The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonicoscillator. Because an arbitrary potential can usually be approximated as a harmonic potentialat the vicinity of a stable equilibrium point, it is one of the most important model systems inquantum mechanics. Consider an electron trapped by a one-dimensional harmonic potentialV (2)==mar.-mox² (where m is the electron mass, @ is a constant angular frequency). In thiscase, the Schrödinger equation takes the following form,h? d'w (x)¸ 1+-mox'y (x)= Ew (x).22m dxThe electron is initially trapped at the ground level. After absorbing a photon, it transits to anexcited level. The wave functions of the ground and excited levels take the following forms,respectively,тох-W;(x) = exp|2h2max?w.(x)=|тоx?-1 exp2hDetermine the energy of the electron at the ground and excited levels, respectively, andtherefore express the wavelength of the incident photon in terms of @.

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