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Il Nuovo Cimento B

Harmonic oscillator in an impenetrable spherical well

Sami M. Al-Jaber

Published Apr 1, 2002• 0 citations
OpenAlex

Abstract

The boundary effects on a quantum system are discussed by examining a harmonic oscillator in an impenetrable spherical well. The boundary corrections to the ground-state energy and wave function are calculated via a linear approximation method which is linear in energy. A simple analytical expression for the asymptotic dependence of the ground-state energy on the well radius is derived.

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The boundary effects on a quantum system are discussed by examining a harmonic oscillator in an impenetrable spherical well. The boundary corrections to the ground-state energy and wave function are calculated via a linear approximation method which is linear in energy. A simple analytical expression for the asymptotic dependence of the ground-state energy on the well radius is derived.
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rcr_crc​ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
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.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.