In Landau and Lipshitz's introductory book on Quanum Mechanics, "Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3", chapter XIV (page 433 in the edition on Amazon) is "Addition of Angular Momenta". Interestingly, the first page is footnoted to the effect that the theory covered only applies in the limit of a binding force sufficiently weak that the angular momentum can be considered separately for the particles:

Strictly speaking, we shall always be considering (without explicitly mentioning the fact each time) a system whose parts interact so weakly that their angular momenta may be regarded as conserved in a first approximation.

All the results given below apply, of course, not only to the addition of the total angular momenta of two particles (or systems) but also to the addition of the orbital angular momentum and spin of the same system, assuming that the spin-orbit coupling is sufficiently weak.

Can someone provide a more complete explanation for this? I'm curious as to what fails as the binding force gets too strong. I'd like an intuitive explanation.

One of the reasons for wanting to know more about this is that I suppose it has something to do with the "proton spin crisis", that is, the question of how angular momentum is distributed among the components (quark, gluon &c.) of a proton. For example see:

A. W. Thomas, Int.J.Mod.Phys.E18:1116-1134,2009, "Spin and orbital angular momentum of the proton" 0904.1735

So it seems like an important part of understanding strongly bound states and it would be nice to better understand the limitations of the usual methods from an intuitive as well as through calculations.


Consider two independent systems, two distant atoms, for example. Each atom has is own angular momentum $J_1$ and $J_2$. If these atoms are different and distant (non interacting), then the QM variables are separated, the total wave function becomes a product of the two atomic wave functions, and the total system angular momentum is a sum of the two. The system energy is determined with the sum of atomic energies and depends on $J_1$ and $J_2$ in a simple way.

If the atoms interact, one cannot introduce two independent atomic angular momenta since the variables are not separated. In case of a weak interaction, one can use the approximation of non-interacting atoms as an initial approximation in the perturbation theory to calculate the interacting system properties.

In case of a strong interaction such an approximation may happen to be too poor, the perturbative corrections large, and the truncated perturbative solution far from the exact one. I think, in this case a quasi-particle approach is more helpful: a strongly coupled system may have low energy excitations like (nearly) independent quasi-particles with effective masses, charges, spins, etc. Then the approximate solution is built differently and there is a chance to guess it right.

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Pinnow model does not suffer of the barionic 'spin crisis':
Douglas Pinnow book 'Our Resonant Universe' (2003) is a monography of a new model of particles, based only on ElectroMagnetism, that has only one parameter: the $m_{e}$ - electron mass, and he derives the particle properties to within 1% of their values (barion masses well bellow 1%), masses, angular momenta, spin. He makes predictions and his model is very simple based in the non-radiation condition and he worked on top of Haus, Goedecke and Kaluza-Klein theory. The author is an 'old' nuclear engineer, and NASA fellow, with a large career in photonics.

Quote from pag 9 (in the Overview):

Chaper 2, Composite particles, is dedicated to the interactions among the standing-wave resonances (elementary, massive particles) which form the composite barions. Models of such structures lead to mass calculations that are accurate to within 1% of observation for all spin$-\frac{1}{2}$ baryons. The magnetic moments of the proton and neutron are also easily calculated from such first principles. An estimate of the magnetic moment of the neutral sigma barion ($\Sigma^{0}$) is offered as a test of this approach.

Overview from here (more to read: Free Preview and About the Author)

Our Resonant Universe presents the resonance model, a ground-breaking vision of the nature of the universe. By applying a simple electromagnetic approach to both elemental matter and nuclear reactions, Pinnow and Miller unite mass, spin, and charge with the four fundamental forces of the universe, without invoking the arbitrary constants of the standard model or the multi-dimensional mathematics of string theory. The results are striking in their accuracy and in their descriptive power. The model also provides a series of predictions that will soon put this conception of the universe to the test.

From this citebase link we can access the sources of all the references cited in the Thomas article as G-scholar or pre-print ;-)

The book Pinnow, low cost, was probably too little read and has not been discussed!

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