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Physics LibreTexts

12.4: Perturbation Expansion

( \newcommand{\kernel}{\mathrm{null}\,}\)

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/. The amplitudes cn(t) satisfy

idcndt=mHnmexp(iωnmt)cm,

where Hnm(t)=n|H1(t)|m and ωnm=(EnEm)/. 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 ωnm, 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 idc(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=it0Hni(t)exp(iωnit)dt.
It follows that, up to first-order in our perturbation expansion,

cn(t)=δniit0Hni(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 Pif(t)=|cf(t)|2=|it0Hfi(t)exp(iωfit)dt|2.
Note, finally, that our perturbative solution is clearly only valid provided Pif(t)1.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 12.4: Perturbation Expansion is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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