Download 10.5.1. Density Operator

10.5.1. Density Operator
When dealing with a large quantum system, we need to take 2 averages, one over the
inherent quantum uncertainties and one over the uninteresting microscopic details.
Consider then an isolated system described, in the Schrodinger picture, by a complete
set of orthonormal eigenstates  n  t 
 m  t   n  t    mn
with
  t 
and
n
 n  t   Iˆ
(10.48)
n
where n stands for the complete set of quantum numbers that can be either discrete or
continuous, or a mixture of both, with the understanding that for the continuous case,
 and  are to be replaced by the appropriate delta functions and integrals,
respectively.
If the probability of finding the system in state  n  t 
is Pn, the expectation value
of an observable A at time t is by definition
At  
P
 n  t  Aˆ  n  t 
n
(10.49a)
n
where, for an isolated system, Pn is independent of time. By means of the
orthonormal condition, we can write (10.49a) as
At  
   t  Aˆ   t 
n
m
Pm  m  t   n  t 
(10.49)
mn
   n  t  Aˆ ˆ  t   n  t 
n
 Tr  Aˆ ˆ  t  
(10.52)
where the density operator ˆ  t  is defined by
ˆ  t    Pm  m  t   m  t 
(10.50)
m
Note that despite the time dependence, ˆ  t  is still a Schrodinger picture operator.
Its effects on the bras and kets are
 ˆ  t    Pm   m  t   m  t 
m
ˆ  t     Pm  m  t   m  t  
m
where Cm  t   Pm  m  t   .
  Cm*  t   m  t 
m
  Cm  t   m  t 
m
(10.51)
Let A and B be square matrices, then
Tr  AB    Amn Bnm   Bmn Anm  Tr  BA
mn
nm
Treating the matrix elements of operators as elements of matrices, we see that
Tr  Aˆ ˆ  t    Tr  ˆ  t  Aˆ 
(10.53)
Taking the trace of (10.50) gives
Tr  ˆ  t     Pm  n  t   m  t   m  t   n  t 
nm
  Pm mn
nm
  Pn  1
(10.54)
n
The Heisenberg picture version of ̂ is given by
ˆ H  t   eiHt ˆ  t  e iHt   Pm  m  m
ˆ
ˆ
m
Thus, contrary to the usual operators, ̂ is time-dependent in the Schrodinger
picture but time-independent in the Heisenberg picture.
In the Schrodinger picture,
the equation of motion for ˆ  t  can be obtained from (10.50) as
i
d ˆ  t 
d
 d


  Pm  i
 m t    m t    m t  i
 m t  
dt
dt

m
 dt

  Pm  Hˆ  m  t   m  t    m  t   m  t  Hˆ 
m
  Hˆ , ˆ  t  
(10.55)
which is the quantum mechanical version of the Liouville equation (10.4). Note that
(10.55) differs by a minus sign from the usual equation of motion for operators in the
Heisenberg picture.