Some applications of programmable calculators

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Abstract
Programmable electronic calculators provide a speedy means of performing scientific calculations, e.g. the evaluation of a given polynomial for many different x values and the performance of iterative procedures for the solution of polynomial equations. Both of these uses appear in variational calculations of the energy of simple quantum mechanical systems such as the perturbed harmonic oscillator. For this system the traditional perturbation method has some drawbacks, end so it is useful to find its energy levels directly by purely numerical methods. The electronic calculator can do this if the relevant Schrödinger differential equation is transformed into a difference equation. The diffusion equation is a partial differential equation, but can be converted to an ordinary differential equation and thence to a pair of difference equations which can be solved on the calculator. This applies even if the diffusion coefficient depends on the concentration, so that the associated ordinary differential equation is non-linear.