For H_0 = (p̂² + ω²x̂²)/2 with spectrum E_n^(0) = ω(n+1/2) (ℏ=m=1 units) and perturbation V = λ·x̂², the first-order RS energy shift is E_n^(1) = λ·⟨n|x̂²|n⟩. Using x̂ = (â+â†)/√(2ω), ⟨n|x̂²|n⟩ = ⟨n|(â†â + â↠+ â² + ↲)/(2ω)|n⟩ = (n +…
For H_0 = (p̂² + ω²x̂²)/2 with spectrum E_n^(0) = ω(n+1/2) (ℏ=m=1 units) and perturbation V = λ·x̂², the first-order RS energy shift is E_n^(1) = λ·⟨n|x̂²|n⟩. Using x̂ = (â+â†)/√(2ω), ⟨n|x̂²|n⟩ = ⟨n|(â†â + â↠+ â² + ↲)/(2ω)|n⟩ = (n +…