- #1
NewGuy
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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
D(\widehat{n},d\phi)=1-i\left(\bold{J}\cdot\widehat{n}/\hbar\right)d\phi
Where J is the angular momentum operator and \widehat{n} is the rotation axis.
In this way he shows that the angular momentum operator J satisfies the canonical commutation relations
[J_i,J_j]=i\hbar \epsilon_{i,j,k}J_k
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 i\hbar times the quantum mechanical commutator of the corresponding operators. For example [x_i,p_j]_{poisson}=\delta_i^j\rightarrow i\hbar[x_i,p_j]_{commutator}=\delta_i^j
D(\widehat{n},d\phi)=1-i\left(\bold{J}\cdot\widehat{n}/\hbar\right)d\phi
Where J is the angular momentum operator and \widehat{n} is the rotation axis.
In this way he shows that the angular momentum operator J satisfies the canonical commutation relations
[J_i,J_j]=i\hbar \epsilon_{i,j,k}J_k
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 i\hbar times the quantum mechanical commutator of the corresponding operators. For example [x_i,p_j]_{poisson}=\delta_i^j\rightarrow i\hbar[x_i,p_j]_{commutator}=\delta_i^j