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Physical Review

Effects of Finite Boundaries on a One-Dimensional Harmonic Oscillator

Richard Vawter

Published Oct 15, 1968• 55 citations
Publisher

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 linear harmonic oscillator located in a potential enclosure. The WKB method is employed to solve this problem for an oscillator whose center is fixed at some arbitrary position inside a potential box. Both numerical and analytical approximations of the WKB method are employed to find the energy eigenvalues over a wide range of the parameters of the problem. Numerical methods are also used on the exact series solutions of a bound oscillator to find the exact eigenvalues for the first few quantum states. A comparison of the two methods show that, in general, the WKB eigenvalues are accurate to much better than 1% except when the classical turning points are near the wall. Here the difference is of the order of 10% for the worst possible cases. The eigenvalues of a bound oscillator are shown to reduce to unbound-oscillator energy eigenvalues if the classical turning points are inside the potential enclosure and not near the walls. At the other extreme the eigenvalues are shown to become plane wave box eigenvalues when the separation of the classical turning points is large compared to the size of the enclosure. Also included is the relationship of the WKB solution to the Bohr-Sommerfeld quantization rule for the bound oscillator.

DOI
10.1103/physrev.174.749

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In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.
WKB analysis of P\mathcal PPT\mathcal TT-symmetric Sturm–Liouville problems
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