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

Mathematical Proceedings of the Cambridge Philosophical Society - Tập 24 Số 1 - Trang 89-110 - 1928
D. R. Hartree1
1St. John's College

Tóm tắt

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).

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Tài liệu tham khảo

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.

Whittaker, op. cit, 135

10.1007/BF01397100

Waller, Zeit. f. Phys, XXXVIII, 635

Whittaker, op. cit, 16

Sugiura, 1927, Phil. Mag, IV, 498

10.1038/120117a0

Whittaker, Modern Analysis

Ann. der Phys, LXXX, 443

Unsöld, Ann. der Phys, LXXXII, 355

Hund, op. cit

1927, Phil. Mag

Klein, 1927, Zeit. f. Phys, XLI, 432

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

1926, Phys. Rev, XXVIII, 1049

10.1002/andp.19263840404

Hund, Linienspektren

10.1038/119090a0

Hund, op. cit, 74

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

Whittaker, op. cit, 147

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.

10.1073/pnas.12.11.629

Born, op. cit, 234

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.

10.1038/120155a0

Whittaker, Calculus of Observations, 35

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Mathematical Proceedings of the Cambridge Philosophical Society

Tập 24 Số 1

89-110

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