FOUNDATIONS OF QUANTUM MECHANICS

72 CHAPTER III. THE POSTULATES
The next two examples concern the center of the Bloch sphere, ⃗w = 0 .
(d) With ⃗w = 0 , we have
( ) 1 0
W = 1 2
. (III. 161)
0 1
The eigenvalues of this mixed state W are degenerate, and various factorizations are possible,
for example
W = 1 2 |x ↑⟩ ⟨x ↑| + 1 2
|x ↓⟩ ⟨x ↓|
= 1 2 |y ↑⟩ ⟨y ↑| + 1 2
|y ↓⟩ ⟨y ↓|
= 1 2 |z ↑⟩ ⟨z ↑| + 1 2
|z ↓⟩ ⟨z ↓|. (III. 162)
(e) Under a rotation R, ⃗w behaves like a vector in R 3 ,
U (R) ( ⃗w · ⃗σ) U − 1 (R) = ⃗w R · ⃗σ (III. 163)
where U (R) is given by (III. 135). Therefore, the only rotation invariant state for a 1 - particle
system is ⃗w = 0 .
The similarity between the set of density matrices W and the 3 - dimensional unit sphere of polarization
vectors is specific for spin 1/2 particles, in which case every pure state is also the eigenstate
for the spin operator in a certain spin direction. For spin 1 bosons and higher spin particles this no
longer applies.
III. 6. 3
TWO SPIN 1/2 PARTICLES
III. 6. 3. 1
SINGLET AND TRIPLET STATES
Consider a composite system of two spin 1/2 fermions. In the direct product space C 2 ⊗ C 2 = C 4
a basis is
|z ↑⟩ ⊗ |z ↑⟩, |z ↑⟩ ⊗ |z ↓⟩, |z ↓⟩ ⊗ |z ↑⟩, |z ↓⟩ ⊗ |z ↓⟩. (III. 164)
From these basis states the simultaneous eigenstates |s, m⟩ of the operators ⃗ S 2 = ( ⃗ S 1 + ⃗ S 2 ) 2
and S z = S 1z + S 2z can be formed, where s can be 0 or 1. The eigenvalues of ⃗ S 2 are 2 s(s + 1), the
eigenvalues of S z are m, as introduced on p. 65.
The singlet state or singlet for short, with s = 0 and therefore m = 0, is the entangled state
|Ψ 0 ⟩ = |0, 0⟩ = 1 2
√
2
(
|z ↑⟩ ⊗ |z ↓⟩ − |z ↓⟩ ⊗ |z ↑⟩
)
, (III. 165)
which looks the same in terms of the eigenstates of S x and S y , having spherical symmetry. The singlet
is a simultaneous eigenstate of S x , S y and S z with eigenvalue 0. Hence the singlet is an eigenstate
of ⃗n · ⃗S with eigenvalue 0, which means that a rotation (III. 133) carries (III. 165) back into itself.