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 (2)==mar. -mox² (where m is the electron mass, @ is a constant angular frequency). In this case, the Schrödinger equation takes the following form, h? d'w (x)¸ 1 +-mox'y (x)= Ew (x). 2 2m dx 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, тох- W;(x) = exp| 2h 2max? w.(x)=| тоx? -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 (2)==mar. -mox² (where m is the electron mass, @ is a constant angular frequency). In this case, the Schrödinger equation takes the following form, h? d'w (x)¸ 1 +-mox'y (x)= Ew (x). 2 2m dx 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, тох- W;(x) = exp| 2h 2max? w.(x)=| тоx? -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 @.