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

H. Jones
J. L. Jain
Journal / Anthology

International Journal of Quantum Chemistry
Year: 2000
Volume: 80
Issue: 4-5
Page range: 842-847

It would seem that limiting computer computations to numbers with a fixed number of decimal digits would inhibit flexibility. Software programs such as Mathematica permit numerical algebra to be done exactly in terms of ratios of integers. Hence, a single Taylor series representation of a function can span the entire range needed for a corresponding independent variable. We decided to find eigenvalues to 14 decimal digits by solving one-dimensional Schrödinger equations by the "shooting method" by employing a single Talylor series in each case. With more terms in the series, higher accuracy may be obtained by evaluations at larger asympotic values. The problems solved were the 1s, 2s, and 2p hydrogen atom; the harmonic oscillator; the quartic potential; and the double-well potential. Noteworthy is the use of the asymptotic condition for the derivative of the eigenfunction as well as its value; this permits the determination of a lower and upper bound for the eigenvalues. The eigenfunctions determined are contiunous rather than evaluated only over a grid, thus permitting easy and accurate evaluations of marix elements by Gaussian quadrature. Also, theoretically accurate normalization constants are found for the eigenfunctions.

*Applied Mathematics > Numerical Methods
*Mathematics > Calculus and Analysis > Differential Equations
*Science > Physics > Quantum Physics

eigenvalues, shooting method, computer algebra, integer arithmetic, one-dimensional Schrödinger equation