Angular Momentum: Generators & Commutation Relations

[Ratzinger]

The angular momentum is the generator of spatial rotations.

Are the commutation relations for angular momentum the result of the fact that rotations (all rotations, also classical) do not commute or are they the result of the quantization rules for quantum mechanical angular momentum?
Are the generators for rotations and their commutation relations originally classical concepts, and when we go over to quantum mechanics then we simply use these generators as operators?

thanks

 

[Physics Monkey]

Hi Ratzinger,

The commutation rules for angular momentum follow directly from the non-commutativity of rotations.

In quantum mechanics you think of angular momentum as an operator generating some unitary transformation that describes rotation.

In classical mechanics you think of angular momentum as a function of momentum and position generating some canonical transformation that again corresponds to rotation.

There is a remarkable parallel between the two approaches. In quantum mechanics the commutation relations read [J_i , J_j ] = i \hbar \epsilon_{i j k} J_k. In classical mechanics the analogue of the commutator is the Poisson bracket \{ L_i , L_j \} = \epsilon_{i j k} L_k. This great similarity is the basis of Dirac's quantization procedure where one simply replaces all Poisson brackets with commutators divided by i \hbar.

Hope this helps.