I'll try to be a little more specific than Daniel:

Interactions (involving localized angular momentum states) tend to couple angular momentum states.

Spin- orbit coupling for example (in absence of anything else) couple states with quantum numbers |S,mS,L,mL> into new ones with quantum numbers |J,mJ>, which are linear combinations of the former ones with well-defined coefficients (named Clebsch-Gordan coefficients). mS and mL then no longer are "good quantum numbers", because (most of) the new eigenstates are linear combinations of states with different mL and mS. However, the rules of angular momentum coupling have it that in each state the expectation (not eigen-) values  and are still proportional to mJ, the new "good" quantum number (to be found in any decent QM or atomic physics text book; the latter statement is also known as Wigner-Eckart theorem). under these circumstances, the orbital moments are thus different but not "quenched".

Now forget soc (for a while :-) and think about angular momentum states of atoms in a lattice. The surrounding atoms create an electrostatic potential of some symmetry. This (point group) symmetry is usually characterized by some axis with the "highest" rotational symmetry (this need not be unique [e.g. in octahedral or tetragonal local symmetry, encountered in many oxides], then pick one at random; In practice the choice can be tricky in sites of low symmetry).

This potential (its angular dependence, to be precise) can then be expanded as a series in the spherical harmonics. The nonvanishing terms are not arbitrary then. If you have a q-fold rotation axis, the expansion will only have nonzero terms in those Ylm where m is an integer multiple of q. As a result, the potential only couples orbital momentum states with quantum numbers mL1 ,mL2  such that mL2-mL1=nq (n being an integer) in so far as they exist. [The strength of coupling is given by the corresponding coefficient of  the potential expansion.] This may then lead to orbital momentum quenching.

Example 1: one d-electron (l=2, ml=-2...2) in a four-fold symmetric site (q=4). The potential couples those orbital angular momentum states which differ in multiples of q=4 in their (orientational) orbital momentum quantum number. only one pair of such states exists, i.e. ml=-2 and ml=2, which are energetically degenerate. [Of course there are two such pairs, one for each mS.] Inevitably then, new eigenstates form and these are (|ml=-2> + ml=2>)/sqrt(2) and (|ml=-2> - ml=2>)/sqrt(2), also known as |xy> and |x2-y2 >.

In cubic sites, their energetic splitting is known as the "t2g-eg"-splitting and is of the order of eV in oxides. In this case you easily see that the expectation value of the orbital momentum in these states is zero, because ml=-2 and ml=+2 contribute with same weight in both states. The orbital moment is said to be quenched.

Note: in reality there will be spin orbit coupling, too, and maybe an external applied field as well, etc. As said above, soc 'prefers' a different  linear combination of the ml states. In compounds with 3d elements, soc is usually much weaker than the electrostatic potential, so that orbital moment quenching remains dominant (soc then only 'unquenches' the orbital momentum a little bit. This is not unimportant, though, since it gives rise to magnetocrystalline anisotropy, consult good magnetism textbooks for more details on this). Typical experimental Zeeman energies are much smaller yet and therefore will not unquench the orbital moments.

Example 2: the "xz" and "yz" states are the other states in the family of t2g states in the above (cubic) situation and are degenerate (by symmetry). They, too, are given by linear combinations of ml states, namely (|ml=-1> + ml=1>)/sqrt(2) and (|ml=-1> - ml=1>)/sqrt(2). These states, however, are not coupled by the potential of fourfold symmetry, the difference between their ml being only 2. In these states, soc has its say, therefore and as soon as one of these is only singly occupied (remember spin multiplicity), there will be some orbital moment. This is why the (relative magnitude of the) orbital moment is oftentimes significantly stronger for Co than for Fe or Ni in otherwise equivalent situations.

Example 3: single f-electron in 6-fold symmetry (typical case: Ce). In the Lanthanides, soc is much stronger than in the 3d transition metals and at the same time, they experience much less of the electrostatic potential around them. We therefore apply soc first, resulting in a sextett of j=5/2 states (Hund's rules). By this coupling, the orbital momenta are smaller than with pure l=3.

At the same time, now the maximum difference in mj between any two of these states is 5. Therefore, these states are not being coupled (or quenched) in sites of 6-fold symmetry. Only the sixfold degeneracy is being lifted.

Sorry for the lengthy post... hope it serves more than just one person.

   Broadly speaking the quenching of orbital angular momentum is when a lack of symmetry ,or other conditions, allows to consider that the orbital angular momentum L=0 (full quenching) or very close (partial quenching) to such value. In such a case the total angular momentum ,J=S, is equal to the one of the electron spin. This simplifies obviously the calculations when the electronic configuration enables to do it with respect to the crystal field.

I'll try to be a little more specific than Daniel:

Interactions (involving localized angular momentum states) tend to couple angular momentum states.

Spin- orbit coupling for example (in absence of anything else) couple states with quantum numbers |S,mS,L,mL> into new ones with quantum numbers |J,mJ>, which are linear combinations of the former ones with well-defined coefficients (named Clebsch-Gordan coefficients). mS and mL then no longer are "good quantum numbers", because (most of) the new eigenstates are linear combinations of states with different mL and mS. However, the rules of angular momentum coupling have it that in each state the expectation (not eigen-) values  and are still proportional to mJ, the new "good" quantum number (to be found in any decent QM or atomic physics text book; the latter statement is also known as Wigner-Eckart theorem). under these circumstances, the orbital moments are thus different but not "quenched".

Now forget soc (for a while :-) and think about angular momentum states of atoms in a lattice. The surrounding atoms create an electrostatic potential of some symmetry. This (point group) symmetry is usually characterized by some axis with the "highest" rotational symmetry (this need not be unique [e.g. in octahedral or tetragonal local symmetry, encountered in many oxides], then pick one at random; In practice the choice can be tricky in sites of low symmetry).

This potential (its angular dependence, to be precise) can then be expanded as a series in the spherical harmonics. The nonvanishing terms are not arbitrary then. If you have a q-fold rotation axis, the expansion will only have nonzero terms in those Ylm where m is an integer multiple of q. As a result, the potential only couples orbital momentum states with quantum numbers mL1 ,mL2  such that mL2-mL1=nq (n being an integer) in so far as they exist. [The strength of coupling is given by the corresponding coefficient of  the potential expansion.] This may then lead to orbital momentum quenching.

Example 1: one d-electron (l=2, ml=-2...2) in a four-fold symmetric site (q=4). The potential couples those orbital angular momentum states which differ in multiples of q=4 in their (orientational) orbital momentum quantum number. only one pair of such states exists, i.e. ml=-2 and ml=2, which are energetically degenerate. [Of course there are two such pairs, one for each mS.] Inevitably then, new eigenstates form and these are (|ml=-2> + ml=2>)/sqrt(2) and (|ml=-2> - ml=2>)/sqrt(2), also known as |xy> and |x2-y2 >.

In cubic sites, their energetic splitting is known as the "t2g-eg"-splitting and is of the order of eV in oxides. In this case you easily see that the expectation value of the orbital momentum in these states is zero, because ml=-2 and ml=+2 contribute with same weight in both states. The orbital moment is said to be quenched.

Note: in reality there will be spin orbit coupling, too, and maybe an external applied field as well, etc. As said above, soc 'prefers' a different  linear combination of the ml states. In compounds with 3d elements, soc is usually much weaker than the electrostatic potential, so that orbital moment quenching remains dominant (soc then only 'unquenches' the orbital momentum a little bit. This is not unimportant, though, since it gives rise to magnetocrystalline anisotropy, consult good magnetism textbooks for more details on this). Typical experimental Zeeman energies are much smaller yet and therefore will not unquench the orbital moments.

Example 2: the "xz" and "yz" states are the other states in the family of t2g states in the above (cubic) situation and are degenerate (by symmetry). They, too, are given by linear combinations of ml states, namely (|ml=-1> + ml=1>)/sqrt(2) and (|ml=-1> - ml=1>)/sqrt(2). These states, however, are not coupled by the potential of fourfold symmetry, the difference between their ml being only 2. In these states, soc has its say, therefore and as soon as one of these is only singly occupied (remember spin multiplicity), there will be some orbital moment. This is why the (relative magnitude of the) orbital moment is oftentimes significantly stronger for Co than for Fe or Ni in otherwise equivalent situations.

Example 3: single f-electron in 6-fold symmetry (typical case: Ce). In the Lanthanides, soc is much stronger than in the 3d transition metals and at the same time, they experience much less of the electrostatic potential around them. We therefore apply soc first, resulting in a sextett of j=5/2 states (Hund's rules). By this coupling, the orbital momenta are smaller than with pure l=3.

At the same time, now the maximum difference in mj between any two of these states is 5. Therefore, these states are not being coupled (or quenched) in sites of 6-fold symmetry. Only the sixfold degeneracy is being lifted.

Sorry for the lengthy post... hope it serves more than just one person.

   Let me try to complement the above excellent explanation, given by Kai, with some practical points:

   1. The angular momentum quenching only works when the crystal field is high, i.e. the atoms are (almost) ionised. In such a case J total angular momentum has magnetic moment with eigenvalues ml+2ms, although we continue to label the states as |l,s,ml,ms> and allow us to calculate a Lande g-factor using only the total angular momentum J=S . Fortunately, we can do that in most of 3d transition metals, fundamentally of those with less than half-filled shells. For instance in Fe with bcc we can take j=1 for obtaining one exchange energy around 8x 10-3 eVs.

   2. For 4-f electrons the crystal field is much lower than the spin-orbit coupling L.S. But in this case the total angular momentum J is a good quantum number and in fact, the 4-f electrons are more shielded of their ligands and their potentials are (in most of the cases) without spherical symmetry and therefore J is not conserved, which needs a group theory to be used for making a quenching approach properly.

See my answer on linear momentum to Peter Enders (above).

All these ideas are purely mathematical (based on the differential time limit); they apply equally to the notion of angular momentum. 

 Here (link below) you can Herb's answer to Peter Enders which he (Herb) was referring to.

Herb, in `traditional' magnetism we have e.g. effects like magnetocrystalline anisotropy or variations in gyromagnetic factor that need some explanation and the math alluded to above works well. (Likewise, one could mention the intense study of optical spectra of rare earth ions doped into transparent crystals, which (I believe) prompted much of the development of the related algebraic tools (applications of point group symmetry).

More recently (well 30 yrs back now), an x-ray magneto-optical effect was discovered, which has the potential to measure the expectation value of the orbital magnetic momentum (its projection onto the direction of propagation of the x-rays, to be more precise, the acronym for the method is XMCD).

I'd be interested in knowing more precisely where you see the physical problem in the present context. Thx.

https://www.researchgate.net/post/Effective_extensions_of_quantum_systems

Magnetism is an electron 'spin' effect which has been over-simplified because dimensionally it is like the dimensionality of angular momentum.  It is a time-based interaction effect between electrons [action also has these dimensions].  The problems are not in the experiments [rarely are] but in the math-based [theoretical] explanations. 

This is covered extensively in my UET5 paper. 

OK, but here we talk about the quenching of orbital angular momentum. Since orbital momentum magnetism rarely shows up with zero spin, spin is usually in the game, sure. And I do vaguely get your point. thanks.

P.S. orbital moment quenching also happens in molecules with no cooperative magnetism in place (single electron (more or less, I know) paramagnetism).

   Dear Kai,

   When you extend the quenching of the angular momentum to molecules, I understand that they have to contain electrons 3d or 4f where the spin-orbit coupling energy is less important than the ionic bonding between the atoms. But in this case you can have also quenching for angular momentum of p electrons at least in one component z component ===0. Could you extend what are the main differences between the quenching in molecules and crystals? I think that this point that you have introduced could be quite interesting for understanding this phenomenon. Thanks.

Hmm Daniel, I am not sure to get your point but I'll try thinking about it.

The motivation of my latest point was Herb's notion of "magnetism as a spin effect" and admittedly I am still not sure what he meant to imply by using the word "magnetism". I was inclined think he thought of magnetism as a cooperative effect.

Now, in most (all) atoms we notice a magnetic moment on, we actually have to do with more than one electron. Hund's rules emanate from the electrostatic interactions of electrons within an atom. And, strictly speaking, there is no single electron state in atoms with more than one electron. Nevertheless, my statement above was addressing the hypothetical case of, say, a d1 configuration in a paramagnetic molecule. If that exists (which I ignore) it could most probably be detected and analyzed in XMCD for the occurrence of quenching.

Putting all of this aside, we usually think of the crystal field as of being equivalent to the interaction of localized electrons with an electrostatic potential (which, as approximation, we may think of being given by a distribution of point charges). So it is a single site interaction. Whether or not there is exchange interaction between sites is secondary (well not for material properties, but for the treatment of above interactions). So, actually, in this respect I see is no fundamental difference between the cases of molecules and lattices.

When you mention p electrons, which ones are you thinking of? Those of the magnetic sites involved in bonding to the ligands or those of the ligands forming the bonds to the magnetic site or both?

Anyway, including bond formation into our considerations brings us from the pure "electrostatic theory" to what is known as "ligand field theory"  (or "cluster calculations") where the interaction with surrounding sites is explicitly taken into account. I am sure you know that this has been around for many years now (the first thing I heard of was in the context of TM oxides (such as NiO) and probably originated from G. Sawatzky's group Later this was exteded to explicitly treating two Ni sites, which resolved some issue in Ni spectra).

Newman and Ng stress in the "crystal field handbook" that actually also in Lanthanides this "covalent contribution" to crystal field splitting is all but negligible. However, it will mostly change our appreciation of what is actually going on and does little to the principles of the math involved. Computations get heavier, though, and need to be more thoughtfully devised than CF calculations.

Concerning my own activity, I am just on the edge of getting involved with CF and LF calculations for molecules. Having done CF before, this part is just the same but I see from comparisons with experiment (doing good ones is my first occupation :-)  that this is most probably not sufficient. In my case we're talking about d7 in a low spin config, so -- very loosely speaking -- we have to do with a single un-paired spin. 

Whatever. Most of what I wrote probably didn't even touch upon what you were interested in, Daniel. Feel free to ask more specifically. If I get your point, I'll do my best to answer up to the level of my understanding...

   Dear Kai,

   Thank you very much for your answer.

   I am physicist and I never worked with molecules, but I have done it clusters of atoms. Thus when you mentioned:

P.S. orbital moment quenching also happens in molecules with no cooperative magnetism in place (single electron (more or less, I know) paramagnetism).

   you opened an interesting question for me.

   First, I didn't know to what kind of magnetism were you speaking on. For instance ferromagnetism or antiferromagnetism are impossible in all that I know.

   Second, the metal p-orbitals lie along the cartesian axes, and so point directly towards the point charges. That is to say, the p-orbitals all have the same degree of interaction with the ligands, and so they remain degenerate, although the extra repulsion between the p-orbital electrons and the point charges of the ligands leads to a raising in energy for the molecule.

   Third, the eg metal d-orbitals point directly towards the point charges of the ligands and therefore they have the greater electrostatic repulsion, while the t2g have lobes which point between the charges, and so have a lesser electrostatic repulsion. This means that the d-orbitals are not degenerate due to the electrostac energy and they are reasonably able to be quenched the orbital angular momentum.

   Fourth, notice that this doesn't mean that, even employing the Hund rules, you can find a kind of ferromagnetism because the exchange interaction is not compensated by a Weiss field, from my humble point of view although I know that there are people which claims to find magnetic molecules and so on.

   This was the kind of discussion that I would like to know your opinion.

OK, Daniel, I satrt getting your point(s)

(i) As I wrote above, moment quenching does not depend on inter-site magnetic interactions. In the case of single "magnetic" site molecules, we have paramagnets. Some XMCD workers put such molecules on magnetic surfaces and study the coupling.

Certain molecules have the capacity to show remanence on not-so-short-timescales at very low temperature, e.g. when there are symmetry restrictions for moment reversal (after all it requires interaction to do that). they need not be long term stable though, because what enters the game then is quantum tunneling of magnetization. I have seen a couple of nice papers on that by Wernsdorfer and Barbara, whom you might also know from their micro-SQUID work on particles.

(ii) The TM 4s,p type electrons spend their lives in the bonds. The crystal field point of view is then fairly inadequate I would say, much more so than for the d's. You have formation of bonding and antibonding states etc.

Sheer degeneracy is not sufficient for orbital moment formation. In order for there to be an orbital moment (say, along z) the state carrying that moment is required to have some finite part of its azimutal wave function to behave like exp(iLz\phi/\hbar). Nonzero orbital moment means: complex azimutal function (just as a linearly propagating state is complex in the spatial variable, while a standing wave [== superposition of k and -k to same amounts] is real [e.g. zone boundary states]). In the earlier examples above this is exactly what is destroyed: xy and x2-y2 are real functions! (actually, so are xz and yz and indeed these have zero orbital momentum. If singly occupied and degenerate, however, even a small field will split them into m=-1 and m=1. That is not the case with your p-orbitals. They're sitting in bonding states (eV involved) and give a **!** about the µeV or maybe meV provided by an external field.

Note though, that this is only the main trend, not absolute mathematical truth. Small XMCD signals have been detected in transition metal oxides (e.g. some work by E. Goering or in RE pnictides [I participated in doing GdN]). But you need good sensitivity to be sure you're not measuring nonsense. More routinely available today than 10 yrs ago...

(iii) not sure to understand what your intention is here. in my view this is fully inline with what I talked about before. The crystal field formalism puts this into the angular dependence of the electrostatic potential and you get that result out. It's just that starting from the |l,ml> basis functions, the xy "belongs" to different ml than xz and yz. Just, who will unquench and not will depend on the direction of the applied field (if that is the mechanism). Usually we whoose the z axis, to keep the language simple.

(iv) if this is a reaction to something I wrote above, I suspect I might have used somewhat cryptic language in the attempt to react to Herb's post. But since you work on clusters you will know the problem of defining a ferromagnet. To me, for example, the defining property is spontaneous magnetization (not hysteresis in an experiment, for example). In this sense, superparamagnets are fluctuating nanoscale ferromagnets to me. But where to go with this... would you consider the Fe2 dimer a ferromagnet etc? I wouldn't quite go as far... Likewise, Hund's rule states represent a cooperative phenomenon in some sense, but I wouldn't go and shout "ferromagnetism" or the like. It's just that the nanoscale puts us between chairs sometimes...

    Dear Kai and Herb,

    The main problem is that sometimes people lose the fundamental idea that the energy in magnetism is not magnetic but electric. The spins, using the Pauli's exclusion principle, what actually do is to forbid the position of the charges in certain places and therefore the elestrostatic energy increases (huge Weiss field). This is very difficult to see for me when people speaks about magnetism in a molecule and to call it paramagnetism is not a form to solve it because paramagnetism=non magnetic structure. There are several reasons which makes difficult this issue as the following:

    1. The molecules can change quite easily their size because they neighbours which don't allow it. Thus electrostatic energy is not so stable than it happens in a crystal.

    2. Quenching of angular momentum means a spatial simmetry which must be stable in time. If you apply an external magnetic or electric field then this condition could change drastically.

    3. The main characteristic of ferro, antiferro,....is that they are phases. Thus there is one fixed critical temperature which always change to paramagnetism above it.

    By the way, at present I don't work in magnetic clusters but in the past we discovered that the Ag which is strongly diamagnetic in crystalline bulck, in a cluster of 14 atoms transformed in a "ferromagnetic cluster". 

Daniel,

(i) totally agree that Hund's rule is the outcome of Coulomb interaction, i.e. it's non-classical term for totally antisymmetric states.

(ii) many molecules are not that flexible. There is an increasing number of works involving many-body physics with (adsorbed) molecules, including Kondo spectroscopy (tunneling spectroscopy), some of this done in your country (e.g. A. Mugarza in Barcelona). And in the end the experiment will tell... if you could modify behaviors by external parameters easily and see that in experiment I would even find that an interesting playground!

(iii) I would tend to say that magnetically ordered phases are the one thing. Magnetic moments, on the other, are simply defined as    -\partial E / \partial B  (which includes diamagnetic moment formation). I include that in my vision of magnetism or magnetic behavior. the paramagnetic response in Kondo lattices, e.g. may be most interesting (but that's one of my personal interests... :-)

   Sorry Kai,

   Frankly, now I don't know what you are saying. Kondo lattice paramagnetism use the Anderson's model and therefore a hybridization term which couples conduction and impurity orbitals. This physical context is very far of what we were previously discussing and this magnetism is out of the ordered magnetism in one phase.

   On the other hand the formula -\partial E / \partial B only works when you have explicitally B in your hamiltonian as it happens with Zeeman terms, but if you loose terms as exchange between spins which are fundamental (hopping t) or Coulomb U interactions.

Daniel,

Kondo physics with molecules was just an example for things done with molecules. And: also impurity states involved in Kondo physics may (and usually are) be subject to a crystal field.

m = -\partial E / \partial B is the definition of magnetic moment (well, at least to me). So you (in principle) find out about whether a state has a magnetic moment (and how large it is) by computing its energy dependence on an external field. That's my take on it.

My thesis, published in Phys. Rev. in 1967, addressed the Kondo problem directly using a new math representation of spin 1/2 and QFT.  It discovered a logarithmic divergence.

It's very interesting and professional discussion.

It would be very useful to add /verify some of this in the topic Magnetochemistry in Wikipedia.

There there is only a paragraph: 

"Orbital angular momentum is generated when an electron in an orbital of a degenerate set of orbitals is moved to another orbital in the set by rotation. In complexes of low symmetry certain rotations are not possible. In that case the orbital angular momentum is said to be "quenched" and L→{\vec {L}} is smaller than might be expected (partial quenching), or zero (complete quenching). There is complete quenching in the following cases. Note that an electron in a degenerate pair of dx2–y2 or dz2 orbitals cannot rotate into the other orbital because of symmetry.[20]"

Dear Daniel: Too much atomic physics assumes that the Coulomb approximation still works at the non-macro scale.  Coulomb's "Law" is very suspect, as an aggregate over billions of charges (time-averaging).  Once again, the static approximation is invoked to simplify the math calculations in a dynamic situation. 

    Dear Herb,

    Fortunately the screening of Lindhardt or the one of Thomas-Fermi work quite well and the charges appears as independent even in the many-body problem. But you are right that if that would no the case as in the strongly correlated electrons, then Coulomb is too much simplified assumption.

Are you not complicating the answer too much?

What would be wrong with the answer given in:

http://physics.stackexchange.com/questions/51211/why-is-orbital-moment-quenched-while-atoms-forming-solid

Au contraire, Daniel.  I believe electrons are aware of each other across ALL of the universe but only select ONE other in each interaction cycle. 

    Dear Herb,

    What do you mean by interaction cycle? How do you select one in a many-body interaction as in a metal (almost a Fermi gas)?

   Dear Jerzy,

   It depends of what level of knowledge you want to go. In my first answer I just tried to give the idea of what was mean by this concept. Kai was much more deeper trying to enter in more details of how it works in a given material and finally we were discussing on molecules where for me the things are not obvious to quench the orbital angular momentum because the crystal field is substituted by other electrostatic fields.

   The answer: "A non-spherically-symmetric potential can couple states with different lz, and so if ψlz were the eigenstate of the spherically symmetric problem with angular momentum lz, then the correct atomic eigenstates in, for example, a cubic or tetragonally symmetric potential separate into linear combinations like ψlz±ψ−lz which measures out to total lz=0".

   Obviously don't tell us how the lz=0 can be reached in certain materials at all and what are factors which allow do it in d or f electrons.

One of my extensions of the properties of all electrons is the abolition of the longstanding Continuum Hypothesis (namely, electrons interact continuously) with the proposal that electrons have an intrinsic interaction cycle (so did de Broglie).  This is the deep explanation for electron's wave effects and QM.

For more details & consequences, see my paper UET6 on Academia.edu.

Dear Daniel:

You asked me about "Interaction Cycle". I gave you a link to a detailed description [UET6] but just realize that all stable systems involve cyclic motion across time so they return to the same spatial configuration after a finite time. This is how the electron stays in the hydrogen atom as I will show soon.

Absence of spin-orbit coupling provides freedom to the orbitals but that is limited by crystal field effect so called orbital angular momentum quenching. On the other hand, large electronic correlations is a cause of this phenomena.

Interesting discussion. Since I have a similar question, I would like to pose it here. So, the way I learned about quenching first was actually by arguments considering the time-reversal operation. Such a discussion can be found, for instance, in the book "Electron Paramagnetic Resonance of Transition Ions" of Abragam & Bleaney. There, it is shown that, if the Hamiltonian is time-even and one considers, basically, any time-odd operator O, then its expectation value in a non-degenerate eigenstate |a> of the Hamiltonian will be zero.

My question is now: If I only consider the kinetic and potential energies (i.e. the general Coulomb term leading to the usual ionic configuration, plus the crystal field, plus the kinetic energy), then I will have a time-even Hamiltonian with split energy levels due to the crystal field. The total angular momentum J = L+S (or similarly, the magnetic moment) are both time-odd operators, which means that, whenever I have a non-degenerate state in this system, the TOTAL momentum is quenched, not just the orbital contribution.

How does that go together with the usual quenching of (only) orbital angular momentum?

Thanks in advance