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Rate of convergence of calculations with one-dimensional Dirichlet wave functions

✍ Scribed by Marco A. Núñez

Publisher: John Wiley and Sons
Year: 1997
Tongue: English
Weight: 236 KB
Volume: 62
Category: Article
ISSN: 0020-7608
DOI: 10.1002/(sici)1097-461x(1997)62:5<449::aid-qua1>3.0.co;2-w

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✦ Synopsis

The convergence of numerical methods to compute the bound states of w . the one-dimensional Schrodinger equation H s E in 0, ϱ by means of numerical ¨w x solutions of the Dirichlet eigenproblem H s E in a box 0, R , is studied. It Rn R R R R Ä 4 ϱ is seen that approximating sequences that converge correctly to in the L norm n ns1 2 may have an intrinsic divergent behavior characterized geometrically by an increasing separation between the asymptotic tails of and as n ª ϱ. It is shown that n numerical Dirichlet wave functions obtained from standard methods cannot exhibit Rn this divergent behavior as R, n ª ϱ, and only rounding errors may affect their Ž . convergence when R is greater than certain distance R N, M that depends on the D method M in question, the precision machine N, and the state . An energy criterion to D Ž . find R N, M is suggested, and an estimation of the convergence rate of expectation D values from the exact Dirichlet function as R ª ϱ is given.

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