Polar Angle Dependence
−n ≤ m ≤ n, (12.152)
which are orthonormal with respect to the polar angle θ .
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 Lx ,Ly , and Lz of quantum mechanics introduced in Section 4.3. Equation (4.68) becomes
and we want to show that
From L+ ψLL = 0,L being the largest M , using the form of L+ given in Exercises 2.5.14 and 12.6.7, we have
Normalizing, we obtain
The θ integral may be evaluated as a beta function (Exercise 8.4.9) and
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 ψL,−L relative to ψL,L .This (−1)L enters from
Combining Eqs. (12.163), (12.163), and (12.166), we obtain
Equations (12.165) and (12.168) agree if
Using Rodrigues’ formula, Eq. (12.65), we have
The last equality follows from Eq. (12.161). We now demand that ψL0 (0, 0) be real and positive. Therefore
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
THE ADDITION THEOREM FOR SPHERICAL HARMONICS
In the following discussion, (θ1 ,ϕ1 ) and (θ2 ,ϕ2 ) denote two different directions in our spherical coordinate system (x1 ,y1 ,z1 ), separated by an angle γ (Fig. 12.16). The polar angles θ1 ,θ2 are measured from the z1 -axis. These angles satisfy the trigonometric identity
which is perhaps most easily proved by vector methods
The addition theorem, then, asserts that
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:
Similarly, we expand Pn (cos γ) in terms of spherical harmonics
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
Now we evaluate using the expansion of Eq. (12.179) and noting that the values of (γ , ξ ) corresponding to (θ1 ,ϕ1 ) = (θ2 ,ϕ2 ) are (0, 0). The result is
all terms with nonzero σ vanishing. Substituting this back into Eq. (12.184), we obtain
Finally, substituting this expression for bnm 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),
Rotating our coordinate system to put the source at (a , θ2 ,ϕ2) and the point of observation at (r, θ1 ,ϕ1), we obtain
1. Two protons are uniformly distributed within the same spherical volume. If the coordinates of one element of charge are (r1 ,θ1 ,ϕ1 ) and the coordinates of the other are (r2 ,θ2 ,ϕ2 ) and r12 is the distance between them, the element of energy of repulsion will be given by
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 N15 .
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 a0 is bthye Bohr radius, h¯ 2 /me2 . 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 r1 not on the z-axis. Expand r12 . Apply the Legendre polynomial addition theorem and show that the angular dependence of V(r1 ) drops out.