The finite-element method for energy eigenvalues of quantum mechanical systems

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Abstract
The finite-element method provides a convenient and flexible procedure for the calculation of energy eigenvalues of quantum mechanical systems. The levels of accuracy that can be attained in the method of finite elements are investigated using various approximations. This is illustrated by first considering two classic examples that form a convenient basis for describing the calculational technique: the radial equation for the hydrogen atom for spherically symmetric states and the simple harmonic oscillator problem in one dimension. These two illustrative examples provide guidelines in the calculation of the energy levels of the hydrogen atom in an arbitrary spatially uniform magnetic field, a problem not solvable by analytical means. The results obtained for the 1s0 and 2s0 levels are the most accurate reported so far. This application shows that finite-element analysis can be employed with advantage for obtaining very accurate results for the energy levels and wavefunctions for quantum mechanical systems.

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