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Journal of the Chemical Society Faraday Transactions 2 Molecular and Chemical Physics

Quantum-mechanical systems restricted by impenetrable barriers studied by a perturbation method

Igor Gonda, B. F. Gray

Published Jan 1, 1975• 18 citations
Publisher

Abstract

A systematic perturbation method to calculate the explicit dependence of energy on the position of the repulsive barrier (or barriers) in one-dimensional non-degenerate systems is presented. The method is also capable of dealing with anisotropic shifts of the boundaries. Two simple applications illustrate the general principles (particle in a box and compression of the ground state of the hydrogen atom).

DOI
10.1039/f29757102016

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A systematic perturbation method to calculate the explicit dependence of energy on the position of the repulsive barrier (or barriers) in one-dimensional non-degenerate systems is presented. The method is also capable of dealing with anisotropic shifts of the boundaries. Two simple applications illustrate the general principles (particle in a box and compression of the ground state of the hydrogen atom).
An atom in a box with penetrable or impenetrable potential walls is shown analytically to move under the influence of the repulsive forces from the boundaries of the cavity when all the atom-medium interactions are described by the Dirichlet boundary conditions or effective one-particle `box' potentials. Simple geometrical methods allow one to find the energetically optimal position of the atom in the cavity. For instance, the ground state energy of the hydrogen atom is proved to be minimal when the nucleus is situated at the centre of the cube or sphere. Some applications for molecules are also mentioned.
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.
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.
We generalize a recently developed perturbation approach [F. M. Fern\'andez and E. A. Castro, Phys. Rev. A 46, 7288 (1992)], extending its applicability to a wider variety of separable quantum-mechanical problems. As an illustrative example, we consider a particle in a one-dimensional box with a finite potential wall perturbed by a linear potential and obtain the perturbation corrections to the wave function and energy of an arbitrary state in terms of the corresponding unperturbed quantities.