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# The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

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The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.

The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).

Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.

For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).

Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).

The method used for integrating the equations numerically is outlined (§ 9).

References
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* See, for example, Born, M., Vortesungen über Atommechanik (or the English translation, The Mechanics of the Atom), Ch. III.

For a general review, see Fowler, R. H., Nature, Vol. CXIX, p. 90 (1927); for a more detailed treatment, Hund, F., Linienspektren, Ch. III.

Schrödinger, E., Ann. der Phys., Vol. LXXIX, pp. 361, 489; Vol. LXXX, p. 437; Vol. LXXXI, p. 109 (1926);Phys. Rev., Vol. XXVIII, p. 1049 (1926).

§ Klein, F., Zeit. f. Phys., Vol. XLI, p. 432 (1927).

* Waller, I., Nature, Vol. CXX, p. 155 (1927);Phil. Mag. 11. 1927 (Supplement).

See Unsöld, A., Ann. der Phys., Vol. LXXXII, p. 355, §§ 5 and 6. Unsöld proves this by considering the energy of an indefinitely small charge in the field of a closed group; it also follows directly for the charge distribution from Unsöld's formulae (67), (69).

* For example, on the orbital mechanics, Rb, Cu, Ag, Au, have 33 X-ray orbits, and for the neutral atoms of these elements the first d term corresponds to a nonpenetrating 33 orbit.

See, for example, Hund, F., op. cit., passim.

* According to Schrödinger's interpretation of ψ (§ 1) the charge lying in an element of volume defined by dr dθ dø is , the integral being over all space, so that the charge lying between radii r and r + dr is

the integration of the spherical harmonic factor cancelling out.

In general the internal and external orbits with the same energy will not both be quantum orbits, but when they occur it is usually possible (always if integral quantum numbers are used) to obtain an internal and an external quantum orbit with the same quantum numbers.

* An outline of the method used for the practical numerical integration of the equation for P is given in § 9.

* Ann. der Phys., Vol. LXXX, p. 443, equation (7′).

Whittaker, E. T. and Watson, G. N., Modern Analysis, Ch. XVI.

Eddington, A. S., Nature, Vol. CXX, p. 117 (1927).

§ Sugiura, Y., Phil. Mag., Ser. 7, Vol. IV, p. 498 (1927).

Whittaker, and Watson, , op. cit., § 16–3.

Epstein, P.. Proc. Nat. Acad. Sci., Vol. XII, p. 629 (1926).

Waller, I., Zeit. f. Phys., Vol. XXXVIII, p. 635. Using atomic units and the notation of this paper, let P be defined as 2C/n times Waller's, τXn, l, i.e.

then it follows from Waller's formulae (32′), (33), (34) that

Now the highest power of x in , is so that, for large τ, P as defined behaves like . Hence if, instead, the arbitrary constant in P is to be chosen so that as (see 7·1), ∫ P2 dp must have the value given here.

See, for example, Born, M., op. cit., p. 234 (English translation, p. 204).

See, for example, Hund, F., op. cit., p. 74, formula (1).

* Heisenberg, W. and Jordan, , Zeit. f. Phys., Vol. XXXVII, p. 263 (1926).

See, for example, Whittaker, Ė. T. and Robinson, G., Calculus of Observations, p. 35.

For the first formula see Whittaker, and Robinson, , op. cit., p. 147 (put r = 1 and express the result in central differences); the second follows directly from the Euler-Maclaurin formula (Whittaker, and Eobinson, , op. cit., p. 135) on putting r = 1 and expressing the differential coefficients in central differences. I am indebted to Mr C. H. Bosanquet for pointing out the advantage of (9·2), involving differences of the derivative of the integrand, with its small fourth order term.

* It is convenient to speak of the process of the numerical integration of a differential equation as ‘stable’ if a small change in the solution at one point (for example, a numerical slip) does not produce greater changes in later values as the integration proceeds, and as ‘unstable’ when the opposite is the case.

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Mathematical Proceedings of the Cambridge Philosophical Society
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