Atomic Structure Theory

1.2 Spin Angular Momentum 9
The Pauli matrices are both hermitian and unitary. Therefore,
σ 2 x = I, σ 2 y = I, σ 2 z = I, (1.46)
where I is the 2×2 identity matrix. Moreover, the Pauli matrices anticommute:
σyσx = −σxσy , σzσy = −σyσz , σxσz = −σzσx . (1.47)
The Pauli matrices also satisfy commutation relations that follow from the
general angular momentum commutation relations (1.4):
[σx,σy] =2iσz , [σy,σz] =2iσx , [σz,σx] =2iσy . (1.48)
The anticommutation relations (1.47) and commutation relations (1.48) can
be combined to give
σxσy = iσz , σyσz = iσx , σzσx = iσy . (1.49)
From the above equations for the Pauli matrices, one can show
σ· aσ· b = a · b + i σ· [a × b], (1.50)
for any two vectors a and b.
In subsequent studies we make use of simultaneous eigenfunctions of L 2 ,
Lz, S 2 and Sz. These eigenfunctions are given by Ylm(θ, φ) χµ.
1.2.2 Infinitesimal Rotations of Vector Fields
Let us consider a rotation about the z axis by a small angle δφ. Under such
a rotation, the components of a vector r =(x, y, z) are transformed to
x ′ = x + δφ y,
y ′ = −δφ x + y,
z ′ = z,
neglecting terms of second and higher order in δφ. The difference δψ(x, y, z) =
ψ(x ′ ,y ′ ,z ′ ) − ψ(x, y, z) between the values of a scalar function ψ evaluated in
the rotated and unrotated coordinate systems is (to lowest order in δφ),
δψ(x, y, z) =−δφ
x ∂
∂
− y ψ(x, y, z) =−iδφ Lz ψ(x, y, z).
∂y ∂x
The operator Lz, in the sense of this equation, generates an infinitesimal
rotation about the z axis. Similarly, Lx and Ly generate infinitesimal rotations
about the x and y axes. Generally, an infinitesimal rotation about an axis in
the direction n is generated by L · n.
Now, let us consider how a vector function,