- #1
Malamala
- 299
- 27
Hello! I came across spherical tensors, and I am a bit confused about the way they are applied. For example, Pauli matrices, can be grouped together to form a rank 1 (vector) spherical tensor as ##(\sigma_-, \sigma_z, \sigma_+)##, which are the raising operator, the z projection operator and the lowering operator. When this acts on a spin state, say ##(1/2,1/2)##, we can think of it as normal angular momentum addition. For example, for ##\sigma_-##, which is, as a tensor, ##(1,-1)##, we would get overall ##(1/2,-1/2)## i.e. the spin down state, which is what you expect. The same applies for the other 2 operators. This makes sense. However, combining a rank 1 tensor with a rank 1/2 tensor would give both a rank 1/2 tensor, which is what I mentioned before i.e. the down state is still part of a rank 1/2, but it should also give a rank 3/2 tensor. What is the mathematical expression and physical meaning of this 3/2 rank tensor? Thank you!