PHY 6646
K. Ingersent
Second-Order Time-Dependent Perturbation Theory
Let us consider the extension of time-dependent perturbation theory to second order
in the interaction H1 (t). The starting point is the set of differential equations
ih̄
dan(j+1) (t) X
= hn|H1 (t)|mieiωnm t a(j)
m (t),
dt
m
j = 0, 1, 2, . . .
(1)
If we assume that the system starts at time t = t0 in an unperturbed stationary state
|ii, then for any t ≥ t0 ,
a(0)
n (t) = δn,i ,
iZt 0
0
a(1)
(t)
=
−
dt hn|H1 (t0 )|iieiωni t ,
n
h̄ t0
i XZ t 0
0
(2)
0
an (t) = −
dt hn|H1 (t0 )|mieiωnm t a(1)
m (t ).
h̄ m t0
(2)
(3)
(4)
The properties of a(2)
n (t) can best be understood by considering several different
time dependences of H1 (t).
Sudden perturbation. Suppose that a perturbation turns on suddenly at time t =
t0 = 0, and is constant thereafter:
H1 (t) = H̃θ(t),
(5)
where H̃ contains no time dependence. In this case, Eqs. (2)–(4) can be used to study
the transient effects of the abrupt change in the Hamiltonian. One finds
1 − eiωni t
,
h̄ωni
i X H̃nm H̃mi Z t 0 iωnm t0
0
(2)
an (t) = −
dt e
− eiωni t
h̄ m h̄ωmi
0
a(1)
n (t) = H̃ni
= −
!
1 − eiωni t 1 − eiωnm t
−
,
ωni
ωnm
X H̃nm H̃mi
m
(6)
h̄2 ωmi
(7)
where H̃nm = hn|H̃|mi.
The most notable aspect of Eq. (4) is that a(2)
n (t) can be nonzero even if H̃ni = 0
(1)
and, hence, an (t) = 0. In effect, the system can get from |ii to |f i through a pair
of “virtual” (energy non-conserving) transitions, the first from |ii to an intermediate
state |mi, the second from |mi to |f i. Even more complicated transitions, involving
multiple intermediate states, are possible at higher orders in H1 .
One can repeat the above for the sudden turn-on of a harmonic perturbation. Although a(2)
n (t) contains many more terms, virtual transitions again feature.
1
Adiabatic perturbation. Now suppose instead that a perturbation turns on very
slowly, starting at t = t0 = −∞, according to
H1 (t) = H̃eηt ,
(8)
where H̃ is again time-independent, and the turn-on rate η is a small, positive real
number. In this case,
a(1)
n (t)
Z t
i
ei(ωni −iη)t
0
= − H̃ni
dt0 ei(ωni −iη)t = −H̃ni
,
h̄
h̄(ωni − iη)
−∞
(9)
and
a(2)
n (t) =
=
i X H̃nm H̃mi Z t
0
dt0 ei(ωni −2iη)t
h̄ m h̄(ωmi − iη) −∞
X
m
H̃nm H̃mi
ei(ωni −2iη)t .
h̄ (ωni − i2η)(ωmi − iη)
(10)
2
This implies that
−iεi t/h̄
|ψ(t)i = e
X
n
!
X
H̃ni eηt
H̃nm H̃mi e2ηt
δn,i −
|ni + . . . (11)
+
2
h̄(ωni − iη)
m h̄ (ωni − i2η)(ωmi − iη)
Here and below, the terms “. . .” are of third order or higher in H1 .
Within time-independent perturbation theory, the effect of H̃1 ≡ H1 (t = 0) is to
convert the stationary state |ni into
X


X H̃mk H̃kn
H̃
H̃ H̃
− mn − mn nn +
 |mi + . . .
|ψn i = |ni +
2 2
2
h̄ω
h̄
ω
h̄
ω
ω
mn
mn
kn
mn
m6=n
k6=n
(12)
Thus, for any n 6= i,
hψn |ψ(0)i = −
+
∗
X
H̃in
H̃ni
H̃nm H̃mi
−
+
2
h̄(ωni − iη)
h̄ωin
m h̄ (ωni − i2η)(ωmi − iη)
∗
∗
∗
∗
X H̃im
H̃mn
H̃mi
H̃nn
H̃mi
H̃in
+
+ ...
−
2
2
2
h̄2 ωin
m6=n h̄ ωmn (ωmi − iη)
m6=n h̄ ωin ωmn
X
(13)
With a little bit of algebra, one can show that in the adiabatic limit, described by an
infinitesimal turn-on rate η → 0+ , the first- and second-order terms on the right-handside of Eq. (13) all cancel, implying that (up to possible third-order corrections)
|hψi |ψ(t)i|2 = 1.
(14)
Equation (14) turns out to be an exact result, which leads to . . .
The adiabatic theorem: Up to an overall phase, any eigenstate |n(H0 )i of an initial
Hamiltonian H0 evolves smoothly under an adiabatic perturbation into the corresponding eigenstate |n(H)i of the Hamiltonian H(t) = H0 + H1 (t).
2
Constant perturbation and level decay. The limit η → 0+ of the slow onset
describes a perturbation that is constant in time. This type of perturbation might
describe the effect of some background interaction which has been left out of the
Hamiltonian H0 . (An example is the effect of gravity on the hydrogen atom.)
Let us examine the effect of such a background interaction on the initial state |ii.
Specializing Eqs. (2), (9), and (10) to the case n = i (keeping η finite for now),
Hence
and
i
eηt
i X |H̃mi |2 e2ηt
ai (t) = 1 − H̃ii
+ 2
+ ...
h
η
h̄ m ωmi − iη 2η
(15)
dai (t)
i X |H̃mi |2 2ηt
i
= − H̃ii eηt + 2
e + ...
dt
h
h̄ m ωmi − iη
(16)
d ln ai (t)
1 dai (t)
i
i X |H̃mi |2 2ηt
=
= − H̃ii eηt + 2
e + ...
dt
ai (t) dt
h
h̄ m6=i ωmi − iη
(17)
Now let us take the limit of a constant perturbation. Recalling that
1
lim+
=P
η→0 ω − iη
1
+ iπδ(ω),
ω
(18)
where P is the Cauchy principal part, we find
d ln ai (t)
i
= − Σi ,
dt
h
(19)
where the (time-independent) self-energy, or complex energy shift, is
Σi = H̃ii − P
X |H̃mi |2
m6=i εm − εi
− iπ
X
|H̃mi |2 δ(εm − εi ).
(20)
m6=i
Equation (19) implies that
ai (t) = ai (0)e−iΣi t/h̄ ,
or
ci (t) = hi|ψ(t)i = ci (0) e−i(εi +Re Σi )t/h̄ eIm Σi t/h̄ .
(21)
This in turn means that the occupation probability decays in time according to
|ci (t)|2 = |ci (0)|2 e−t/τi ,
(22)
with a decay rate (inverse lifetime)
2
τi−1 = − Im Σi ≥ 0.
h̄
3
(23)