QuantumPhysics.dvi

13.3 Time-dependent perturbation theory

A systematic way of organizing time-dependent perturbation theory is through the use the evolution
operator in the interaction picture. After all, we have

|ψa;t〉I=UI(t,t 0 )|ψa;t 0 〉I (13.29)

where

i ̄h

d

dt

UI(t,t 0 ) = VI(t)UI(t,t 0 )

UI(t 0 ,t 0 ) = I (13.30)

Notice that, sinceVIis time dependent, we need to keep the initial and final times inUI, and not
just their differencet−t 0 , as we had done in the time independent case.

The differential equation, together with the initial value condition att 0 may be reformulated
in terms of a single integral equation,

UI(t,t 0 ) =I−

i

̄h

∫t

t 0

dt′VI(t′)UI(t′,t 0 ) (13.31)

Clearly, the initial value problem is satisfied by this equation, and differentiation intimmediately
reproduces the differential equation. Its solution may be obtained order by order in perturbation
theory by successive substitution and iteration. A first iteration gives

UI(t,t 0 ) =I−

i

̄h

∫t

t 0

dt′VI(t′)−

1

̄h^2

∫t

t 0

dt′

∫t′

t 0

dt′′VI(t′)VI(t′′)UI(t′′,t 0 ) (13.32)

Iterating again will yield the solution to second order inVI,

UI(t,t 0 ) =I−

i

̄h

∫t

t 0

dt′VI(t′)−

1

̄h^2

∫t

t 0

dt′

∫t′

t 0

dt′′VI(t′)VI(t′′) +··· (13.33)

This is referred to as theDyson series. Alternatively, we may derive an analogous formula directly
on the coefficientscn, and we get

cn(t) =cn(0)−

i

̄h

∫t

t 0

dt′

∑

m

〈n|VI(t′)|m〉cm(t′) (13.34)

and then this formula may be iterated in turn.

The difficulty of this integral solution is that the potentialVI(t) is an operator, and in general,
we will have

[VI(t′),VI(t′′)] 6 = 0 (13.35)

Generally, it is difficult to go beyond the first order in any expliciit way. For short time intervals
t−t 0 , the expansion should yield reliable results given by the contributions of the leading orders.
For long time evolution, however, predictions are much harder to come by. Below, we show that
it is possible to obtain new results already just from the first order approximation to this formula.
Below, we shall treat a special case to illuminate the procedure.