Let |ΨS(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
S(t)> = U(t,t0)|ΨS(t0)> = U(t0,t)ΨS(t0)> .
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,
H> = |ΨS(t0)> = U(t0,t)|ΨS(t)>, |ΨH> = U(t,t0)|ΨS(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
AH = U(t0,t)AsU(t0,t) = U(t,t0)AsU(t,t0).
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.
H> = U(t0,t)|ΨS(t)>, |ΨH> = U(t,t0)|ΨS(t)> = |ΨS(t0)>.
S(t)|ASS(t)> = <ΨS(t)|U(t,t0)U(t,t0)ASU(t,t0)U(t,t0)|ΦS(t)>
= <ΨH(t)|U(t,t0)ASU(t,t0)|ΦH(t)> = <ΨH(t)|AHH(t)>.
The matrix elements of any operator A are independent of the representation.
The predictions of quantum mechanics are independent of the representation.

Problem:

Consider the Schroedinger representation, (often called the Schroedinger picture), defined by the Schroedinger equation
iħ∂|Ψ(q,t)>/∂t = H(p,q)|Ψ(q,t)>.
(a)  Write the transformation equations for the wave function Ψ 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 H(p,q) = p2/(2m) + U(q).   Show that the Heisenberg equation of motion for the operator is  ∂H/∂p = p/m = dq/dt,  which has the same form as the corresponding classical equation of motion.

Solution:
(a) |ΨH> = U(t,t0)|ΨS(t)> = |ΨS(t0)>.
QH = U(t0,t)QsU(t0,t) = U(t,t0)QsU(t,t0).
Note: QS is not a function of time unless it contains time explicitly.
If H does not explicitly depend on time then
U(t,t0) = exp(-iH(t - t0)/ħ), U(t0,t) = exp(-iH(t0 - t)/ħ)
If H  =  H then U(t,t0) = exp(+iH(t - t0)/ħ) = U(t0,t),
and U(t,t0)U(t,t0) = U(t0,t)U(t0,t) = I.  Therefore U is unitary.
S(t)|QSS(t)> = <ΨS(t)|U(t,t0)U(t,t0)QSU(t,t0)U(t,t0)|ΦS(t)>
= <ΨH(t)|U(t,t0)QSU(t,t0)|ΦH(t)> = <ΨH(t)|QHH(t)>.

(b) In the Schroedinger picture we write for an arbitrary operator A
(d/dt)<ΨS(t)|ASS(t)>
= [(∂/∂t)<ΨS(t)|]ASS(t)> + <ΨS(t)|∂AS/∂tS(t)> + <ΨS(t)|AS[(∂/∂t)|ΦS(t)>]
= <ΨS(t)|∂AS/∂t|ΦS(t)> + (iħ)-1S(t)|ASHs - HsAsS(t)>
= <ΨS(t)|∂AS/∂t|ΦS(t)> + (iħ)-1S(t)|[AS,Hs]|ΦS(t)>.
Here we have used the Schroedinger equation,
(iħ∂/∂t)|ΨS> = H|ΨS>,  (-iħ∂/∂t)<ΨS| = H<ΨS|.

Matrix elements are independent of the representation.
In the Heisenberg picture this therefore becomes
(d/dt)<ΨH(t)|AHH(t)> = <ΨH(t)|∂AH/∂t|ΦH(t)>
= <ΨS(t)|∂AS/∂t|ΦS(t)> + (iħ)-1S(t)|[AS,Hs]|ΦS(t)>
= <ΨH|U(t,t0)(∂AS/∂t)U(t,t0)|ΦH> + (iħ)-1S(t)|ASHs - HsAsS(t)>
= <ΨH|(∂AS/∂t)HH> + (iħ)-1H(t)|UASUUHsU - UHsUUAsU|ΦH(t)>
= <ΨH|(∂AS/∂t)HH> + (iħ)-1H|AHHH - HHAHH>,
for all kets |ΨH|> and |ΦH>.
We therefore have dAH/dt = (∂AS/∂t)H +  (iħ)-1[AH,HH].

If A does not explicitly depend on time then dAH/dt = (iħ)-1[AH,HH]  describes the evolution of the operators in the Heisenberg picture.

• (i) Unless AS(t) explicitly depends on time ∂AS/∂t = 0.
• (ii) If the operator A commutes with the Hamiltonian H and ∂AS/∂t = 0, then AH does not depend on time dAH/dt = 0.  Such an operator is called a constant of motion.  The matrix elements of A, <Φ|A|Ψ>, then do not change with time, i.e. the results of measurements of A do not depend on time.

Consider the Hamiltonian  H(P,Q) = P2/(2m) + U(Q).
In the Heisenberg picture we write  HH(PH,QH) = PH2/(2m) + U(QH).   We have
dQH/dt =  (iħ)-1[QH,HH] = (iħ)-1[QH,PH2/(2m)] = (iħ2m)-1[QH,PH2] = PH/m.
dPH/dt =  (iħ)-1[PH,HH] = (iħ)-1[PH,U(QH)] = -∂U(QH)/∂QH.
Here we use [P,F(Q)] =  [P,∑nfnQn] = ∑nfn[P,Qn] = -∑nfnniħQn-1] = -iħ∂f(Q)/∂Q.
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)  HH(PH,QH) = PH2/(2m) + U(QH), ∂H/∂PH = PH/m = dQH/dt,
dQH/dt = (iħ)-1[QH,HH] = (iħ)-1[QH,PH2/(2m)] = (iħ2m)-1[QH,PH2] = PH/m.