 First-order ground state of the quartic oscillator

ID: first-order-ground-state-of-the-quartic-oscillator

λ X^{4}

, write

X = ℏ/ (2 mω) (A + A^{†})

. Since

(A + A^{†})^{4} ∣0 ⟩ = 3∣0 ⟩ + 62 ∣2 ⟩ + 26 ∣4 ⟩,

∣ 0_{λ} ⟩ = ∣0 ⟩ - \frac{ℏ λ}{4 m ^{2} ω ^{3}} (32 ∣2 ⟩ + \frac{3}{2} ∣4 ⟩) + O (λ^{2}) .

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