Energy eigenvalues of a bounded centrally located harmonic oscillator

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Abstract
In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one‐dimensional harmonic oscillator located in a potential enclosure. Series expansion techniques are applied to solve this problem for a harmonic oscillator located at the center of an infinitely high potential well. An analytical expression for the energy eigenvalues is found as a function of the size potential enclosure L, the quantum state n, oscillator frequency ω, and the mass of the particle m. The first order approximation of this expression is given by E = Eosccoth(Eosc/Ebox) where Eosc = ℏ ω(n + 1/2) is the energy eigenvalues of an unbound harmonic oscillator and Ebox = (2m/ℏ2) (n + 1)2π2/L2 is the energy eigenvalues of a free particle in an infinitely high well. For the ground state this approximation is better than 1% for all values of L, ω, and m.

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