Brillouin-Wigner Perturbation Theory - Quantum Mechanics II | PHY 6646, Study notes of Quantum Mechanics

Download Brillouin-Wigner Perturbation Theory - Quantum Mechanics II | PHY 6646 and more Study notes Quantum Mechanics in PDF only on Docsity! PHY 6646 K. Ingersent Brillouin-Wigner Perturbation Theory Brillouin-Wigner (BW) perturbation theory is less widely used than the Rayleigh-Schrödinger (RS) version. At first order in the perturbation, the two theories are equivalent. However, BW perturbation theory extends more easily to higher orders, and avoids the need for sep- arate treatment of non-degenerate and degenerate levels. • Assume that we know the stationary states of the unperturbed HamiltonianH0, namely the kets |n〉 satisfying H0|n〉 = εn|n〉. For each unperturbed state, we can define a projection operator Qn = I − |n〉〈n| = ∑ m6=n |m〉〈m|. Using the spectral representation H0 = ∑ n εn|n〉〈n|, one can see that [H0, Qn] = 0. • Let us write the perturbed eigenproblem in the form (En −H0)|ψn〉 = H1|ψn〉. (1) Acting with Qn from the left, and taking advantage of [H0, Qn] = 0, Qn|ψn〉 = RnH1|ψn〉, where Rn = (En −H0)−1Qn = Qn(En −H0)−1, which has a spectral representation Rn = ∑ m6=n |m〉〈m| En − εm . Note that Rn is not a projection operator because R 2 n 6= Rn. • Adopting the usual convention 〈n|ψn〉 = 1, we can write |ψn〉 = |n〉 +Qn|ψn〉 = |n〉 +RnH1|ψn〉. (2) Assuming that H1 is small, this equation can be solved iteratively in powers of H1. The wave function correct through order j, ψ(j)n 〉, can be obtained from Eq. (2) by inserting |ψ(j−1)n 〉 on the right-hand side: |ψ(0)n 〉 = |n〉, |ψ(1)n 〉 = |n〉 +RnH1|n〉, |ψ(2)n 〉 = |n〉 +RnH1|n〉 + (RnH1)2|n〉, etc 1