Discussion Overview
The discussion revolves around the calculation of helicity in non-relativistic quantum mechanics using Pauli matrices. Participants explore the implications of the helicity operator, its representation in spinor space, and the conditions under which different states may or may not be eigenstates of this operator.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants state that helicity is given by the expression ##\sigma \cdot p / |p|##, where ##\sigma## are the Pauli matrices and ##p## is the momentum.
- It is noted that in spinor space, the Pauli matrices are 2x2 matrices, leading to helicity being a 2x2 quantity, while in 3D physical space, it is treated as an inner product between three-vectors, suggesting helicity is a 1x1 quantity.
- A participant questions how to transform the Pauli matrices so that each matrix becomes a number, possibly by taking their determinant.
- Another participant emphasizes that the Pauli matrices and the helicity operator are operators, not scalars, and mentions that the scalar helicities ##\pm\frac{1}{2}\hbar## are eigenstates of the helicity operator.
- Concerns are raised about general states with spin and momentum not aligned, indicating that such states will not be eigenstates of the helicity operator.
- A later reply agrees that non-aligned states will result in a superposition or a spin eigenstate along another axis.
Areas of Agreement / Disagreement
Participants express differing views on the transformation of Pauli matrices and the nature of helicity in various states. There is no consensus on how to handle the transformation of the matrices or the implications for non-aligned states.
Contextual Notes
Participants do not resolve the question of how to transform the Pauli matrices into numbers, nor do they clarify the implications of the helicity operator in the context of general states with non-aligned spin and momentum.