We normally think of the one-dimensional harmonic oscillator as lying horizontally. Suppose that it is held vertically so that it experiences a perturbation mgx. Calculate the first-order correction to the energy of the harmonic oscillator in state |v〉; specialize to the ground-state oscillator.

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Table 2.2 Properties of the harmonic oscillator E₁ = (v + ¹) hw, w = = (kg/m) ¹/2 y₂(x) = N₂H₂(ax) e-α²x²12 1/4 mk ħ² (v + 1|x|v) = Energies: Wavefunctions: Matrix elements: Virial theorem: α= N₁ (v + 1|px|v) = i = 方 2mw hmo 2 (Ek) = (Ep) for all v α 2°v!π¹/2 1/2 1/2 (v +1)¹/2 (v− 1|x|v) = ħ 2mw 1/2 (v + 1)¹2 (v1|px|v) = −i| 1/2 v1/2 ħmw 2 1/2 V112 Table 2.2 Properties of the harmonic oscillator E₁ = (v + ¹) hw, w = = (kg/m) ¹/2 y₂(x) = N₂H₂(ax) e-α²x²12 1/4 mk ħ² (v + 1|x|v) = Energies: Wavefunctions: Matrix elements: Virial theorem: α= N₁ (v + 1|px|v) = i = 方 2mw hmo 2 (Ek) = (Ep) for all v α 2°v!π¹/2 1/2 1/2 (v +1)¹/2 (v− 1|x|v) = ħ 2mw 1/2 (v + 1)¹2 (v1|px|v) = −i| 1/2 v1/2 ħmw 2 1/2 V112

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