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Introduction of the density operator: the pure case
A. Description by a state vector
Consider a system whose state vector at instant t is:
 (t )   cn (t ) un where { un } form an orthonorma l basis (3)
n
The coefficien ts cn satisfy :
 c (t )
n
2
1
(4)
n
A is an observable , with matrix elements :
un A u p  Anp
(5)
The mean (" expectatio n" ) value of A at instant t is :
A (t )   (t ) A  (t )   cn* (t )c p (t ) Anp
(6)
n, p
Finally, the evolution of  (t) is described by TDSE :
i
d
 (t )  H (t )  (t )
dt
(7)
Mathematical tools of crucial importance in quantum approach
to thermal physics are the density operator op and the
mixed state operator M. They are similar, but not identical.
Dr. Wasserman in his text, when introducing quantum thermal
physics, often “switches” from op to M or vice versa, and one
has to be alert when reading and always know which operator
the text is talking about at a given moment.
I thought it would help if you could learn about the density
operator not only from Dr. Wasserman’s text, but also from
another source, and therefore I made a short “auxiliary”
slide presentation about the density operator and its
significance, based on another book (“Quantum Mechanics”
by Cohen-Tannoudji et al.). The pages I used for preparing this
presentation will be given to you as a handout. Cohen-Tannoudji
uses a slightly different notation than Dr. Wasserman, but I
decided not to change it.
B. Description by a density operator
Relation (6) shows that the coefficients c(t) enter into the
*
c
mean values through quadratic expressions of the type n (t )c p (t )
These are simply the matrix elements of the
 (t )  (t )
operator, the projector onto the ket  (t ) as it easy to
show using (3):
*
u p  (t )  (t ) un  cn (t )c p (t )
(8)
It is therefore natural to introduce ther density operator ρ(t)
defined by:
 (t )   (t )  (t )
(9)
The density operator is represented in the {|un} basis by a
matrix called density matrix whose elements are:
 pn (t )  u p  (t ) un  c (t )c p (t )
*
n
(10)
We are going to show that the specification of ρ(t) suffices
to characterize the quantum state of the system: that is, it
enables us to obtain all the physical predictions that can
be calculated from  (t ) . To do this, let us write formulas
(4), (6) and (7) in terms of the operator ρ(t). According to (10),
relation (4) indicates that the sum of the diagonal elements
of the density matrix is equal to 1:
 c (t )
n
n
2
  nn (t )  Tr  (t )  1
(11)
n
In addition, using (5) and (10), formula (6) becomes:
A (t )   u p  (t ) un un A u p 
n, p
u
p
p
 (t ) u p  Tr  (t ) A
Tr  (t )  1
Finally, the time evolution of the ρ(t) operator can be deduced
from the TDSE:
d
d

d

 (t )    (t )   (t )   (t )   (t ) 
dt
 dt

 dt

1
1
 H (t )  (t )  (t ) 
 (t )  (t ) H (t )
i
(i)
1
 H (t ),  (t )
i
SUMMARY
A (t )  Tr  (t ) A
Tr  (t )  1
Also important :  (t)   † (t )
d
i  (t)  H (t ),  (t )
dt