The mixed state
Example: using a Stern-Gerlach apparatus, a beam of ions
with spin ½ can be split into two separate beams in which
the ions are in pure quantum states, “spin up” and “spin down”.
Let us denote these pure states as
We can now introduce
“pure spin state operators”:

and  
    and    
Then, with another apparatus, the beams can be re-deflected
in some fixed proportions, say:
f
and
f  f
f
f  f
to reconstitute a new beam that is, literally, an incoherent
mixture of the two pure states.
In that case it makes no physical sense to try to assign
a single state operator   to this reconstituted beam.
But it does make sense to define a Mixed State Operator
as:

1
M 
f      f    
 f  f 

The general case of such an incoherent mixture is the
Mixed State Operator:
( j)
( j)
M   wj 

j
where wj is the fraction of the mixed system made up from
each of the pure states

( j)

( j)
with
w
j
j
 1.
For example, a reconstituted beam with ¾ of the ions in
the (↑) state and ¼ of the ions in the (↓) state would have
the mixed state operator:
3
1
M          
4
4
It follows straightforwardly that even for the mixed
case: Tr{  }  1 . But it may be instructive to prove
M
it in a formal way:
Now, let’s go back to the “ordinary” density operator we
discussed in the QM2 file. It was shown there that the
expectation value of an A operator can be expressed in
terms of the density operator as: A (t )  Tr  (t ) A
It was also shown that: Tr  (t )  1
Normally, these two equations are combined, and the
formula for the expectation value is written as:
Tr  (t ) A
A (t ) 
Tr  (t )
But the denominator is 1, so why do we write it? Well,
we do that in order to stress that the density operator
has a proper normalization.
Dr. Wasserman uses a slightly different notation – in
His book the same equation [Eq. (5.43)] has the form:
 
Tr op op
Tr op
Dr. Wasserman also used a different method of deriving
that formula. It may be instructive to take a look:
On Page 77 in Dr. Wasserman’s text it is shown that for an
 op i
  op , H op using a
“ordinary” density operator:
t

slightly different calculation scheme than we used in the
QM2 file. But the two procedures are equivalent, of course.


The equations in the preceding slide apply to a system which
is in a pure quantum state, i.e., its state function is a linear
combination of eigenstates. Let’s now return to mixed state
systems.
On Page 78 in Dr. Wasserman’s text it is stated: It is equally
straightforward to show that the average value of the observables ωs for the mixture is:
Tr M  op
 
Tr M
We will accept this without a proof, but it may be a good
exercise for you to prove that.