Understanding Spherical Tensors & Their Applications

[Malamala]

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!

 

[eys_physics]

So, in your example you have one tensor $S$ of angular momentum (rank) $j_1=1/2$ and another one $D$ with $j_2 = 1$. You can form the new tensors (using Clebsch-Jordan symbols)

$$T_{J=1/2, M} = \sum_{m_1 m_2} (1/2 m_1 1 m_2 | 1/2 M) S_{1/2, m_1}D_{1, m_2}$$,

and

$$T_{J=3/2, M} = \sum_{m_1 m_2} (1/2 m_1 1 m_2 | 3/2 M) S_{1/2, m_1}D_{1, m_2}$$,

where $(1/2 m_1 1 m_2 | 3/2 M)$ is a Clebsch-Jordan symbol.

The only example of spherical tensors of non-integer rank which I can think of are creation and annihilation operators.

 

[Malamala]

So, in your example you have one tensor $S$ of angular momentum (rank) $j_1=1/2$ and another one $D$ with $j_2 = 1$. You can form the new tensors (using Clebsch-Jordan symbols)

$$T_{J=1/2, M} = \sum_{m_1 m_2} (1/2 m_1 1 m_2 | 1/2 M) S_{1/2, m_1}D_{1, m_2}$$,

and

$$T_{J=3/2, M} = \sum_{m_1 m_2} (1/2 m_1 1 m_2 | 3/2 M) S_{1/2, m_1}D_{1, m_2}$$,

where $(1/2 m_1 1 m_2 | 3/2 M)$ is a Clebsch-Jordan symbol.

The only example of spherical tensors of non-integer rank which I can think of are creation and annihilation operators.

Thank you for your reply. So in the first case, I understand the meaning of the expression from a physical point of view, in terms of quantum mechanics. For example, for $M=1/2$, it says that you can get that either by applying $\sigma_z = \sigma_0$ on the state $(1/2,1/2)$ or applying $\sigma_+$ on $(1/2,-1/2)$. I.e. you have the same particle, of spin $1/2$ (say an electron), but you can change its spin orientation. I am not sure what is the physical meaning in the second case. It looks like you start with a particle of spin $1/2$ and end up with a particle of spin $3/2$ (and some direction of the spin on the z axis). So you basically change the identity of the particle. What is the physical meaning of this?