The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V (x)=–mo²x² (where m is the electron mass, ø is a constant angular frequency). In this case, the Schrödinger equation takes the following form, ħ? d°w (x) ¸ 1 -moxy (x)= Ey (x). 2m dx? 2 The electron is initially trapped at the ground level. After absorbing a photon, it transits to an excited level. The wave functions of the ground and excited levels take the following forms, respectively, mox? y,(x)= exp| 2h 2mox? v.(x)=| mox -1 exp| 2h Determine the energy of the electron at the ground and excited levels, respectively, and therefore express the wavelength of the incident photon in terms of @.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V (x)=–mo²x² (where m is the electron mass, ø is a constant angular frequency). In this case, the Schrödinger equation takes the following form, ħ? d°w (x) ¸ 1 -moxy (x)= Ey (x). 2m dx? 2 The electron is initially trapped at the ground level. After absorbing a photon, it transits to an excited level. The wave functions of the ground and excited levels take the following forms, respectively, mox? y,(x)= exp| 2h 2mox? v.(x)=| mox -1 exp| 2h Determine the energy of the electron at the ground and excited levels, respectively, and therefore express the wavelength of the incident photon in terms of @.