(a) Expanding the state vector|ψ(t)>
Let H(t) = H0 + W(t).
Let {|Φn>}
be an orthonormal eigenbasis of H0, H0|Φn>
= En|Φn>.
Let ωmn
= (Em - En)/ħ
and Wmn(t) = <Φm|W(t)|Φm>.
Assume that at t = 0 the system is in the state |Φi>.
We can expand |ψ(t)> in terms of the eigenfunctions od H0,
|ψ(t)> = ∑ncn(t)exp(-iEnt/ħ)|Φn>.
The Schroedinger equation yields
iħ∂/∂t(∑ncn(t)exp(-iEnt/ħ)|Φn>)
= (H0 + W(t))∑ncn(t)exp(-iEnt/ħ)|Φn>
or
iħ(∑n(dcn(t)/dt)exp(-iEnt/ħ)|Φn>)
= W(t)∑ncn(t)exp(-iEnt/ħ)|Φn>.
Taking the inner product with <Φm|exp(iEmt/ħ)
yields
dcm(t)/dt = (-i/ħ)∑n<Φm|W(t)|Φn>cn(t)exp(iωmnt).
Expressing this equation in integral form we have
cm(t) = cm(0) + (-i/ħ)∑n∫0t<Φm|W(t')|Φn>cn(t')exp(iωmnt')dt'.
At t = 0 we have cn(0) = δni.
Assume that at t > 0 but very small we have ci ≈ 1 and cn≠i
very small. Then
cm≠i(t) ≈ (-i/ħ)∫0t<Φm|W(t')|Φi>exp(iωmit')dt'.
The probability of finding the system in the state |Φf>
(f ≠ i) at time t is
Pif(t)
= |cf≠i(t)|2= (1/ħ2)|∫0texp(iωfit')Wfi(t')dt'|2.
(b) Changing the representation to the interaction picture
In the Schroedinger picture let
H(t) = H0 + W(t)
and let U0(t,t0) = exp(iH0t/ħ) be the
evolution operator of a system with Hamiltonian H0.
Let {|Φn>}
be an orthonormal eigenbasis of H0, H0|Φn>
= En|Φn>.
Let ωmn
= (Em - En)/ħ
and Wmn(t) = <Φm|W(t)|Φm>.
Assume that at t = 0 the system is in the state |Φi>.
Define the state vector |ΨI(t)>
in the interaction picture from the state vector in the Schroedinger picture through
|ΨI(t)>
= U0†(t,t0)|ΨS(t)> =
exp(iH0t/ħ)|ΨS(t)>.
The state vector in the interaction picture evolves according to
iħ∂|ΨI(t)>/∂t = WI(t))|ΨI(t)>,
with WI(t) = exp(iH0t/ħ) WS(t) exp(-iH0t/ħ).
We can rewrite this differential equation in the form of an integral equation.
d|ΨI(t)> = (iħ)-1WI(t))|ΨI(t)>dt.
∫t0t d|ΨI(t)> = (iħ)-1∫t0t
WI(t')|ΨI(t')>dt'.
|ΨI(t)> = |ΨI(t0)> + (iħ)-1∫t0t
WI(t')|ΨI(t')>dt'
This integral equation can be solved by iteration.
The ket |ΨI(t)> can be
expanded in a power series of the form
|ΨI(t)> = {I + (iħ)-1∫t0t
dt'WI(t') + (iħ)-2∫t0t
dt'WI(t') ∫t0t' dt''WI(t'')
+ ... }|Ψ0(t)>.
Assume W(t) is a small correction to H0, and W(t) = 0 for t < 0.
Then |ΨI(0)> = |ΨS(0)>.
Neglecting higher order terms we have
|ΨI(t)> = |ΨI(t0)> + (iħ)-1∫t0t
WI(t')|ΨI(0>dt'.
The probability
Pif(t) of finding the system in the eigenstate |Φf>
(f ≠ i) of H0 at time t is |<Φf|ΨI(t)>|2.
(The predictions of quantum mechanics are independent of the representation.)
<Φf|ΨI(t)> = <Φf|Φi> + (iħ)-1∫0t
<Φf|WI(t')|Φi>dt'
= (iħ)-1∫0t
exp(i(Ef - Ef)t/ħ) <Φf|WS(t')|Φi>dt'.
The probability of finding the system in the state |Φf>
(f ≠ i) at time t is
Pif(t) = (1/ħ2)|∫0texp(iωfit')Wfi(t')dt'|2.