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Journal of Physics A Mathematical and General

Converging bounds for the eigenvalues of multiminima potentials in two-dimensional space

Hasan Taşeli, R. Eid

Published Nov 7, 1996• 16 citations
Publisher

Abstract

The eigenvalue problem in of Schrödinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls. It is shown that the Dirichlet and the von Neumann problems in generate upper and lower bounds, respectively, to the eigenvalues of the unbounded system. To illustrate the method, rapidly converging two-sided bounds for the energy levels of two- and three-well oscillators are presented, using simple trigonometric basis functions.

DOI
10.1088/0305-4470/29/21/027

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The eigenvalue problem in of Schrödinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls. It is shown that the Dirichlet and the von Neumann problems in generate upper and lower bounds, respectively, to the eigenvalues of the unbounded system. To illustrate the method, rapidly converging two-sided bounds for the energy levels of two- and three-well oscillators are presented, using simple trigonometric basis functions.
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.
Recent investigations by Bender and Boettcher and by Mezincescu have argued that the discrete spectrum of the non-Hermitian potential V ( x ) = -i x 3 should be real. We give further evidence for this through a novel formulation which transforms the general one-dimensional Schrodinger equation (with complex potential) into a fourth-order linear differential equation for |Ψ( x )| 2 . This permits the application of the eigenvalue moment method, developed by Handy, Bessis and coworkers, yielding rapidly converging lower and upper bounds to the low-lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.
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.