Derivation of an angular momentum expression

[Malamala]

Hello! I found this formula in several places for the total angular momentum of a particle with intrinsic spin 1/2 and angular momentum l=1 in the non-relativistic limit:
$$\frac{1}{\sqrt{4 \pi}}(-\sigma r /r )\chi$$
where $\sigma$ are the Pauli matrices and $\chi$ is the spinor. Can someone point me towards a derivation of this? Also can this be generalized to other s and l (or at least other l for s=1/2), can someone point me towards that, too, if that is the case? Thank you!

 

[eljose79]

Hello! Thank you for bringing up this formula for the total angular momentum of a particle with intrinsic spin 1/2 and angular momentum l=1 in the non-relativistic limit. This formula is commonly known as the "spin-orbit coupling" term and it plays a crucial role in understanding the behavior of atoms and molecules.

To derive this formula, we first need to understand the concept of spin angular momentum. In quantum mechanics, particles have both orbital angular momentum (associated with their motion around a central point) and intrinsic spin angular momentum (associated with their intrinsic properties). The total angular momentum of a particle is given by the sum of these two components.

Now, for a particle with intrinsic spin 1/2 and angular momentum l=1, we can represent its spin state using a 2x2 matrix called the spinor. This spinor is represented by the symbol $\chi$ in the formula you mentioned. The Pauli matrices, denoted by $\sigma$, are used to describe the spin state of this particle.

To derive the formula, we start with the total angular momentum operator $\hat{J}$, which is given by the sum of the orbital angular momentum operator $\hat{L}$ and the spin angular momentum operator $\hat{S}$. In the non-relativistic limit, the spin angular momentum operator can be approximated by the Pauli matrices multiplied by the spinor.

Next, we use the commutation relation between the orbital and spin angular momentum operators to obtain the expression for the total angular momentum operator. This expression can then be simplified to get the formula you mentioned.

This formula can be generalized to other values of s and l by using the appropriate spin and orbital angular momentum operators. However, for s=1/2, the Pauli matrices and the spinor will still be used.

I hope this helps in understanding the derivation of the formula for the total angular momentum of a particle with intrinsic spin 1/2 and angular momentum l=1 in the non-relativistic limit. If you have any further questions, please let me know. Thank you.