Angular momentum quantization

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.

There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term angular momentum operator can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always conserved, see Noether's theorem.

In quantum mechanics, angular momentum can refer to one of three different, but related things.

The classical definition of angular momentum is ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$. The quantum-mechanical counterparts of these objects share the same relationship:

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