FOUNDATIONS OF QUANTUM MECHANICS

ut the probability to find the system in |u k ⟩ is
µ W (U k ) = Tr U k W = Tr
III. 3. THE INTERPRETATION OF MIXED STATES 53
K∑
m=1
U k p m |u m ⟩ ⟨u m | =
K∑
m=1
p m |⟨u k | u m ⟩| 2 (III. 53)
Although the result (III. 52) is in accordance with the behavior of an ensemble of systems being in
the state |u k ⟩ with probability p k , we see that for (III. 53), contrary to (III. 47), the outcome, i.e. the
probability to find in (III. 51) the state |u k ⟩, is in general not p k , which is a consequence of the non -
orthogonality of the states |u k ⟩. On the other hand, (III. 51) can always be written in the form (III. 41),
in terms of the orthonormal set of eigenvectors of W , which leads to the conclusion that ensembles
which are interpreted as being physically completely different, are described by the same operator W .
This can be compared with the fact that a pure state |ψ⟩ can be written in numerous ways as a
superposition of other pure states, which corresponds to different ways of preparation of |ψ⟩ by superposition
of other states, for instance in a tilted Stern - Gerlach apparatus in case of measurement of
spin. We can no longer see if |ψ⟩ is, for example, a superposition of spin up and down in the z - direction,
or of spin up and down in the x - direction.
For pure states this seems completely natural; it is a direct consequence of the state postulate
which forms a vector space of states. In case of mixed states the situation is less clear. It can be
maintained that an ensemble, of which each system is in the state |u k ⟩ with probability p k , really
differs from an ensemble of systems which are in a state |w i , j⟩ with probability w i , even though the
expectation values of all physical quantities are equal for both ensembles. In that case, from
W =
M∑
i=1
∑n i
j=1
w i |w i , j⟩ ⟨w i , j| =
K∑
p k |u k ⟩ ⟨u k | (III. 54)
k=1
it has to be concluded that the state operator W characterizes these ensembles incompletely. There is
no postulate in quantum mechanics by which this is prohibited.
Another view is, however, that the state operator is a complete description of a state, the different
possible ways of preparation are not retrievable from the state W . Consequently, the conclusion has
to be that W , in (III. 51), does not characterize an ensemble which exists of a mixture of systems in
pure states |u k ⟩, but an ensemble characterized by W only presents itself as such an ensemble upon
measurement. Here we see again that, in quantum mechanics, we get in trouble if we speak in terms
of what really exists. In section III. 5 we will return to this discussion in the context of improperly
mixed states.
The dynamics of mixed states follows, as in the classical case, from the pure states. Define,
analogously to (III. 41),
W (t) :=
M∑
i=1
∑n i
j=1
w i W i,j (t). (III. 55)
According to the Schrödinger postulate, (III. 2),
|w i , j, t⟩ := U (t − t 0 ) |w i , j, t 0 ⟩, (III. 56)