536 Quantum time-dependent statistical mechanics
14.2.2 Iterative solution for the interaction-picture state vector
The solution to eqn. (14.2.7) can be expressed in terms of a unitary operatorUˆI(t;t 0 ),
which is the interaction-picture propagator. The initial state|Φ(t 0 )〉evolves in time
according to
|Φ(t)〉=UˆI(t;t 0 )|Φ(t 0 )〉=UˆI(t;t 0 )|Ψ(t 0 )〉. (14.2.8)
Substitution of eqn. (14.2.8) into eqn. (14.2.7) yields an evolution equation for the
propagatorUˆI(t;t 0 ):
HˆI(t)UˆI(t;t 0 ) =i ̄h∂
∂tUˆI(t;t 0 ). (14.2.9)Eqn. (14.2.9) has the initial conditionUˆI(t 0 ;t 0 ) = Iˆ. In developing a solution to
eqn. (14.2.9), we assume thatHˆI(t) is a small perturbation so that the solution can
be constructed in terms of a power series inHˆI(t). Such a solution is generated by
rewriting eqn. (14.2.9) as an integral equation:
UˆI(t;t 0 ) =UˆI(t 0 ;t 0 )−i
̄h∫tt 0dt′HˆI(t′)UˆI(t′;t 0 )=Iˆ−
i
̄h∫tt 0dt′HˆI(t′)UˆI(t′;t 0 ). (14.2.10)We can easily verify that eqn. (14.2.10) is the solution forUˆI(t;t 0 ). Taking the time
derivative of both sides of eqn. (14.2.10) gives
i ̄h∂
∂tUˆI(t;t 0 ) =−i ̄hi
̄h∂
∂t∫tt 0dt′HˆI(t′)UˆI(t′;t 0 )=HˆI(t)UˆI(t;t 0 ). (14.2.11)Eqn. (14.2.10) allows a perturbation series solution to be developed systematically. We
start with a zeroth-order solution by settingHˆI(t) = 0 in eqn. (14.2.10), which gives
the trivial result
Uˆ(0)
I (t;t^0 ) =
I.ˆ (14.2.12)
This solution is now fed back into the right side of eqn. (14.2.10) to develop a first-order
solution:
UˆI(1)(t;t 0 ) =Iˆ−i
̄h∫tt 0dt′HˆI(t′)UˆI(0)(t′;t 0 )=Iˆ−
i
̄h∫tt 0dt′HˆI(t′). (14.2.13)This first-order solution is now substituted back into the right side of eqn. (14.2.10)
to generate a second-order solution
Uˆ(2)
I (t;t^0 ) =
Iˆ−i
̄h∫tt 0dt′HˆI(t′)UˆI(1)(t′;t 0 )