Let *|y _{S}(t_{0})>* be a state
vector in the

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,

*|y _{H}>=|y_{S}(t_{0})>=U(t_{0},t)|y_{S}(t)>, |y_{H}>=U^{T}(t,t_{0})|y_{S}(t)>*.

However the basis vectors are changing, and
therefore the operators are changing. The operator *A _{H}(t)* which does to
the new basis vectors what

.

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

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

*|y _{H}>=U(t_{0},t)|y_{S}(t)>, |y_{H}>=U^{T}(t,t_{0})|y_{S}(t)>=|y_{S}(t_{0})>.*

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

=.

The predictions of quantum mechanics are independent of the representation.

(a) Write the transformation equations for the wave function

(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.

**Solution:**(a)

*|y*_{H}>=U(t_{0},t)|y_{S}(t)>, |y_{H}>=U^{T}(t,t_{0})|y_{S}(t)>=|y_{S}(t_{0})>..

Note:

*Q*is not a function of time unless it contains time explicitly._{S}

If*H*does not explicitly depend on time then .If

(b) In the Schroedinger picture we write for an arbitrary operator*H = H*then and^{T}

. Therefore*U*is unitary.

=.

*A*

.

(Here we have used the Schroedinger equation, )In the Heisenberg picture this becomes

for all ketsWe therefore have .

If

*A*does not explicitly depend on time then describes the evolution of the operators in the Heisenberg picture.- (i) Unless
*A*explicitly depends on time_{S}(t) - (ii) If the operator
*A*commutes with the Hamiltonian*H*and , then*A*does not depend on time Such an operator is called a_{H}**constant of motion**. The matrix elements of*A, <f|A|y>*, then do not change with time, i.e. the results of measurements of*A*do not depend on time.

Consider the Hamiltonian .

In the Heisenberg picture we write We have

.

Here we use .

The Heisenberg picture leads to equations similar to the classical equations of motion. The Heisenberg picture is often used to explore general properties of quantum systems and the formal analogy between classical and quantum theory.

The Schroedinger picture is most often used when doing calculations. The Schroedinger equation, an equation between vectors, is, in general, easier to solve than the Heisenberg equation, an equation between operators.

(c) , ..

- (i) Unless