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Total angular momentum as generator of rotations

As mentioned above, orbital angular momentum L is defined as in classical mechanics: L = r × p {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }, but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator R (n ^, ϕ) ≡ exp (− i ℏ ϕ J ⋅ n ^) {\displaystyle R({\hat {n}},\phi)\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)} is the rotation operator that takes any system and rotates it by angle ϕ {\displaystyle \phi } about the axis n ^ {\displaystyle {\hat {\mathbf {n} }}}. (The "exp" in the formula refers to 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 algebras and Lie groups 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.