FOUNDATIONS OF QUANTUM MECHANICS

68 CHAPTER III. THE POSTULATES
We illustrate (III. 136) using a rotation of ⃗n in the x z - plane, ϕ = 0, over an angle α around
the y - axis as in diagram III. 2.
⃗n
z
θ
α
⃗n R
x
y
Figure III. 2: A rotated unit vector in the xz - plane
For ⃗n and ⃗n R we have
⎛ ⎞ ⎛ ⎞
sin θ
sin(θ + α)
⃗n = ⎝ 0 ⎠ , ⃗n R = ⎝ 0 ⎠ . (III. 137)
cos θ
cos(θ + α)
The eigenstates of ⃗n · ⃗σ, using (III. 132), are
( cos
1
|⃗n, +⟩ = 2 θ )
sin 1 2 θ = cos 1 2 θ |z ↑⟩ + sin 1 2θ |z ↓⟩ (III. 138)
and
|⃗n, −⟩ =
( − sin
1
2 θ )
cos 1 2 θ
= − sin 1 2 θ |z ↑⟩ + cos 1 2θ |z ↓⟩. (III. 139)
Rotating around the y - axis and therefore
(
U (⃗e y , α) = (cos 1 2 α 11 − i ⃗e y · ⃗σ sin 1 cos
1
2 α) = 2 α − sin 1 2 α )
sin 1 2 α cos 1 2 α , (III. 140)
we have
U (⃗e y , α) |⃗n, +⟩ =
( )
cos
1
2
(θ + α)
sin 1 2 (θ + α)
and
U (⃗e y , α) |⃗n, −⟩ =
= cos 1 2 (θ + α) |z ↑⟩ + sin 1 2
(θ + α) |z ↓⟩ (III. 141)
( )
− sin
1
2
(θ + α)
cos 1 2 (θ + α)
= − sin 1 2 (θ + α) |z ↑⟩ + cos 1 2
(θ + α) |z ↓⟩, (III. 142)