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**Polar Angle Dependence**

−n ≤ m ≤ n, (12.152)

which are orthonormal with respect to the polar angle θ .

**Spherical Harmonics**

**The function ****m (ϕ ) (Eq. (12.150)) is orthonormal with respect to the azimuthal angle ϕ . We take the product of ****m (ϕ ) and the orthonormal function in polar angle from Eq. (12.152) and deﬁne**

to obtain functions of two angles (and two indices) that are orthonormal over the spherical surface. These (θ , ϕ ) are spherical harmonics, of which the ﬁrst few are plotted in Fig. 12.15. The complete orthogonality integral becomes

The extra (−1)^{m} included in the deﬁning equation of (θ , ϕ ) deserves some comment.

It is clearly legitimate, since Eq. (12.144) is linear and homogeneous. It is not necessary, but in moving on to certain quantum mechanical calculations, particularly in the quantum theory of angular momentum (Section 12.7), it is most convenient. The factor (−1)^{m} is a phase factor, often called the Condon–Shortley phase, after the authors of a classic text on atomic spectroscopy. The effect of this (−1)^{m} (Eq. (12.153)) and the (−1)^{m} of Eq. (12.73c) for (cos θ) is to introduce an alternation of sign among the positive m spherical harmonics. This is shown in Table 12.3.

The functions (θ , ϕ ) acquired the name spherical harmonics ﬁrst because they are deﬁned over the surface of a sphere with θ the polar angle and ϕ the azimuth. The harmonic was included because solutions of Laplace’s equation were called harmonic functions and (cos,ϕ ) is the angular part of such a solution.

Table 12.3 Spherical Harmonics (Condon–Shortley Phase)

In the framework of quantum mechanics Eq. (12.145) becomes an orbital angular momentum equation and the solution (θ , ϕ ) (n replaced by L, m replaced by M )isan angular momentum eigenfunction, L being the angular momentum quantum number and M the z-axis projection of L.

**Laplace Series, Expansion Theorem **

Part of the importance of spherical harmonics lies in the completeness property, a consequence of the Sturm–Liouville form of Laplace’s equation. This property, in this case, means that any function f(θ , ϕ ) (with sufﬁcient continuity properties) evaluated over the surface of the sphere can be expanded in a uniformly convergent double series of spherical harmonics (Laplace’s series):

If f(θ , ϕ ) is known, the coefﬁcients can be immediately found by the use of the orthogonality integral.

**Table 12.4 Gravity Field Coefﬁcients, Eq. (12.156)**

**Example : LAPLACE SERIES —GRAVITY FIELDS **

The gravity ﬁelds of the Earth, the Moon, and Mars have been described by a Laplace series with real eigenfunctions:

**ORBITAL ANGULAR MOMENTUM OPERATORS**

Now we return to the speciﬁc orbital angular momentum operators L_{x} ,L_{y} , and L_{z} of quantum mechanics introduced in Section 4.3. Equation (4.68) becomes

and we want to show that

From L+ ψ_{LL} = 0,L being the largest M , using the form of L_{+ }given in Exercises 2.5.14 and 12.6.7, we have

(12.158)

and thus

(12.159)

Normalizing, we obtain

(12.160)

The θ integral may be evaluated as a beta function (Exercise 8.4.9) and

(12.161)

This completes our ﬁrst step.

Again, note that the relative phases are set by the ladder operators. L_{+ }and L_{−} operating

Repeating these operations n times yields

From Eq. (12.163),

and for M =−L:

Note the characteristic (−1)^{L} phase of ψ_{L,−L} relative to ψ_{L,L} .This (−1)^{L} enters from

(12.167)

Combining Eqs. (12.163), (12.163), and (12.166), we obtain

Equations (12.165) and (12.168) agree if

(12.169)

Using Rodrigues’ formula, Eq. (12.65), we have

(12.170)

The last equality follows from Eq. (12.161). We now demand that ψL0 (0, 0) be real and positive. Therefore

(12.171)

The expression in the curly bracket is identiﬁed as the associated Legendre function (Eq. (12.151), and we have

in complete agreement with Section 12.6. Then by Eq. (12.73c), for negative superscript is given by

(12.174)

- Our angular momentum eigenfunctions ψ
_{LM}(θ , ϕ ) are identiﬁed with the spherical harmonics. The phase factor (−1)^{M}is associated with the positive values of M and is seen to be a consequence of the ladder operators. - Our development of spherical harmonics here may be considered a portion of Lie algebra.

**THE ADDITION THEOREM FOR SPHERICAL HARMONICS Trigonometric Identity**

**In the following discussion, (θ _{1} ,ϕ_{1} ) and (θ_{2} ,ϕ_{2} ) denote two different directions in our spherical coordinate system (x_{1} ,y_{1} ,z_{1} ), separated by an angle γ (Fig. 12.16). The polar angles θ_{1} ,θ_{2 }are measured from the z_{1} -axis. These angles satisfy the trigonometric identity**

(12.175)

which is perhaps most easily proved by vector methods

The addition theorem, then, asserts that

or equivalently,

In terms of the associated Legendre functions, the addition theorem is

Equation (12.175) is a special case of Eq. (12.178), n = 1.

**Derivation of Addition Theorem**

We need for our proof only the coefﬁcient which we get by multiplying Eq. (12.179) by and integrating over the sphere:

(12.180)

Similarly, we expand P_{n }(cos γ) in terms of spherical harmonics

(12.181)

In terms of spherical harmonics Eq. (12.182) becomes

Note that the subscripts have been dropped from the solid angle element dΩ Since the range of integration is over all solid angles, the choice of polar axis is irrelevant. Then comparing Eqs. (12.180) and (12.183), we see that

(12.184)

Now we evaluate using the expansion of Eq. (12.179) and noting that the values of (γ , ξ ) corresponding to (θ_{1} ,ϕ_{1 }) = (θ_{2} ,ϕ_{2} ) are (0, 0). The result is

(12.185)

all terms with nonzero σ vanishing. Substituting this back into Eq. (12.184), we obtain

(12.186)

Finally, substituting this expression for b_{nm} into the summation, Eq. (12.181) yields Eq. (12.177), thus proving our addition theorem.

Those familiar with group theory will ﬁnd a much more elegant proof of Eq. (12.177) by using the rotation group.

One application of the addition theorem is in the construction of a Green’s function for the three-dimensional Laplace equation in spherical polar coordinates. If the source is on

the polar axis at the point (r = a, θ = 0,ϕ = 0), then, by Eq. (12.4a),

(12.187)

Rotating our coordinate system to put the source at (a , θ_{2} ,ϕ_{2}) and the point of observation at (r, θ_{1} ,ϕ_{1}), we obtain

**Exercises**

1. Two protons are uniformly distributed within the same spherical volume. If the coordinates of one element of charge are (r_{1} ,θ_{1} ,ϕ_{1} ) and the coordinates of the other are (r_{2} ,θ_{2} ,ϕ_{2} ) and r_{12} is the distance between them, the element of energy of repulsion will be given by

Here

Calculate the total electrostatic energy (of repulsion) of the two protons. This calculation is used in accounting for the mass difference in “mirror” nuclei, such as O ^{15} and N^{15 }.

This is double that required to create a uniformly charged sphere because we have two separate cloud charges interacting, not one charge interacting with itself (with permutation of pairs not considered).

2. Each of the two 1S electrons in helium may be described by a hydrogenic wave function

in the absence of the other electron. Here Z , the atomic number, is 2. The symbol a_{0} is bthye Bohr radius, h¯^{ 2} /me^{2} . Find the mutual potential energy of the two electrons, given

3. The probability of ﬁnding a 1 S hydrogen electron in a volume element r ^{2} dr sin θd θd ϕ is

Find the corresponding electrostatic potential. Calculate the potential from

with r_{1} not on the z-axis. Expand r_{12} . Apply the Legendre polynomial addition theorem and show that the angular dependence of V(r_{1} ) drops out.