FOUNDATIONS OF QUANTUM MECHANICS

III. 6. SPIN 1/2 PARTICLES 67
we have
⃗n · ⃗σ =
( cos θ e
− i ϕ )
sin θ
e i ϕ , (III. 131)
sin θ − cos θ
with eigenvectors
|⃗n, +⟩ =
(
)
e − i 2 ϕ cos 1 2 θ
e i 2 ϕ sin 1 2 θ
and |⃗n, −⟩ =
(
)
− e − i 2 ϕ sin 1 2 θ
e i 2 ϕ cos 1 2 θ
(III. 132)
for eigenvalues ±1.
EXERCISE 22. Verify (III. 132)
III. 6. 1
SPIN 1/2 AND ROTATIONS IN SPIN SPACE
A rotation over an angle α ∈ [0, π) around an axis in the direction of the unit vector ⃗m,
with ⃗m ∈ R 3 , can be written as a unitary matrix
U (⃗m, α) = e − i α ( ⃗m · ⃗J) , (III. 133)
where the total angular momentum J ⃗ = L ⃗ + S ⃗ is the infinitesimal generator of rotations. With L ⃗ = 0
and writing S i = 1 2 σ i, which is, using (III. 124), in accordance to (III. 120) and the still unfounded
(III. 129), the Pauli matrices are the generators of rotations in C 2 , leading to
U (⃗m, α) = e − i 2 α ( ⃗m · ⃗σ) , (III. 134)
where ∥⃗m∥ is again 1. Using Taylor expansions, with (III. 128) we find for (III. 134)
∞∑ (− i) k (⃗m · ⃗σ) k (
U(⃗m, α) =
1
k!
2 α) k
=
k=0
∞∑
k=0
k=even
(− 1) 1 2 k ( 1
k!
2 α) ∑
k ∞ 11 + i (⃗m · ⃗σ)
k=1
k=odd
(− 1) 1 2 (k+1) ( 1
k!
2 α) k
= cos 1 2 α 11 − i (⃗m · ⃗σ) sin 1 2α. (III. 135)
It can be verified that, under a rotation around an axis ⃗m over an angle α, with ⃗n R the unit vector
in the rotated direction, the eigenstates of ⃗n · ⃗σ, (III. 132), transform into the eigenstates of ⃗n R · ⃗σ,
obeying the rotational transformation rules
U (⃗m, α) |⃗n, ±⟩ = |⃗n R , ±⟩. (III. 136)