1549380323-Statistical Mechanics Theory and Molecular Simulation

Time-dependent perturbation theory 541

UˆI(t;t 0 ) =T

[

exp

(

−

i

̄h

∫t

t 0

dt′HˆI(t′)

)]

, (14.2.31)

which is known as atime-ordered exponential. Since eqn. (14.2.31) is just a shorthand
for eqn. (14.2.30), it is understood that the time-ordering operator orders the operators
in each term of an expansion of the exponential.
Given the formalism of time-dependent perturbation theory, we now seek to answer
the following question: If the system is initially in an eigenstate ofHˆ 0 with energyEi,
what is the probability as a function of timetthat the system will undergo a transition
to a new eigenstate ofHˆ 0 with energyEf? To answer this question, we first set the
initial state vector|Ψ(t 0 )〉equal to the eigenvector|Ei〉ofHˆ 0. Then, the amplitude
as a function of time that the system will undergo a transition to theeigenstate|Ef〉

is obtained by propagating this initial state to timetwith the propagatorUˆ(t;t 0 ) and
then taking the overlap of the resulting state with the eigenstate|Ef〉:

Afi(t) =〈Ef|Uˆ(t;t 0 )|Ei〉. (14.2.32)

The probability is just the square modulus of this complex amplitude:

Pfi(t) =

∣

∣

∣〈Ef|Uˆ(t;t 0 )|Ei〉

∣

∣

∣

2

. (14.2.33)

Consider first the amplitude at zeroth order in perturbation theory. At this order,
Uˆ(t;t 0 ) =Uˆ 0 (t;t 0 ), and the amplitude is simply

A(0)fi(t) =〈Ef|e−i

Hˆ 0 (t−t 0 )/ ̄h

|Ei〉

= e−iEi(t−t^0 )/ ̄h〈Ef|Ei〉, (14.2.34)

which clearly vanishes by orthogonality ifEi 6 =Ef. Thus, at zeroth order, the only
possibility is a trivial one in which no transition occurs.
The lowest nontrivial (Ei 6 =Ef) result occurs at first order, where the transition
amplitude is given by

Afi(1)(t) =〈Ef|Uˆ(1)(t;t 0 )|Ei〉

=−

i

̄h

∫t

t 0

dt′〈Ef|Uˆ 0 (t;t′)Hˆ 1 (t′)Uˆ 0 (t′;t 0 )|Ei〉

=−

i

̄h

∫t

t 0

dt′〈Ef|e−i

Hˆ 0 (t−t′)/ ̄hˆ

H 1 (t′)e−i

Hˆ 0 (t′−t 0 )/ ̄h

|Ei〉

=−

i

̄h

∫t

t 0

dt′e−iEf(t−t

′)/ ̄h

e−iEi(t

′−t 0 )/ ̄h

〈Ef|Hˆ 1 (t′)|Ei〉