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.

Table 2.2 Properties of the harmonic oscillator E = (v + ) hw, w = = (kg/m) /2 y(x) = NH(ax) e-x12 1/4 mk (v + 1|x|v) = Energies: Wavefunctions: Matrix elements: Virial theorem: = N (v + 1|px|v) = i = 2mw hmo 2 (Ek) = (Ep) for all v 2v!/2 1/2 1/2 (v +1)/2 (v 1|x|v) = 2mw 1/2 (v + 1)2 (v1|px|v) = i| 1/2 v1/2 mw 2 1/2 V112