Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The document Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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INTEGRALS OF PRODUCTS OF THREE SPHERICAL
 HARMONICS

Frequently in quantum mechanics we encounter integrals of the general form

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

in which all spherical harmonics depend on θ, ϕ . The first factor in the integrand may come from the wave function of a final state and the third factor from an initial state, whereas the middle factor may represent an operator that is being evaluated or whose “matrix element” is being determined.

By using group theoretical methods, as in the quantum theory of angular momentum, we may give a general expression for the forms listed. The analysis involves the vector– addition or Clebsch–Gordan coefficients from Section 4.4, which are tabulated. Three general restrictions appear.

1. The integral vanishes unless the triangle condition of the L’s (angular momentum) is zero, |L1 − L3|≤ L2 ≤ L1 + L3 .

2. The integral vanishes unless M2 + M3 = M1 . Here we have the theoretical foundation of the vector model of atomic spectroscopy.

3. Finally, the integral vanishes unless the product Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is even, that is, unless L1 + L2 + L3 is an even integer. This is a parity conservation law.

The key to the determination of the integral in Eq. (12.189) is the expansion of the product of two spherical harmonics depending on the same angles (in contrast to the addition theorem), which are coupled by Clebsch–Gordan coefficients to angular momentum L, M , which, from its rotational transformation properties, must be proportional to Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(θ , ϕ ); that is,

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Let us outline some of the steps of this general and powerful approach .
The Wigner–Eckart theorem applied to the matrix element in Eq. (12.189) yields

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

where the double bars denote the reduced matrix element, which no longer depends on the M. Selection rules (1) and (2) mentioned earlier follow directly from the Clebsch– Gordan coefficient in Eq. (12.190). Next we use Eq. (12.190) for M= M2 = M3 = 0in conjunction with Eq. (12.153) for m = 0, which yields

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

where x = cos θ . By elementary methods it can be shown that

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Substituting Eq. (12.192) into (12.191) we obtain

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The aforementioned parity selection rule (3) above follows from Eq. (12.193) in conjunction with the phase relation

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note that the vector-addition coefficients are developed in terms of the Condon–Shortley phase convention,23 in which the (−1)m of Eq. (12.153) is associated with the positive m.
It is possible to evaluate many of the commonly encountered integrals of this form with the techniques already developed. The integration over azimuth may be carried out by inspection:

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 Physically this corresponds to the conservation of the z component of angular momentum.

 

Application of Recurrence Relations 

A glance at Table 12.3 will show that the θ -dependence of Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev , that is, Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (θ ), can be expressed in terms of cos θ and sin θ . However, a factor of cos θ or sin θ may be combined with the Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev factor by using the associated Legendre polynomial recurrence relations. For

instance, from Eqs. (12.92) and (12.93) we get

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Using these equations, we obtain

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The occurrence of the Kronecker delta (L1 ,L ± 1) is an aspect of the conservation of angular momentum. Physically, this integral arises in a consideration of ordinary atomic electromagnetic radiation (electric dipole). It leads to the familiar selection rule that transitions to an atomic level with orbital angular momentum quantum number L1 can originate only from atomic levels with quantum numbers L1 − 1or L1 + 1. The application to expressions such as

quadrupole momentLegendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is more involved but perfectly straightforward.

 

LEGENDRE FUNCTIONS OF THE SECOND KIND

In all the analysis so far in this chapter we have been dealing with one solution of Legendre’s equation, the solution Pn (cos θ), which is regular (finite) at the two singular points of the differential equation, cos θ =±1. From the general theory of differential equations it is known that a second solution exists. We develop this second solution, Qn , with nonnegative integer n (because Qn in applications will occur in conjunction with Pn ), byaseries solution of Legendre’s equation. Later a closed form will be obtained.

 

Series Solutions of Legendre’s Equation

To solve

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev    (12.200)

we proceed as in Chapter 9, letting24

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(12.201)

with

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Substitution into the original differential equation gives

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The indicial equation is

k(k − 1) = 0,                        (12.205)

with solutions k = 0, 1. We try first k = 0 with a= 1,a= 0. Then our series is described by the recurrence relation

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      (12.206)

which becomes

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev            (12.207)

Labeling this series, from Eq. (12.201), y(x) = pn (x ),wehave

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The second solution of the indicial equation, k = 1, with a0 = 0,a1 = 1, leads to the recurrence relation

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (12.209)

Labeling this series, from Eq. (12.201), y(x) = qn (x ), we obtain

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Our general solution of Eq. (12.200), then, is

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (12.211)

provided we have convergence. we do not have convergence at x =±1. To get out of this difficulty, we set the separation constant n equal to an integer (Exercise 9.5.5) and convert the infinite series into a polynomial.
For n a positive even integer (or zero), series pn terminates, and with a proper choice of a normalizing factor 

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

If n is a positive odd integer, series qn terminates after a finite number of terms, and we write

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note that these expressions hold for all real values of x, −∞ <x < ∞, and for complex values in the finite complex plane. The constants that multiply pn and qn are chosen to make Pn agree with Legendre polynomials given by the generating function.
Equations (12.208) and (12.210) may still be used with n = ν , not an integer, but now the series no longer terminates, and the range of convergence becomes −1 <x < 1. The  

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

so that Eqs. (12.212) and (12.213) become

Legendre Special Function (Part - 7)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (12.214)

where the upper limit s = n/2 (for n even) or (n − 1)/2(for n odd). This reproduces Eq. (12.8), which is obtained directly from the generating function. This agreement with Eq. (12.8) is the reason for the particular choice of normalization in Eqs. (12.212) and (12.213).

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