Recovering the generator of rotation from canonical commutation relations

[NewGuy]

I'm having a course in advanced quantum mechanics, and we're using the book by Sakurai. In his definition of angular momentum he argues from what the classical generator of angular momentum is, and such he defines the generator for infitesimal rotations as

[tex]D(\widehat{n},d\phi)=1-i\left(\bold{J}\cdot\widehat{n}/\hbar\right)d\phi[/tex]

Where J is the angular momentum operator and [tex]\widehat{n}[/tex] is the rotation axis.
In this way he shows that the angular momentum operator J satisfies the canonical commutation relations

[tex][J_i,J_j]=i\hbar \epsilon_{i,j,k}J_k[/tex]

My question is this:
Is it possible to define the commutation relations as above, and then derive the expression of the generator of infintesimal relations to be the above expression? That is: Is defining angular momentum by it's commutation relations fully equivalent to defining it as a generator of infinitesimal rotations?

I like the definition of angular momentum by it's commutation relations, since you can make a reasonable connection to classical machanics by the rule that the poisson bracket of classical quantities gets replaced by [tex]i\hbar[/tex] times the quantum mechanical commutator of the corresponding operators. For example [tex][x_i,p_j]_{poisson}=\delta_i^j\rightarrow i\hbar[x_i,p_j]_{commutator}=\delta_i^j[/tex]

 

[zetafunction]

I think you must FIRST give the Lie Group, once we have the LIe group can obtain its generators, and from the generators you can get the commutation relation.

 

[Entropia]

I would say that both definitions are valid and can be used to define angular momentum in quantum mechanics. However, the definition using the commutation relations is more fundamental and has a deeper mathematical basis.

The canonical commutation relations are a fundamental part of quantum mechanics, and they describe the relationship between observables in a quantum system. In this case, the angular momentum operator J is an observable, and the commutation relations [J_i,J_j]=i\hbar \epsilon_{i,j,k}J_k define its behavior under rotations.

On the other hand, the concept of a generator of rotations is a classical one, and it is used to describe how a physical system changes under rotations. In quantum mechanics, this concept is extended to operators that generate infinitesimal rotations, and the expression for the generator of infitesimal rotations that you have provided is derived from the commutation relations.

Therefore, it is possible to define the commutation relations and then derive the expression for the generator of infintesimal rotations, as Sakurai has done. This shows the equivalence between the two definitions, and it also highlights the connection between classical and quantum mechanics.

In conclusion, both definitions of angular momentum are valid and can be used in quantum mechanics. However, the definition using the commutation relations is more fundamental and provides a deeper understanding of the behavior of angular momentum in quantum systems.