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.

 

[denian]

for your question. The commutation relations for angular momentum are a result of both the fact that rotations do not commute and the quantization rules for quantum mechanical angular momentum. The classical concept of rotations not commuting is extended to the quantum mechanical realm, where angular momentum is quantized and represented by operators. These operators, which are the generators of spatial rotations, have specific commutation relations that are a direct consequence of the quantization rules for angular momentum.

In classical mechanics, rotations are described by the Poisson bracket, which is a type of commutation relation. In quantum mechanics, this concept is extended to the commutator, which quantifies the non-commutativity of operators. The specific commutation relations for angular momentum operators are a result of the fact that rotations do not commute, but they are also influenced by the quantization rules for angular momentum.

So, to answer your question, the generators for rotations and their commutation relations are originally classical concepts, but they are adapted and extended to quantum mechanics in order to describe the quantized nature of angular momentum. We use these generators as operators in quantum mechanics because they accurately represent the behavior of angular momentum and its commutation relations.