Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T01:14:41.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalues of differential equations by finite-difference methods

Published online by Cambridge University Press:  24 October 2008

H. C. Bolton ,
H. I. Scoins  and
G. S. Rushbrooke
H. C. Bolton
Affiliation:
Department of PhysicsKing's CollegeNewcastle upon Tyne
H. I. Scoins
Affiliation:
Department of PhysicsKing's CollegeNewcastle upon Tyne
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown that

where h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 2 , April 1956 , pp. 215 - 229
Copyright
Copyright © Cambridge Philosophical Society 1956

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)Aitken, A. C.Determinants and matrices (Edinburgh, 1939).Google Scholar
(2)Auluck, F. C. and Kothari, D. S.Proc. Camb. phil. Soc. 41 (1945), 175.CrossRefGoogle Scholar
(3)Bickley, W. G.Math. Gaz. 25 (1941), 19.CrossRefGoogle Scholar
(4)Bückner, H.Math. Z. 51 (1948), 423.CrossRefGoogle Scholar
(5)Chandrasekhar, S.Astrophys. J. 97 (1943), 268.Google Scholar
(6)Collatz, L.Eigenwert-aufgaben mit technischen Anwendungen (Leipzig, 1949).Google Scholar
(7)Courant, R., Friedrichs, K. and Lewy, H.Math. Ann. 100 (1928), 42.CrossRefGoogle Scholar
(8)Flanders, D. A. and Shortley, G.J. appl. Phys. 21 (1950), 1326.CrossRefGoogle Scholar
(9)Fox, L.Proc. roy. Soc. A, 190 (1947), 31.Google Scholar
(10)Fox, L.Proc. Camb. phil. Soc. 45 (1949), 50.CrossRefGoogle Scholar
(11)Hartree, D. R.Numerical analysis (Oxford, 1952).Google Scholar
(12)Kimball, G. and Shortley, G.Phys. Rev. (2), 45 (1934), 815.CrossRefGoogle Scholar
(13)Milne, W. E.J. Res. nat. Bur. Stand. 45 (1950), 245.CrossRefGoogle Scholar
(14)Milne, W. E.Numerical solution of differential equations (New York, 1953).Google Scholar
(15)Richardson, L. F.Phil. Trans. A, 210 (1910), 307.Google Scholar
(16)Richardson, L. F.Phil. Trans. A, 226 (1927), 299.Google Scholar
(17)Rutherford, D. E.Proc. roy. Soc. Edinb. 62 (1947), 229.Google Scholar
(18)Schiff, L. I.Quantum mechanics (New York, 1949).Google Scholar
(19)Southwell, R. V., Allen, D. N. de G., Fox, L. and bMotz, H.Phil.Trans. A, 239 (1942), 488.Google Scholar
(20)Weinstein, D. H.Proc. nat. Acad. Sci., Wash., 20 (1934), 529.CrossRefGoogle Scholar