Eigenvalues to arbitrary precision for one-dimensional Schrodinger equations by the shooting method using integer arithmetic

It would seem that limiting computer computations to numbers with a fixed n umber of decimal digits would inhibit flexibility. Software programs such a s Mathematica permit numerical algebra to be done exactly in terms of the r atio of integers. Hence, a single Taylor series representation of a functio n can span the entire range needed for a corresponding independent variable . We decided to find eigenvalues to 14 decimal digits by solving one-dimens ional Schrodinger equations by the "shooting method" by employing a single Taylor series in each case. With more terms in the series, higher accuracy may be obtained by evaluations at larger asymptotic values. The problems so lved were the Is, 2s, and 2p hydrogen atom, the harmonic oscillator, the qu artic potential, and the double-well potential. Noteworthy is the use of th e asymptotic condition for the derivative of the eigenfunction as well as i ts value; this permits the determination of a lower and upper bound for the eigenvalues. The eigenfunctions determined are continuous rather than eval uated only over a grid, thus permitting easy and accurate evaluations of ma trix elements by Gaussian quadrature. Also, theoretically accurate normaliz ation constants are found for the eigenfunctions. (C) 2000 John Wiley & Son s, Inc.