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**INTEGRALS OF PRODUCTS OF THREE SPHERICAL HARMONICS**

Frequently in quantum mechanics we encounter integrals of the general form

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, |L_{1} − L_{3}|≤ L_{2} ≤ L_{1} + L_{3} .

2. The integral vanishes unless M_{2} + M_{3} = M_{1} . Here we have the theoretical foundation of the vector model of atomic spectroscopy.

3. Finally, the integral vanishes unless the product is even, that is, unless L_{1} + L_{2} + L_{3} 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 (θ , ϕ ); that is,

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

where the double bars denote the reduced matrix element, which no longer depends on the M_{i }. 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_{1 }= M_{2} = M_{3} = 0in conjunction with Eq. (12.153) for m = 0, which yields

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

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

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

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:

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 , that is, (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 factor by using the associated Legendre polynomial recurrence relations. For

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

Using these equations, we obtain

The occurrence of the Kronecker delta (L_{1} ,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 L_{1} can originate only from atomic levels with quantum numbers L_{1} − 1or L_{1} + 1. The application to expressions such as

quadrupole moment 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, Q_{n} , with nonnegative integer n (because Q_{n} in applications will occur in conjunction with P_{n} ), byaseries solution of Legendre’s equation. Later a closed form will be obtained.

**Series Solutions of Legendre’s Equation**

To solve

(12.200)

we proceed as in Chapter 9, letting^{24}

(12.201)

with

Substitution into the original differential equation gives

The indicial equation is

k(k − 1) = 0, (12.205)

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

(12.206)

which becomes

(12.207)

Labeling this series, from Eq. (12.201), y(x) = p_{n} (x ),wehave

The second solution of the indicial equation, k = 1, with a_{0} = 0,a_{1} = 1, leads to the recurrence relation

(12.209)

Labeling this series, from Eq. (12.201), y(x) = q_{n} (x ), we obtain

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

(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 p_{n} terminates, and with a proper choice of a normalizing factor

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

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 p_{n} and q_{n} are chosen to make P_{n} 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

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

(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).