QuantumPhysics.dvi

then we define the corresponding state|ψa;t〉Iin the interaction picture by factoring out the time
dependence induced byH 0 alone,

|ψa;t〉I≡eitH^0 / ̄h|ψa;t〉S (13.22)

Correspondingly, any observableAS in the Schr ̈odinger picture (which is by construction time
independent), maps to an observableAIin the interaction picture, by

AI(t) =eitH^0 / ̄hASe−itH^0 / ̄h (13.23)

Thus, observables, in general, also become time dependent. The interaction picture is therefore
intermediate between the Heisenberg and Schr ̈odinger pictures.

The time evolution of the states|ψa;t〉Iin the interaction picture may be deduced from their
definition and from the Schr ̈odinger equation, and we find,

i ̄h

d

dt

|ψa;t〉I=VI(t)|ψa;t〉I (13.24)

where the potential in the interaction picture is given by the relation between observables given
above,

VI(t) =eitH^0 / ̄hV(t)e−itH^0 / ̄h (13.25)

The time-evolution of the coefficients cn is now simply given by the matrix elements ofVI(t).
Indeed, we have

|ψa;t〉I=

∑

n

cn(t)|n〉 (13.26)

and

i ̄h

∑

n

c ̇n(t)|n〉=

∑

m

cm(t)VI(t)|m〉 (13.27)

Taking the matrix elements with〈n|,

i ̄hc ̇n(t) =

∑

m

cm(t)〈n|VI(t)|m〉

=

∑

m

cm(t)e−it(Em−En)/ ̄h〈n|V(t)|m〉 (13.28)

In general, it is not possible to solve the resulting system of differential equations exactly (except
in very special cases, such as magnetic resonnace treated earlier). Thus, we resort to carrying out
a calculation which is perturbative in powers ofVI.