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The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra

Published online by Cambridge University Press:  24 October 2008

D. R. Hartree
D. R. Hartree
Affiliation:
Christ's College
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Extract

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the form

and require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 3 , July 1928 , pp. 426 - 437
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* Proc. Camb. Phil. Soc. vol. 24, pp. 89, 111 (1928). These papers will be referred to as I and II. The same notation will be used here as in those papers, with the exception that the effective quantum number will be written n instead of n*.CrossRefGoogle Scholar

† Each solution contains an arbitrary multiplying constant, hence the form of the condition for the solutions to fit.Google Scholar

‡ This solution cannot be treated as a variation on the solution for ɛ=0, as for any ɛ, however small, ɛ is the leading term in P″/P for r sufficiently large, and the solution for such values of r for any ɛ ≠ 0 is essentially different from that for ɛ = 0.Google Scholar

* Eddington, A. S., Nature, vol. 120, p. 117 (1927).CrossRefGoogle Scholar

† Sugiura, Y., Phil. Mag. Ser. 7, vol. 4, p. 498 (1927).CrossRefGoogle Scholar

‡ See Whittaker, E. T. and Watson, G. N., Modern Analysis, ch. 16.Google Scholar

§ Op. cit. § 16·12.Google Scholar

‖ Op. cit. § 16·1.Google Scholar

Op. cit. § 16·41.Google Scholar

* See, for example, Piaggio, H. T. H., Differential Equations, § 98.Google Scholar

* Cf. Whittaker, and Watson, , op. cit. ch. 12. Writing γ for Euler's Constant, we have the special valuesψ(1)=−γ, (n integral).Google Scholar

† Cf. Whittaker, and Watson, , op. cit. § 12–33.Google Scholar

* See Jahnke, E. and Emde, F., Funktionentafeln, p. 166. [Put β = 2, γ = ½, p = 2l + 1 in the second equation of § 13.]Google Scholar

† Watson, G. N., Treatise on Bessel Functions, § 3·52 (3), and § 3·54 (2).Google Scholar

* Op. cit. § 16·41.Google Scholar

* Dashes denote differentiation with respect to r.Google Scholar

* See Whittaker, and Watson, , op. cit. § 16·3.Google Scholar

* The square of the matrix amplitude for the transition is [cf. Sugiura, loc. cit. equation (4)] multiplied by an absolute constant and by a factor depending on lA and lB, arising from the dependence of the wave functions on direction of the radius r. The factor v 4 arises from the relation between doublet amplitude and intensity of emitted radiation.Google Scholar