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.

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Download 10.5.1. Density Operator