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American Journal of Physics

On the use of direct variational methods to study confined quantum systems

J. L. Marin, Salvador A. Cruz

Published Oct 1, 1991• 64 citations
Publisher

Abstract

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.

DOI
10.1119/1.16674

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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.
This chapter evaluates some of the simplified Schrödinger equations obtained in the previous chapter, and presents explicit solutions for the energy levels and the corresponding wavefunctions. It begins by looking at the free particle and the particle in a box. The particle in a box is the simplest example of a confined particle, and enables us to understand the origin of many features of such systems, such as the appearance of discrete energy levels. It is a general feature of particles confined by a potential to some region of space that solutions to the Schrödinger equation can only be found at certain discrete energies. The chapter then considers the harmonic oscillator; angular momentum; rotations of a polyatomic molecule; and the hydrogen atom.
Abstract It is a common practice to approach quantum mechanics through the study of a few, exactly soluble examples using the Schrödinger equation in position-space. The number of potential energy functions for which such closed-form solutions are available is, however, quite small. Luckily, many of them actually correspond reasonably well to actual physical systems; examples include the infinite well as a model of a free particle ina “box “ (Chapters 5 and 7), the harmonic oscillator (Chapter 9), the rigid rotator (Chapter 16), and the Coulomb potential for the hydrogen atom (Chapter
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.
The authors propose a new method of solving the Schrodinger equation for a quantum system enclosed in a box with infinite potential walls. The method combines the variational technique with boundary perturbation theory and can be applied to a general non-separable case. When the Schrodinger equation for an enclosed system separates into an ordinary differential equation, the method gives the exact energy and wavefunction. As an application of the method the authors obtain the ground-state energy of the hydrogen atom placed off-centre in a spherical cavity. A generalisation of the variational boundary perturbation technique for finite potential walls is suggested.
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.