Legendre functions associated

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

The functions Ztmn , = cos 0) are a generalization of associated Legendre functions and the coefficients of the series V(mn are given by... 

Substitution into the Rodriques formula gives the associated Legendre functions in the form... 

To normalize a function such as P m(cos0) it is necessary to equate the integral / [P,m( cos 9)]2dx = 1. Starting from the Rodriques formula and integrating by parts it can be shown that the normalized associated Legendre functions are ... 

The variation of Pn(ft) with ft for a few values of n is sliown in Fig. 5. Since, in most physical problems, the Legendre polynomial involved is usually / tl(cos 0) we have shown in Fig. fi the variation of this function with 0. Numerical values may be obtained from Tables of Associated Legendre Functions (Columbia University Press, 194-5),... 

Use of Associated Legendre Functions in Wave Mechanics. To illustrate the use of associated Legendre functions in wave mechanics, we shall consider one of the simplest problems in that subject — that of solving Sehrodinger s equation... 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

The normalised associated Legendre functions constitute our fixed basis representation (FBR) [92]. The transformation matrix from the DVR to the FBR representation is given by... 

The theta integration involves the associated Legendre functions. Using4... 

In these coordinates Laplace s equation separates such functions of the form P ( )P. (7l)(g )m(J satisfy it where are the associated Legendre functions. Thus for this interior problem (i.e. inside the spheroid, not outside it) general solution of the linear problem is of the form... 

P 1 denotes the associated Legendre function of degree / and order m. Nt m is a normalization constant. 

Figure 3.8 Associated Legendre functions P , P 1, P2 1 plotted as polar diagrams.

The sum over wave numbers is written as an integral over wave number spaee, with a density of states in wave number space of 2/(2rc)- = Naii/ 6n ). The exponent k d is simply kd cos 0, and the spherical harmonics can be written out in terms of the associated Legendre functions, PtJ (cos (Schiff, 1968, p. 80),... 

The substitution of equation 18.33 in 18.18 gives the equation for 0, The functions (6) are known as associated Legendre functions. We shall not derive the general solution of the equation but shall give the solutions which may be shown to be correct by substitution in equation 18.18 (for full details of the method see Pauling and Wilson page 125 et seq). 

Consulting our friendly neighborhood mathematician (or Supplement 6B), we learn that the single-valued, finite solutions to Eq (6.28) are known as associated Legendre functions. The parameters A. and m are restricted to the values... 

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