This is expressed as [H, J] = 0, the total angular momentum commutes with the hamiltonian. The total angular momentum is a vector operator defined by:
J = L + 2S
where L is the orbital and S is the spin angular momentum. The L vector operator and S vector operator do not commute with the hamiltonian in general and are not separately conserved in general. This is why the Lande analysis was introduced for the Zeeman effect. However Lande carried out this analysis empirically, just before Pauli inferred the Pauli matrices. It has been greatly refined by computational quantum chemistry packages for supercomputers. I was the 1988 lead writer for the first of these by IBM: MOTECC (“Modern Techniques in Computational Chemistry”, Ed. Clementi). Quickly after Pauli inferred the matrices, the Schroedinger Pauli equation was inferred, and after that the Dirac equation, which has been developed into the fermion equation. The theory of combinations of angular momenta in quantum mechanics is also very highly developed and quickly becomes very intricate, so supercomputers are used almost immediately. It develops into the Clebsch Gordan series, coupled and uncoupled representations, the Wigner and Racah symbols, 3- j, 6 – j, 9 – j etc, irreducible tensorial sets, and so on. Howver the basics remain as they were in the twenties and thirties, more or less. In UFT250 onward we have found new basics. The revolution in thought introduced by ECE is to base quantum mechanics on Cartan geometry, and refuting the Copernhagen interpretation. The ECE ideas have made a huge and permanent impact as is well known. It is impossible to cover up this impact, if people try to do that they will ose all credibility and the subject of standard will be reduced to rubble by radical new thinkers. In the Copenhagen context I strongly recomemend teh work of Prof Jose Croca, his books and papers. He has refuted Copnehagen in many ways.
1) Jose Croca, “Towards a Non Linear Quantum Physics” in my World Scientific Sereis of 27 or 28 volumes to date, “Contemporary Chemical Physics”.
2) Jose Croca, a new and important dialogue format book to be published by CISP (Cambridge International Science Publishing, www.cisp-publishing,com)
It is important to know that the Croca group has refuted Heisenberg experimentally by many orders of magnitude. Horst Eckardt gave a definitive and universally accepted theoretical refutation in UFT175, which also introduced the quantum hamilton equations. My Civil List predecessor Rowan Hamilton introduced them at Trinity College Dublin as is well known. (Sir William Rowan Hamilton).