Chapter 9 The Two-Body Problem

1.4. Spin Angular Momentum
Some particles such as electrons and protons have intrinsic angular momentum in addition to the orbital angular momentum which is associated with their motion. The spin may be compared to the angular momentum of a rigid body about its center of mass, but it arises from the internal degrees of freedom of the particle.
Let us denote the components of the spin angular momentum of a particle by the Hermitian operator with the components and . These components satisfy the fundamental commutation relation

A trivial solution of (9.111) is found when all of the components of are zero. This is the case of spin zero particle such as -meson. But particles like electrons and protons are spin particles. This means that the eigenvalues of the components of along a given direction can take on the values . Thus for any direction , there are two eigenstates and corresponding to the eigenvalues and . We can choose the eigenstates of as the basis for spin space. Once this choice is made the operator can be written in this basis as

where is the unit vector in the -direction. Now for this representation we have

and therefore the matrix becomes

The other two components, and , are also Hermitian matrices and these can be written as

(see Eqs. (9.83) and (9.84)). We can easily verify that these three matrices satisfy the fundamental commutation relation (9.111).
The eigenstates and obtained from (9.114) are given by

Generally the spin operator is expressed in terms of Pauli matrices and , where

and

Some of the properties of the Pauli spin matrices are as follows [8]:
(a) - They obey the commutation relation

which is the same as (9.111) but written in terms of .
(b) - The square of each is a unit matrix

(c) - By direct calculation we can show that when the Pauli matrices satisfy the anti-commutation relation

We can combine (9.120) and (9.121) into a single relation

(d) - We can also combine (9.119) and (9.122) and write

(e) - If and are two -number vectors we have

(f) - From (9.117) and (9.120) we find to be

(g) - The total angular momentum of a spinning particle is the sum of its orbital angular momentum and its spin angular momentum