Questions about tensor operator

[kjjtr]

Hi.
Before question, sorry about my bad english. It's not my mother tongue.

My QM textbook(Schiff) adopt

J x J = i(h bar)J.

as the defining equations for the rotation group generators in the general case.
My question is, then tensor J must have one index which has three component? (e.g. x-y-z or rho-theta-z or r-theta-pi)

And, i also have one question about position operator. In wikipedia, its eigen'value' is said to be particle's position 'vector'. What this mean is that position operator is (rank 3) tensor operator?

Thanks for reading.

 

[Bill_K]

My QM textbook(Schiff) adopt

J x J = i(h bar)J.

as the defining equations for the rotation group generators in the general case.

Writing cross products of operators can be confusing, it's better to use the form in the previous equation, which uses commutation relations. Also, I would call this a property of angular momentum, not its definition. Schiff gives the definition a few equations earlier - the three components of angular momentum are defined to be the generators of infinitesimal rotations.

My question is, then tensor J must have one index which has three component? (e.g. x-y-z or rho-theta-z or r-theta-pi)

Yes. A tensor operator with one index we call a vector operator.

And, i also have one question about position operator. In wikipedia, its eigen'value' is said to be particle's position 'vector'. What this mean is that position operator is (rank 3) tensor operator?

A vector operator, or rank 1 tensor. The rank of a tensor is the number of indices it has.