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 (x)=–mo²x² (where m is the electron mass, ø is a constant angular frequency). In thiscase, the Schrödinger equation takes the following form,ħ? d°w (x) ¸ 1-moxy (x)= Ey (x).2m dx?2The 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,mox?y,(x)= exp|2h2mox?v.(x)=|mox-1 exp|2hDetermine 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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