12.4: Perturbation Expansion
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Apr 1, 2025
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Let us recall the analysis of Section 1.2. The ψn are the stationary orthonormal eigenstates of the time-independent unperturbed Hamiltonian, H0. Thus, H0ψn=Enψn, where the En are the unperturbed energy levels, and ⟨n|m⟩=δnm. Now, in the presence of a small time-dependent perturbation to the Hamiltonian, H1(t), the wavefunction of the system takes the form ψ(t)=∑ncn(t)exp(−iωnt)ψn,
where ωn=En/ℏ. The amplitudes cn(t) satisfy
iℏdcndt=∑mHnmexp(iωnmt)cm,
where Hnm(t)=⟨n|H1(t)|m⟩ and ωnm=(En−Em)/ℏ. Finally, the probability of finding the system in the nth eigenstate at time t is simply Pn(t)=|cn(t)|2
(assuming that, initially, ∑n|cn|2=1).
Suppose that at t=0 the system is in some initial energy eigenstate labeled i. Equation ([e13.42]) is, thus, subject to the initial condition cn(0)=δni.
Let us attempt a perturbative solution of Equation ([e13.42]) using the ratio of H1 to H0 (or Hnm to ℏωnm, to be more exact) as our expansion parameter. Now, according to Equation ([e13.42]), the cn are constant in time in the absence of the perturbation. Hence, the zeroth-order solution is simply c(0)n(t)=δni.
The first-order solution is obtained, via iteration, by substituting the zeroth-order solution into the right-hand side of Equation ([e13.42]). Thus, we obtain iℏdc(1)ndt=∑mHnmexp(iωnmt)c(0)m=Hniexp(iωnit),
subject to the boundary condition c(1)n(0)=0. The solution to the previous equation is c(1)n=−iℏ∫t0Hni(t′)exp(iωnit′)dt′.
It follows that, up to first-order in our perturbation expansion,
cn(t)=δni−iℏ∫t0Hni(t′)exp(iωnit′)dt′.
Hence, the probability of finding the system in some final energy eigenstate labeled f at time t, given that it is definitely in a different initial energy eigenstate labeled i at time t=0, is Pi→f(t)=|cf(t)|2=|−iℏ∫t0Hfi(t′)exp(iωfit′)dt′|2.
Note, finally, that our perturbative solution is clearly only valid provided Pi→f(t)≪1.
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