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Mathematical Proceedings of the Cambridge Philosophical Society

Notes on the two-centre problem in wave mechanics. I. The hyperbolic nodes of the wave equation

P. E. Marrack

Published Jan 1, 1939• 3 citations
Publisher

Abstract

Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.

DOI
10.1017/s0305004100020727

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Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.
Neglecting the spin, two electrons are described in quantum mechanics by means of a wave equation in six variables. It is shown that well-known relations between angular momentum operators make it possible to determine the dependence of the wave function on three variables. The problem is thus reduced from six to three dimensions. For a state with an assigned "orbital" angular momentum lll, say an SSS, PPP, DDD state the dependence of the wave function on three Euler angles is determined by the value of lll. The wave function is a linear combination of products of distance and angle functions, the latter depending only on the three Euler angles. The angle functions are well-known solutions of the wave equation for a symmetrical top. The distance functions satisfy wave equations in three variables r1{r}_{1}r1​, r2{r}_{2}r2​, r12{r}_{12}r12​ or r1{r}_{1}r1​, r2{r}_{2}r2​, $\ensuremath{\theta}$. The case of PPP terms is worked out in detail. Equations (10), (25) apply to two electrons having the same asimuthal quantum number. Equations (18), (20), (24) describe all the other cases, for instance SSS and PPP electrons combining to give 1P^{1}P1P and 3P^{3}P3P. Triplets are described by (18) and singlets by (20).
In the model of a compressed atom (or ion) considered in the present paper the boundary condition associated with the corresponding uncompressed atom, i.e., the condition that the radial wave function must vanish at r=∞, is replaced by the boundary condition that the radial wave function must have a node at the finite distance r=a. The treatment of the problem of obtaining the energy shift due to the compression is based on the phase-integral method developed by Fröman and Fröman, an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage. The treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by Furry [Phys. Rev. 71, 360 (1947)] and Yngve [J. Math. Phys. 13, 324 (1972)], in which one also uses exact formulas in the calculations and makes all approximations in the final stage. Since compression of an atom gives rise to very subtle effects, rigorous methods are indispensible for obtaining accurate and reliable analytical final formulas. As an application, the resulting general formulas are particularized to the case of a hydrogenic atom, and a numerical illustration of the accuracy of the formulas is given.
from the Hamiltonian function üof classical mechanics by the operator substitution h d liri dqiwhere h is Planck's constant.For instance, for a hydrogen atom with fixed nucleus H is the sum of the kinetic energy of the one electron and the potential energy of the electron with respect to the nucleus :The Hamiltonian operator correspondingly involves the Laplacian operator in a term representing the kinetic energy of the electron and a scalar multiplier representing the potential energy:The wave equation is thenIn these equations £ is a number, the energy of the system, and *, the corresponding solution, is the wave function which describes the state of the system when it has that energy, x// is obviously a function of all the spatial coordinates needed to define a configuration of the system.In the case of a simple molecule like Cl 2 , for instance, \p would depend on 108 variables.It is this multiplicity of variables which makes it so hard for physicists to interest mathematicians in some problems they would like very much to have solved.Now not all the solutions of the wave equation are of interest, but only those which can be physically interpreted.This requirement of physical interpretability places certain auxiliary conditions on the functions \//, often merely that they be quadratically integrable or that they remain bounded.Between the physically interprétable wave functions and the stationary states in which the system may be found there is a one-to-one correspondence.Thus in many cases physically satisfactory solutions of the wave equation exist only for certain discrete values of E, and in such cases one finds that the physical system also can possess only these discrete energies.The physically satisfactory solutions \p of such an equation and the 194i]