RATE OF CONVERGENCE OF CALCULATIONS WITH ONE-DIMENSIONAL DIRICHLET WAVE FUNCTIONS

摘要

The convergence of numerical methods to compute the bound states of the one-dimensional Schrodinger equation H psi = E psi in [0, infinity) by means of numerical solutions psi(Rn) of the Dirichlet eigenproblem H-R psi(R) = E(R) psi(R) in a box [0,R], is studied. It is seen that approximating sequences {psi(n)}(n=1)(infinity) that converge correctly to psi in the L(2) norm may have an intrinsic divergent behavior characterized geometrically by an increasing separation between the asymptotic tails of psi(n) and psi as n --> infinity. It is shown that numerical Dirichlet wave functions psi(Rn) obtained from standard methods cannot exhibit this divergent behavior as R, n --> infinity, and only rounding errors may affect their convergence when R is greater than certain distance R(N, M(D)) that depends on the method M(D) in question, the precision machine N, and the state psi. An energy criterion to find R(N, M(D)) is suggested, and an estimation of the convergence rate of expectation values from the exact Dirichlet function psi(R) as R --> infinity is given. (C) 1997 John Wiley & Sons, Inc. [References: 54]