Rayleigh-Ritz method: Difference between revisions
imported>Paul Wormer (New page: In quantum mechanics, the '''Rayleigh-Ritz method''', also known as the '''linear variation method''' is a method to obtain (approximate) solutions of the time-independent [[Schröding...) |
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==History== | ==History== | ||
In the older quantum mechanics literature the method is known as the Ritz method, called after the | In the older quantum mechanics literature the method is known as the Ritz method, called after the mathematical physicist [[Walter Ritz]],<ref>W. Ritz, Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik, | ||
[On a new method for the solution of certain variational problems of mathematical physics], Journal für reine und angewandte Mathematik vol. '''135''' pp. 1–61 (1908)</ref> who first devised it. | [On a new method for the solution of certain variational problems of mathematical physics], Journal für reine und angewandte Mathematik vol. '''135''' pp. 1–61 (1908)</ref> who first devised it. | ||
In prewar quantum mechanics it was customary to follow | In prewar quantum mechanics texts it was customary to follow the highly influential book by Courant and Hilbert,<ref>R. Courant and D. Hilbert, ''Methoden der mathematischen Physik, (two volumes)'', Springer Verlag, Berlin (1968)</ref> who were contemporaries of Ritz and speak of the Ritz procedure (''Ritzsches Verfahren''). It is parenthetically amusing to note that the majority of these old quantum mechanics texts quote the wrong year, 1909 instead of 1908, an error first made in the Courant-Hilbert treatise. | ||
In the numerical analysis literature one usually prefixes the name of [[Lord Rayleigh]] to the method, and lately it has become common in quantum mechanics, too, to use the two names. Leissa<ref>A.W. Leissa, ''The historical bases of the Rayleigh and Ritz methods'', Journal of Sound and Vibration '''287''', pp. 961–978 (2005).</ref> recently became intrigued by the name | In the numerical analysis literature one usually prefixes the name of [[Lord Rayleigh]] to the method, and lately it has become common in quantum mechanics, too, to use the two names. Leissa<ref>A.W. Leissa, ''The historical bases of the Rayleigh and Ritz methods'', Journal of Sound and Vibration '''287''', pp. 961–978 (2005).</ref> recently became intrigued by the name and after reading the original sources discovered that the methods of the two workers differ considerably, although Rayleigh himself believed<ref>Lord Rayleigh, On the calculation of Chladni’s figures for a square plate, Philosophical Magazine Sixth Series '''22''' 225–229 (1911)</ref> that the methods were very similar and that his own method predated the one of Ritz by several decades. However, according to Leissa's convincing conclusion, Rayleigh was mistaken and the method now known as Rayleigh-Ritz method is solely due to Ritz. Leissa states: ''Therefore, the present writer concludes that Rayleigh’s name should not be attached to the Ritz method; that is, the Rayleigh–Ritz method is an improper designation.'' | ||
==References== | ==References== | ||
Revision as of 10:34, 28 January 2008
In quantum mechanics, the Rayleigh-Ritz method, also known as the linear variation method is a method to obtain (approximate) solutions of the time-independent Schrödinger equation. In numerical analysis, it is a method of solving differential equations with boundary conditions. In the latter field it is sometimes called the Rayleigh-Ritz-Galerkin procedure.
History
In the older quantum mechanics literature the method is known as the Ritz method, called after the mathematical physicist Walter Ritz,[1] who first devised it. In prewar quantum mechanics texts it was customary to follow the highly influential book by Courant and Hilbert,[2] who were contemporaries of Ritz and speak of the Ritz procedure (Ritzsches Verfahren). It is parenthetically amusing to note that the majority of these old quantum mechanics texts quote the wrong year, 1909 instead of 1908, an error first made in the Courant-Hilbert treatise.
In the numerical analysis literature one usually prefixes the name of Lord Rayleigh to the method, and lately it has become common in quantum mechanics, too, to use the two names. Leissa[3] recently became intrigued by the name and after reading the original sources discovered that the methods of the two workers differ considerably, although Rayleigh himself believed[4] that the methods were very similar and that his own method predated the one of Ritz by several decades. However, according to Leissa's convincing conclusion, Rayleigh was mistaken and the method now known as Rayleigh-Ritz method is solely due to Ritz. Leissa states: Therefore, the present writer concludes that Rayleigh’s name should not be attached to the Ritz method; that is, the Rayleigh–Ritz method is an improper designation.
References
- ↑ W. Ritz, Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik, [On a new method for the solution of certain variational problems of mathematical physics], Journal für reine und angewandte Mathematik vol. 135 pp. 1–61 (1908)
- ↑ R. Courant and D. Hilbert, Methoden der mathematischen Physik, (two volumes), Springer Verlag, Berlin (1968)
- ↑ A.W. Leissa, The historical bases of the Rayleigh and Ritz methods, Journal of Sound and Vibration 287, pp. 961–978 (2005).
- ↑ Lord Rayleigh, On the calculation of Chladni’s figures for a square plate, Philosophical Magazine Sixth Series 22 225–229 (1911)
(To be continued)