- Polarized along theSzdirection with eigenvalue− ̄h/2, the state is|z−〉, and the correspond-
ing density operator isρ=|z−〉〈z−|. - Polarized along theSxdirection with eigenvalue± ̄h/2, the state is|x±〉= (|z+〉±|z−〉)/
√
2,
and the corresponding density operator isρ= (|z+〉±|z−〉)(〈z+|±〈z−|)/2.From these pure states, we construct mixed states, and theirassociated density operators. First,
we compute the density operators for unpolarized mixtures.Mixing 50% of|z+〉and 50% of|z−〉
states produces the density matrix
ρ =1
2
|z+〉〈z+|+1
2
|z−〉〈z−|=1
2
I (16.14)
Mixing 50% of|x+〉and 50% of|x−〉states produces the density matrix
ρ =1
2
|x+〉〈x+|+1
2
|x−〉〈x−|=1
4
(
|z+〉+|z−〉)(
〈z+|+〈z−|)
+1
4
(
|z+〉−|z−〉)(
〈z+|−〈z−|)=
1
2
I (16.15)
We see that equal mixture of|z±〉states produce the same quantum mechanical state as the
equal mixture of|x±〉states. This is good new, because after all, we aimed at constructing a
characterization of an unpolarized beam. The ensemble averages ofSall vanish tr(ρS) = 0 in the
unpolarized state.
Finally, let us derive the density matrix for the mixture of 50% of|z+〉and 50% of|x+〉pure
states,
ρ=1
2
|z+〉〈z+|+1
2
|x+〉〈x+|=( 3
41
1 4
41
4)
(16.16)Even though the pure states|z+〉and|x+〉are not orthogonal to one another, the resulting density
matrix is perfectly fine. The eigenvalues ofρare (2±
√
2)/4 and represent the population fractions
in an orthonormal basis. The ensemble averages are tr(ρSx) = ̄h/4, tr(ρSy) = 0, and tr(ρSz) = ̄h/4,
indicating that the mixture is partially polarized.
The most general density matrix for the two-state system maybe constructed by solving for its
axioms one by one. Hermiticity of a 2×2 matrixρallows us to put it in the following
ρ=1
2
(
a 0 I+a 1 σ^1 +a 2 σ^2 +a 3 σ^3)
=1
2
(
a 0 I+a·σ)
(16.17)whereσ^1 ,^2 ,^3 are the three Pauli matrices, anda 0 ,a 1 ,a 2 ,a 3 are real. The unit trace condition forces
a 0 = 1. Finally, positivity of the the matrix is equivalent to positivity of its eigenvalues. Since