QM in 3D position-space

Orbital angular momentum operator

The orbital angular momentum operators are defined by , or in cartesian coördinates

correspond to the observable orbital angular momentum for a single particle in 3D position space. The total orbital angular momentum operator has eigenvalues quantized by while ¹ is quantized by .

Spherical coördinates

In spherical coördinates, they may be expressed as²

Properties

1. The operators obey the commutation relation , hence they form a Lie algebra isomorphic to 𝔰𝔬(3).
2. The eigenfunctions are the spherical harmonics which are related via raising and lowering operators (see Irreps of ). It follows only integer values of are allowed.

#state/develop | #lang/en | #SemBr