Textbook Bender-Wu problem: H_0 = (p̂² + ω²x̂²)/2 plus V = λ·x̂⁴. Evaluate ⟨0|x̂⁴|0⟩ using x̂ = (â+â†)/√(2mω): x̂⁴ = (â+â†)⁴ / (4m²ω²) = (terms). Expanding (â+â†)⁴ and taking ⟨0|·|0⟩ keeps only terms with equal numbers of â's and â†'s:…
Textbook Bender-Wu problem: H_0 = (p̂² + ω²x̂²)/2 plus V = λ·x̂⁴. Evaluate ⟨0|x̂⁴|0⟩ using x̂ = (â+â†)/√(2mω): x̂⁴ = (â+â†)⁴ / (4m²ω²) = (terms). Expanding (â+â†)⁴ and taking ⟨0|·|0⟩ keeps only terms with equal numbers of â's and â†'s:…