Actually, the time-evolution equations in both pictures may be solved with the use of thetime
evolution operatorU(t). This operator is defined to beunitary, and to satisfy,^13
i ̄h
d
dt
U(t) =H(t)U(t) U(t)†U(t) =I (14.3)It is often convenient to give an initial condition onU such that at timeta, the operator is unity.
We shall denote this operator byU(t;ta). The solution to both equations in (14.2) may then be
presented as follows,
|φ(t)〉 = U(t;ta)|φ(ta)〉 U(ta;ta) =I
A(t) = U(t;ta)†A(ta)U(t;ta) (14.4)This result may be verified by explicit calculation; to do so in the Heisenberg picture, the following
intermediate step will be helpful,
i ̄hd
dt
U(t)†=−U(t)†H(t) (14.5)For a time-independent HamiltonianH, the evolution operator is given by the exponential,
U(t;ta) =U(t−ta) U(t) =e−itH/ ̄h (14.6)The evolution operatorU(t) now commutes withH at all timest, and manifestly satisfies the
following composition formula,
U(t 1 +t 2 ) =U(t 1 )U(t 2 ) (14.7)14.2 The evolution operator for quantum mechanical systems
Let us consider a quantum system which is associated with a classical mechanical one. As discussed
in the preceding section, such systems may be formulated in terms of the positionQand momentum
P operators, in terms of which the Hamiltonian is expressed,H(Q,P). In general, we may have
several such pairs,Qi,Piwithi= 1,···,N. For simplicity, we shall start with the caseN= 1; the
generalization to higherNwill be straightforward. Recall the bases defined by these operators,
Q|q〉=q|q〉 [Q,P] =i ̄h
P|p〉=p|p〉 (14.8)Orthonormality and completeness hold as follows,
〈q′|q〉=δ(q−q′) I=∫
dq|q〉〈q|〈p′|p〉= 2π ̄hδ(p−p′) I=∫
dp
2 π ̄h|p〉〈p| (14.9)(^13) Note the definite order in the product ofH(t) andU(t); generally these operators do not commute.