QFT e.g. seeing Coulomb interaction through photon exchange ... but this is our approximation - fundamental question should be for non-perturbative.

A common alternative is quantization through emitter/absorber - usually being atoms of quantized energy states. E.g. cosmic microwave background radiation seems just a thermal noise of EM degrees of freedom - quantized when absorbing its energy by atoms.

The problem starts with antennas e.g. linear - producing cylindrically-symmetric EM waves. Assuming such wave consists of a finite number of photons, going with distance to infinity such discrete photons would become infinitely diluted, large ...

"Magic happens" is not in any textbook I have ever seen.

I stress the idea of precession is a semi-classical analogue for what happens quantum-mechanically. As I say, my understanding is that some of the atoms gain energy from the applied field and some lose it to the field, depending on which M state they are in. This is reflected in the idea of kinetic energy of precession but, as that is a semiclassical analogue, you can't take too far. The analogy works intuitively for those states that are raised to higher energy by the field, but doesn't work so well for those that are lowered in energy. 

If you take a single atom, the field changes the potential experienced by the electrons in the atom. They no longer just experience a spherically symmetrical electric potential from the nucleus, but also a linearly polarising magnetic potential as well. They either gain potential energy from the field or  give up potential energy to the field, depending on orientation. This is presumably what is responsible for the force and consequent deflection they experience as they pass through the field. I don't think there is any need to presume they have to radiate. 

 

 

[joigus]

The whole point of the Stern-Gerlach experiment is to disprove the classical theory of radiation. You get two or more (2S+1), separated, clearly defined spots where the charged particles end up, corresponding to the different values of spin. If the phenomenon could be interpreted classically, you would get a continuous range of arrival positions, which is never the case. That's why the SG experiment is considered to be one --among many-- confimation of quantum dynamics, as opposed to the expectations of classical-field dynamics.

[Duda Jarek]

4 minutes ago, exchemist said:

I don't think there is any need to presume they have to radiate. 

Such need is suggested by conservation laws - especially of energy and angular momentum.

Magnetic dipole of random direction in external magnetic field has abundant energy (kinetic of precession) - in Stern-Gerlach somehow lost by aligning, so what happens with this energy difference? Could turn into heat, through EM interactions.

Also angular momentum is different for a random initial spin and aligned final spin - what has happened with this difference?

Quote

linearly polarising magnetic potential as well

There are two effects here - "V cdot mu" energy as in Zeeman effect, and kinetic energy from precession of unaligned spin.

In Stern-Gerlach the latter seems to dominate, but there should be also statistical difference of population of two beams (?) - although, it might be extremely tiny.

5 minutes ago, joigus said:

The whole point of the Stern-Gerlach experiment is to disprove the classical theory of radiation.

The problem is that classical theory of radiation predicts exactly the same outcome - magnetic dipole in external magnetic field gets torque, additional kinetic energy of precession - as oscillating dipole should should EM radiate energy, until reaching minimum: when it is aligned ... exactly as seen in Stern-Gerlach.

So what is the difference between such classical behavior of magnet in magnetic field, and what they observe in Stern-Gerlach?

[joigus]

10 minutes ago, Duda Jarek said:

The problem is that classical theory of radiation predicts exactly the same outcome - magnetic dipole in external magnetic field gets torque, additional kinetic energy of precession - as oscillating dipole should should EM radiate energy, until reaching minimum: when it is aligned ... exactly as seen in Stern-Gerlach.

It only does that if you assume quantum mechanics is valid for the charged particles and classical field theory is valid for the EM field. If you combine quantum and classical attributes, it's possible to obtain relatively satisfactory models for some quantum behaviours. You have to put in QM by hand at some point.

If charged particles can adopt any orientation --they are classical too--, it's obvious that you would get the continuous range of deflections I was talking about.

[swansont]

12 hours ago, Duda Jarek said:

Using which I get power like 10^−3 W, suggesting radiation of atomic scale energies (∼10^−18 J) in e.g. femtoseconds

Why femtoseconds? 

If there was radiation why wouldn’t it be at the precession frequency?

 

26 minutes ago, Duda Jarek said:

Such need is suggested by conservation laws - especially of energy and angular momentum.

Magnetic dipole of random direction in external magnetic field has abundant energy (kinetic of precession) - in Stern-Gerlach somehow lost by aligning, so what happens with this energy difference? Could turn into heat, through EM interactions.

Also angular momentum is different for a random initial spin and aligned final spin - what has happened with this difference?

That assumes that there is precession, as if this were a classical system.

QM says that you only have two possible spin orientations - once you have a quantization axis, as provided by the magnetic field, you only have these two choices. The random spin orientation is a probability that it will be one spin or the other. 

[exchemist]

23 minutes ago, Duda Jarek said:

Such need is suggested by conservation laws - especially of energy and angular momentum.

Magnetic dipole of random direction in external magnetic field has abundant energy (kinetic of precession) - in Stern-Gerlach somehow lost by aligning, so what happens with this energy difference? Could turn into heat, through EM interactions.

Also angular momentum is different for a random initial spin and aligned final spin - what has happened with this difference?

There are two effects here - "V cdot mu" energy as in Zeeman effect, and kinetic energy from precession of unaligned spin.

In Stern-Gerlach the latter seems to dominate, but there should be also statistical difference of population of two beams (?) - although, it might be extremely tiny.

 

Ah but what "kinetic energy of precession" is this? On reflection I don't think there is any. I think the phenomenon of precession merely re-allocates the existing angular momentum into motion about 2 axes, doesn't it?

The energy change results from the change in potential, due to the alignment/anti-alignment of the magnetic moment to the field.

Edited January 23, 2023 by exchemist

[Duda Jarek]

6 minutes ago, joigus said:

It only does that if you assume quantum mechanics is valid for the charged particles and classical field theory is valid for the EM field. If you combine quantum and classical attributes, it's possible to obtain relatively satisfactory models for some quantum behaviours. You have to put in QM by hand at some point.

I have described classical radiation explanation leading to the same conclusion as Stern-Gerlach: of finally aligned spins.

3 minutes ago, swansont said:

Why femtoseconds? 

If there was radiation why wouldn’t it be at the precession frequency?

I have used the formula for EM radiation energy of antenna as oscillating dipole in the first post here.

This is a complicated problem - it would be great if somebody experienced in antennas could make a better calculation.

5 minutes ago, exchemist said:

Ah but what kinetic energy of precession is this? On reflection I don't think there is any. I think the phenomenon of precession merely re-allocates the angular momentum into motion about 2 axes, doesn't it? 

If it would be a macroscopic magnet, torque should lead to precession. Exactly the same argument is used for electron in https://en.wikipedia.org/wiki/Larmor_precession

So why there shouldn't be precession in intermediate scale: of atom?

And precession means additional kinetic energy - contributions with time derivative, which can be minimalized by just aligning spin - what they experimentally observe e.g. in Stern-Gerlach.

[exchemist]

Just now, Duda Jarek said:

I have described classical radiation explanation leading to the same conclusion as Stern-Gerlach: of finally aligned spins.

I have used the formula for EM radiation energy of antenna as oscillating dipole in the first post here.

This is a complicated problem - it would be great if somebody experienced in antennas could make a better calculation.

If it would be a macroscopic magnet, torque should lead to precession. Exactly the same argument is used for electron in https://en.wikipedia.org/wiki/Larmor_precession

So why there shouldn't be precession in intermediate scale: of atom?

And precession means additional kinetic energy - contributions with time derivative, which can be minimalized by just aligning spin - what they experimentally observe e.g. in Stern-Gerlach.

OK, why do you think precession means additional kinetic energy, rather than just partitioning the existing kinetic energy between motion about two axes? What work is done? Take me through the logic. 

[Duda Jarek]

11 minutes ago, exchemist said:

OK, why do you think precession means additional kinetic energy, rather than just partitioning the existing kinetic energy between motion about two axes? What work is done? Take me through the logic. 

Imagine you have some object e.g. atom, and put it into precessing coordinates - it would introduce additional time derivative terms (kinetic), until stopping this precession.

[swansont]

40 minutes ago, Duda Jarek said:

I have used the formula for EM radiation energy of antenna as oscillating dipole in the first post here.

Yes. I was responding to that post.

40 minutes ago, Duda Jarek said:

This is a complicated problem - it would be great if somebody experienced in antennas could make a better calculation.

You didn’t answer my questions, which don’t pertain to your calculation.

[exchemist]

1 hour ago, Duda Jarek said:

Imagine you have some object e.g. atom, and put it into precessing coordinates - it would introduce additional time derivative terms (kinetic), until stopping this precession.

Why can't it just divert a proportion of the existing angular momentum - and kinetic energy - to the precession axis?

My understanding is that is what happens in a lossless precessing gyroscope. (Though maybe a proper physicist can comment on whether that is the case.)

Edited January 23, 2023 by exchemist

[Duda Jarek]

3 hours ago, swansont said:

You didn’t answer my questions, which don’t pertain to your calculation.

There is this formula for power of rotating electric dipole: http://www.phys.boun.edu.tr/~sevgena/p202/docs/Electric dipole radiation.pdf

Inserting k = 10^6 Hz and p ~ 10^-23, you get power ~10^-4 W ... proper calculations would require someone experienced with antennas, but generally we are talking about  ~femtosecond scales.

2 hours ago, exchemist said:

Why can't it just divert a proportion of the existing angular momentum - and kinetic energy - to the precession axis?

My understanding is that is what happens in a lossless precessing gyroscope. (Though maybe a proper physicist can comment on whether that is the case.)

We are talking about rotating dipole and acceleration of charges/dipoles generally leads to radiation of energy as EM waves, e.g. in bremsstrahlung.

The above formula is for oscillating dipole, getting the details is difficult I will think about, but generally these are ~femtosecond scales.

... and this radiation says that magnetic dipoles should align in magnetic filed - what is exactly what they observe e.g. in Stern-Gerlach.

[exchemist]

1 hour ago, Duda Jarek said:

There is this formula for power of rotating electric dipole: http://www.phys.boun.edu.tr/~sevgena/p202/docs/Electric dipole radiation.pdf

Inserting k = 10^6 Hz and p ~ 10^-23, you get power ~10^-4 W ... proper calculations would require someone experienced with antennas, but generally we are talking about  ~femtosecond scales.

We are talking about rotating dipole and acceleration of charges/dipoles generally leads to radiation of energy as EM waves, e.g. in bremsstrahlung.

The above formula is for oscillating dipole, getting the details is difficult I will think about, but generally these are ~femtosecond scales.

... and this radiation says that magnetic dipoles should align in magnetic filed - what is exactly what they observe e.g. in Stern-Gerlach.

No, that is wrong for this situation, because it is a bound system and therefore quantised. I pointed out to you earlier that electrons in atoms do not radiate and fall into the nucleus. That's because it is a bound system, which restricts the states the electron can occupy. 

Forget brehmsstralung. It's irrelevant. That is for free, i.e. unbound, particles. The Wiki article on brehmsstralung makes the point: https://en.wikipedia.org/wiki/Bremsstrahlung. 

An unbound particle can occupy a continuum of states: they are not quantised. So the particle can radiate and lose energy in a classical manner.  Electrons in a bound system cannot do this. 

[swansont]

2 hours ago, Duda Jarek said:

generally we are talking about  ~femtosecond scales.

Yes, this is what I asked you about. How did you arrive at this conclusion? I want your reasoning, not just a repetition of the statement.

2 hours ago, Duda Jarek said:

... and this radiation says that magnetic dipoles should align in magnetic filed - what is exactly what they observe e.g. in Stern-Gerlach.

But other effects are not classical (the deviation of the beam), so why should the alignment be classical? The discrete deviation is an indication that you do not have randomly-aligned spins that come into alignment over some period of time.