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,
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.
The Heisenberg picture can be obtained from the Schroedinger picture by a unitary transformation at any time t.
The matrix elements of any operator A are independent of the representation.
The predictions of quantum mechanics are independent of the representation.
(a) |yH>=U(t0,t)|yS(t)>, |yH>=UT(t,t0)|yS(t)>=|yS(t0)>.
Note: QS is not a function of time
unless it contains time explicitly.
If H does not explicitly depend on time then .
If H = HT then and
. Therefore U is unitary.
In the Heisenberg picture this becomes
for all kets
We therefore have .
If A does not explicitly depend on time then describes the evolution of the operators in the Heisenberg picture.
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) , .