Degenerate Rayleigh-Schrödinger Perturbation Theory: Power Series of Stationary Solutions, Study notes of Quantum Mechanics

Download Degenerate Rayleigh-Schrödinger Perturbation Theory: Power Series of Stationary Solutions and more Study notes Quantum Mechanics in PDF only on Docsity! PHY 6646 K. Ingersent Degenerate Rayleigh-Schrödinger Perturbation Theory • Assume that we know the stationary states of the unperturbed HamiltonianH0, namely the kets |n, r〉 satisfying H0|n, r〉 = εn|n, r〉. The integer index r (1 ≤ r ≤ gn) is used to distinguish among the gn eigenstates of energy εn. (For simplicity, we assume that the vector space is has a finite or countably infinite dimension. The extension to continuous vector spaces is straightforward.) • We seek stationary solutions |ψn,r〉 of the perturbed problem (H0 + λH1)|ψn,r〉 = En,r|ψn,r〉 (1) in the form of power-series expansions |ψn,r〉 = ∞∑ j=0 λj|ψ(j)n,r〉, En,r = ∞∑ j=0 λjE(j)n,r. (2) Let us insert Eqs. (2) into Eq. (1), and collect terms having the same power of λ. • At order λ0 we have (H0−E(0)n,r)|ψ(0)n,r〉 = 0, which is satisfied by any linear combination of the unperturbed eigenkets of energy εn, i.e., |ψ(0)n,r〉 = gn∑ t=1 (cn,r)t|n, t〉, with E(0)n,r = εn. Orthonormality requires that 〈ψ(0n,r|ψ(0)n,s〉 = ∑gn t=1(cn,r) ∗ t (cn,s)t = δr,s. • At order λ1 we find (H0 − E(0)n,r)|ψ(1)n,r〉 = (E(1)n,r − H1)|ψ(0)n,r〉. Acting from the left with 〈m, s|, we obtain (εm − εn)〈m, s|ψ(1)n,r〉 = δm,nE(1)n,r(cn,r)s − gn∑ t=1 〈m, s|H1|n, t〉(cn,r)t. (3) • For m = n, Eq. (3) yields gn∑ t=1 〈n, s|H1|n, t〉(cn,r)t = E(1)n,r(cn,r)s, which is the eigenequation for H1 in the gn-dimensional subspace spanned by the unperturbed states of energy εn. It is perfectly consistent with the λ 0 result to choose the |ψ(0)n,r〉’s to be the eigenkets of this problem. We will assume henceforth that this is the case, so that 〈ψ(0)n,s|H0 + λH1|ψ(0)n,r〉 = (εn + λE(1)n,r)δs,r, (4) where the first-order correction to the unperturbed energy is E(1)n,r = 〈ψ(0)n,r|H1|ψ(0)n,r〉. (5) 1