Calculating Helicity in Non-Relativistic Quantum Mechanics using Pauli Matrices

[Tio Barnabe]

The helicity in non relativistic quantum mechanics is given by $\sigma \cdot p / |p|$ where $\sigma$ are the pauli matrices and $p$ the momentum. In spinor space, the $\sigma$ are 2x2 matrices, and thus, the helicity, if we calculate it, is a 2x2 quantity. But in 3d physical space, the above equation is an inner product between two three-vectors, because we have three pauli matrices. It's like $\sigma_i p^i$ and thus the helicity is a 1x1 quantity.

My question is, How can I transform the pauli matrices, such that each matrix $\sigma_i$ becomes a number, such that I get the latter result I mentioned above? Maybe taking their determinant?

 

[mikeyork]

The helicity in non relativistic quantum mechanics is given by $\sigma \cdot p / |p|$ where $\sigma$ are the pauli matrices and $p$ the momentum. In spinor space, the $\sigma$ are 2x2 matrices, and thus, the helicity, if we calculate it, is a 2x2 quantity. But in 3d physical space, the above equation is an inner product between two three-vectors, because we have three pauli matrices. It's like $\sigma_i p^i$ and thus the helicity is a 1x1 quantity.

My question is, How can I transform the pauli matrices, such that each matrix $\sigma_i$ becomes a number, such that I get the latter result I mentioned above? Maybe taking their determinant?

The Pauli matrices and the corresponding helicity operator $\sigma \cdot p / |p|$ are operators, not scalars. The scalar helicities $\pm\frac{1}{2}\hbar$ are eigenstates of the helicity operator.

 

[Tio Barnabe]

The scalar helicities $\pm\frac{1}{2}\hbar$ are eigenstates of the helicity operator.

what if we have a general state with spin and momentum not aligned. In this case it will not be an eigenstate of the helicity operator.

 

[mikeyork]

what if we have a general state with spin and momentum not aligned. In this case it will not be an eigenstate of the helicity operator.

That's right. It will be a superposition -- or a spin eigenstate along another axis.