Editing Angular momentum (section)

=== Total angular momentum as generator of rotations ===
As mentioned above, orbital angular momentum '''L''' is defined as in classical mechanics: <math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>, but ''total'' angular momentum '''J''' is defined in a different, more basic way: '''J''' is defined as the "generator of rotations".<ref name=littlejohn>{{cite web|url=http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf |archive-date=2022-10-09 |url-status=live|author-link1=Robert Grayson Littlejohn|title= Lecture notes on rotations in quantum mechanics|first= Robert |last=Littlejohn|access-date= 13 Jan 2012|work=Physics 221B Spring 2011|year=2011}}</ref> More specifically, '''J''' is defined so that the operator
<math display="block">R(\hat{n},\phi) \equiv \exp\left(-\frac{i}{\hbar}\phi\, \mathbf{J}\cdot \hat{\mathbf{n} }\right)</math>
is the [[Rotation operator (quantum mechanics)|rotation operator]] that takes any system and rotates it by angle <math>\phi</math> about the axis <math>\hat{\mathbf{n}}</math>. (The "exp" in the formula refers to [[matrix exponential|operator exponential]].) To put this the other way around, whatever our quantum Hilbert space is, we expect that the [[rotation group SO(3)]] will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between [[Lie algebra]]s and [[Lie group]]s in mathematics. The close relationship between angular momentum and rotations is reflected in [[Noether's theorem]] that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.