Schroedinger and Heisenberg Pictures

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 


The Schroedinger picture and the Heisenberg picture are two different representations.

Switching between representations

The Heisenberg picture can be obtained from the Schroedinger picture by a unitary transformation at any time t.

|yH>=U(t0,t)|yS(t)>, |yH>=UT(t,t0)|yS(t)>=|yS(t0)>.

The matrix elements of any operator A are independent of the representation.


The predictions of quantum mechanics are independent of the representation.


Consider the Schroedinger representation, (often called the Schroedinger picture), defined by the Schroedinger equation

(a) Write the transformation equations for the wave function y and arbitrary operators Q to the Heisenberg representation (or picture), in which the operators are time dependent but the wave functions are not.  Show that this transformation is unitary if H is Hermitian and that matrix elements have the same value in the two pictures.
(b) From the transformation equations in part (a) derive the Heisenberg equation of motion for the operators in the Heisenberg representation.  (Assume the operators in the Schroedinger picture have no explicit time dependence).
(c) Consider a Hamiltonian operator   Show that the Heisenberg equation of motion for the operator is  which has the same form as the corresponding classical equation of motion.