FOUNDATIONS OF QUANTUM MECHANICS

64 CHAPTER III. THE POSTULATES
but otherwise they are free to choose. As far as being in one of the states |u i ⟩ or |v j ⟩ can be interpreted
as a property the subsystems possess, all correlations between these properties in the total state can
be expressed by the p ij . If there are no correlations, p ij = u i v j .
But we see that W ′ is of the special form (III. 93), and therefore in general not equal to the arbitrary
state operator W we started with, it cannot be said that the individual subsystems are in the pure
states |u i ⟩ and |v j ⟩. Although W I and W II are state operators, they cannot be interpreted as mixtures of
pure states. The mixed states W I and W II are called improper mixtures by B. d’Espagnat (1989, p.61).
Proper mixed states can in principle be taken as an ensemble of systems which are in pure states,
where improper states cannot.
The foregoing shows that the concept of mixed states is forced upon us by the theory of composite
systems as a natural extension of the concept of pure states. Even if the composite system is in a pure
state, the subsystems are generally not pure, it is not correct to understand mixed states in general as
simple mixtures of pure states, in the way the mixture of pieces in the box of a game of chess consists
of black and white pieces.
Finally, we make an observation about similar, or identical, particles. A system of similar particles
is described in quantum mechanics by symmetrized states. Consider the following symmetrized
two - particle state
|Ψ(1, 2)⟩ = 1 2
√
2
(
|u⟩ ⊗ |v⟩ ± |v⟩ ⊗ |u⟩
)
, (III. 109)
where the first factor in each direct product is related to particle 1, and the second to particle 2. In
this case the two subspaces are identical and |u⟩ and |v⟩ can represent states in both one and the other
subspace. The corresponding state operator is
W = |Ψ(1, 2)⟩ ⟨Ψ(1, 2)| = 1 (
2 |u⟩ ⟨u| ⊗ |v⟩ ⟨v| ± |u⟩ ⟨v| ⊗ |v⟩ ⟨u|
the partial traces are
and
W I = Tr II W = 1 2
W II = Tr I W = 1 2
± |v⟩ ⟨u| ⊗ |u⟩ ⟨v| + |v⟩ ⟨v| ⊗ |u⟩ ⟨u| ) , (III. 110)
(
|u⟩ ⟨u| + |v⟩ ⟨v|
)
, (III. 111)
(
|v⟩ ⟨v| + |u⟩ ⟨u|
)
, (III. 112)
and we see that the partial traces are identical. We have to say that both particles are in the same state,
we certainly can not say that one particle is in the state |u⟩ and the other in |v⟩. We cannot assign a
pure state to the separate particles, although the state of the composite system is pure.
III. 6
SPIN 1/2 PARTICLES
The time - dependent Schrödinger equation for the wave function Ψ(q, t) is given by
i ∂Ψ
∂t
= − 2
2m ∇2 Ψ + V Ψ. (III. 113)