The Schroedinger and the Heisenberg picture

Let |yS(t0)> be a state vector in the Schroedinger picture, i.e. let it evolve in time and let its evolution be described by the Schroedinger equation.  Then   The Schroedinger picture implies an active unitary transformation.  The state vector is transformed, but all operators are constant in time unless they contain time explicitly.  The basis vectors are not changing.  The operators are defined through their action on the basis vectors.

The Heisenberg picture implies the equivalent passive unitary transformation.  The state vector is constant, |yH>=|yS(t0)>=U(t0,t)|yS(t)>, |yH>=UT(t,t0)|yS(t)>.  However the basis vectors are changing, and therefore the operators are changing.  The operator AH(t) which does to the new basis vectors what AS(t) does to the old basis vectors is given by  .

 

describes the evolution of the operators in the Heisenberg picture.

The interaction picture

Assume the Hamiltonian of an arbitrary system is H0(t), and the corresponding evolution operator is U0(t,t0).  Assume that the Hamiltonian of a given system is H(t)=H0(t)+W(t).  The state vector for this system in the interaction picture is |yI(t)>=U0T(t,t0)|yS(t)>.

 describes the evolution of |yI(t)>.  In integral form we have

¯_________________________­

This integral equation can be solved by iteration.

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The interaction picture assigns part of the time dependence to the state vectors, and part to the operators.  The equation for the evolution of operators in the interaction picture is

.

The evolution of the mean value of an observable

In the Schroedinger picture the evolution of the mean value of an observable is given by

Ehrenfest’s theorem:

The density operator

Let  We introduce the pure state density operator , i.e. the projector onto the state , and the density matrix with matrix elements .  In terms of the density operator we express:

  1. conservation of probability, ,
  2. the mean value of the observable A, ,
  3. the time evolution of the system, ,
  4. the probability that a measurement of A will yield the eigenvalue an,   , where Pn is the projector into the eigensubspace of A associated with the eigenvalue an.

We are dealing with a statistical mixture of states, if we have incomplete information and only know that a system may be in a state |yk> with probability pk, with .  For such a system we introduce the density operator and the density matrix with matrix elements .  In terms of the density operator we express:

  1. conservation of probability, ,
  2. the mean value of the observable A, ,
  3. the time evolution of the system, ,
  4. the probability that a measurement of A will yield the eigenvalue an,   , where Pn is the projector into the eigensubspace of A associated with the eigenvalue an,