I ran into a problem reading Sakurais book about advanced quantum mechanics. I understand what a spherical tensor operator is, it's just an odd number of operators that transform in a nice way under rotation (or equivalently has some nice commutation relations with angular momentum). Sakurai defines a vector operator in several ways: by it's commutation relation [tex][V_i,J_j]=i\epsilon_{ijk}\hbar V_k[/tex], and he also defines a vector operator simply as a spherical tensor operator of rank 1.(adsbygoogle = window.adsbygoogle || []).push({});

In my mind this is quite confusing however. By the first definition [tex]\textbf{J}=(J_x,J_y,J_z)[/tex] is clearly a vector operator, butJis not a spherical tensor operator (how would you define the three components in the first place?). It is true that you can construct 3 new operators from it: [tex]T_0^1=V_z, T_{\pm1}^1=\mp1/\sqrt{2}(V_x\pm iV_y)[/tex], and these 3 operators constitue a vector operator by definition 1. However, I can't understand why Sakurai claims this new vector operator to be equal toJ. It could be that it was possible to make 3 other operators fromJ, and that these 3 operators also would constitute a vector operator.

I guess my basic question is: Is the total angular momentum operatorJa spherical tensor operator?

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# A spherical tensor operator

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