Mathematical Methods for Physics and Engineering - Matematica.NET

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATESHelmholtz’s equation in spherical polars is thusF(r, θ, φ) =r −1/2 [AJ l+1/2 (kr)+BY l+1/2 (kr)](C cos mφ + D sin mφ)×[EPl m (cos θ)+FQ m l (cos θ)]. (21.57)For solutions that are finite at the origin we require B = 0, and for solutionsthat are finite on the polar axis we require F = 0. It is worth mentioning thatthe solutions proportional to r −1/2 J l+1/2 (kr) andr −1/2 Y l+1/2 (kr), when suitablynormalised, are called spherical Bessel functions of the first and second kind,respectively, and are denoted by j l (kr) andn l (µ) (see section 18.6).As mentioned at the beginning of this subsection, the separated solution ofthe wave equation in spherical polars is the product of a time-dependent part(21.54) and a spatial part (21.57). It will be noticed that, although this solutioncorresponds to a solution of definite frequency ω = kc, the zeros of the radialfunction j l (kr) are not equally spaced in r, except for the case l = 0 involvingj 0 (kr), and so there is no precise wavelength associated with the solution.To conclude this subsection, let us mention briefly the Schrödinger equationfor the electron in a hydrogen atom, the nucleus of which is taken at the originand is assumed massive compared with the electron. Under these circumstancesthe Schrödinger equation is− 22m ∇2 u −e2 u4πɛ 0 r = i∂u ∂t .For a ‘stationary-state’ solution, for which the energy is a constant E and the timedependentfactor T in u is given by T (t) =A exp(−iEt/), the above equation issimilar to, but not quite the same as, the Helmholtz equation. § However, as withthe wave equation, the angular parts of the solution are identical to those forLaplace’s equation and are expressed in terms of spherical harmonics.The important point to note is that for any equation involving ∇ 2 , provided θand φ do not appear in the equation other than as part of ∇ 2 , a separated-variablesolution in spherical polars will always lead to spherical harmonic solutions. Thisis the case for the Schrödinger equation describing an atomic electron in a centralpotential V (r).21.3.3 Solution by expansionIt is sometimes possible to use the uniqueness theorem discussed in the previouschapter, together with the results of the last few subsections, in which Laplace’sequation (and other equations) were considered in polar coordinates, to obtainsolutions of such equations appropriate to particular physical situations.§ For the solution by series of the r-equation in this case the reader may consult, for example, L.Schiff, Quantum Mechanics (New York: McGraw-Hill, 1955), p. 82.741