4 ms·
> so he knows what he's talking about; I'm not so sure. For example: > The commutator of the Hamiltonian with the angular momentum of an electron does not va
by jpmattia 10y ago
> so he knows what he's talking about;
I'm not so sure. For example:
> The commutator of the Hamiltonian with the angular momentum of an electron does not vanish.8 Its eigenvalue is, therefore, not a constant of motion.
In fact, the Hamiltonian is invariant under rotations (the electron does not care about which way the protons in the nucleus are rotated), which is why the angular momentum even exists. (From the fact that the Lagrangian is invariant under rotation: Noether's theorem tells you that there is a conserved quantity associated with that invariance. It turns out that rotational invariance gives rise to angular momentum.)
I haven't checked his reference 8, but "Boston Studies Series in the Philosophy of Science" is not the first place I'd go to understand constants of the motion and how they relate to things that commute with the Hamiltonian.
- thisrod 10y agoThe key word is an electron. The total angular momentum is conserved, but the way the electrons share it between them is not. As I recall, actual atomic Hamiltonians have terms like σ₁·σ₂. The reason involves some tricky correspondence between exchange and spin that I haven't thought about in a long time, and would have to look up. I agree with the broader point that this essay is odd and confusing.
- pdonis 10y ago> the Hamiltonian is invariant under rotations Yes, but that doesn't contradict what the article said. The article said, unpacking slightly, that the Hamiltonian operator does not commute with the "angular momentum of a single electron" operator. It only commutes with the "angular momentum of all the electrons taken together" operator.