Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T16:32:55.692Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalue problems treated by finite-difference methods. II. Two-dimensional Schrödinger equations

Published online by Cambridge University Press:  24 October 2008

H. C. Bolton  and
H. I. Scoins
Get access Rights & Permissions [Opens in a new window]

Abstract

A discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown that

where h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 1 , January 1957 , pp. 150 - 161
Copyright
Copyright © Cambridge Philosophical Society 1957

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Allen, D. N. de G., Fox, L., Motz, H. and Southwell, R. V.Phil. Trans. A, 239 (1946), 488.Google Scholar
(2)Bartlett, J. H.Phys. Rev. 88 (1952), 525.CrossRefGoogle Scholar
(3)Bickley, W. G.Quart. J. Mech. 1 (1948), 35.CrossRefGoogle Scholar
(4)Bolton, H. C. and Scoens, H. I.Proc. Camb. Phil. Soc. 52 (1956), 215.CrossRefGoogle Scholar
(5)Collatz, L.Z. angew. Math. Mech. 19 (1939), 224 and 297CrossRefGoogle Scholar
(6)Collatz, L.Eigenwert-Aufgaben mit technischen Anwendungen (Leipzig, 1949).Google Scholar
(7)Coulson, C. A.Valence (Oxford, 1952), p. 56.Google Scholar
(8)Culver, R.Brit. J. Appl. Phys. 3 (1952), 376.CrossRefGoogle Scholar
(9)Flanders, G. and Shortley, G.J. Appl. Phys. 21 (1950), 1326.CrossRefGoogle Scholar
(10)Fox, L.Proc. Roy. Soc. A, 190 (1947), 31.Google Scholar
(11)Frost, A. A.J. Chem. Phys. 10 (1942), 240.CrossRefGoogle Scholar
(12)Glazer, H. and Reiss, H.J. Chem. Phys. 21 (1953), 903.CrossRefGoogle Scholar
(13)Groot, S. R. de and Ten, Seldam C. A.Physica, 's Grav. 18 (1952), 891.Google Scholar
(14)Hartree, D. R.Numerical analysis (Oxford, 1952), p. 85.Google Scholar
(15)Hawgood, J. Thesis (Oxford, 1955).Google Scholar
(16)Hestenes, M. R. and Karush, W.J. Res. Nat. Bur. Stand. 47 (1951), 471.CrossRefGoogle Scholar
(17)Kato, T.J. Phys. Soc., Japan, 4 (1949), 334.CrossRefGoogle Scholar
(18)Kimball, G. and Shortley, G.Phys. Rev. (2), 45 (1934), 815.CrossRefGoogle Scholar
(19)Luke, P. J., Meyerott, R. B. and Clendenin, W. W.Phys. Rev. 85 (1952), 401.CrossRefGoogle Scholar
(20)McWeeny, R.Proc. Roy. Soc. A, 235 (1956), 496.Google Scholar
(21)Mohrenstein, A. V.Z. Phys. 134 (1953), 488.CrossRefGoogle Scholar
(22)Salvadori, M. G.Proc. Amer. Soc. Civ. Engrs, 75 (1949), 1441.Google Scholar
(23)Schulte, A. M.Appl. Sci. Res., Hague, A, 2 (1949), 93.CrossRefGoogle Scholar
(24)Taylor, G. R. and Parr, R. G.Proc. Nat. Acad. Sci., Wash., 38 (1952), 154.CrossRefGoogle Scholar
(25)Temple, G.Proc. Lond. Math. Soc. (2), 29 (1929), 257.CrossRefGoogle Scholar
(26)Temple, G.Proc. Roy. Soc. A, 211 (1952), 204.Google Scholar
(27)Thomas, L. H.Phys. Rev. 51 (1937), 202.CrossRefGoogle Scholar
(28)Weinstein, D. H.Proc. Nat. Acad. Sci., Wash., 20 (1934), 529.CrossRefGoogle Scholar
(29)Woods, L. C.Quart. J. Mech. 3 (1950), 349.CrossRefGoogle Scholar