Postulates of quantum mechanics 379[
−̄h^2
2 m∇^2 +U(r)]
ψk(r) =Ekψk(r), (9.2.55)where the labelk= (kx,ky,kz) indicates that three quantum numbers are needed to
characterize the states.
9.2.6 The Heisenberg picture
An important fact about quantum mechanics is that it supports multiple equivalent
formulations, which allows us to choose the formulation that is most convenient for the
problem at hand. The picture of quantum mechanics we have been describing postu-
lates that the state vector|Ψ(t)〉evolves in time according to the Schr ̈odinger equation
and the operators corresponding to physical observables are static. This formulation
is known as theSchr ̈odinger pictureof quantum mechanics. In fact, there exists a
perfectly equivalent alternative formulation in which the state vector is taken to be
static and the operators evolve in time. This formulation is known as theHeisenberg
picture.
In the Heisenberg picture, an operatorAˆcorresponding to an observable evolves
in time according to theHeisenberg equation of motion:
dAˆ
dt=
1
i ̄h[A,ˆHˆ]. (9.2.56)
Note the mathematical similarity to the evolution of a classical phasespace function:
dA
dt={A,H}. (9.2.57)
This similarity suggests that the commutator [A,ˆHˆ]/i ̄hbecomes the Poisson bracket
{A,H}in the classical limit. Like the Schr ̈odinger equation, the Heisenbergequation
can be solved formally to yield
Aˆ(t) = eiHˆt/ ̄hAˆ(0)e−iHˆt/ ̄h=Uˆ†(t)Aˆ(0)Uˆ(t). (9.2.58)The initial valueAˆ(0) that appears in eqn. (9.2.58) is the operatorAˆin the Schr ̈odinger
picture. Thus, given a state vector|Ψ〉, the expectation value of the operatorAˆ(t) in
the Heisenberg picture is simply
〈Aˆ(t)〉=〈Ψ|Aˆ(t)|Ψ〉. (9.2.59)The Heisenberg picture makes clear that any operatorAˆthat commutes with the
Hamiltonian satisfies dA/ˆdt= 0 and, hence, does not evolve in time. Such an operator
is referred to as aconstant of the motion. In the Schr ̈odinger picture, if an operator
is a constant of the motion, the probabilities associated with the eigenvalues of the
operator do not evolve in time. To see this, consider the evolution ofthe state vector
in the Schr ̈odinger picture:
|Ψ(t)〉= e−i
Hˆt/ ̄h
|Ψ(0)〉. (9.2.60)