# The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator

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)=0.5 mw 2 x 2 (where m is the electron mass, w is a constant angular frequency). In this case, the Schrödinger equation takes the following form, 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, 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 w. Transcribed Image Text: 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)=mox² (where m is the electron mass, o is a constant angular frequency). In this
case, the Schrödinger equation takes the following form,
h? d’w (x) , 1
mox’y (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
W;(x) = exp|
2h
2max?
тоx
w.(x) =
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 @.

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