Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

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Physics : Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The document Legendre Special Function (Part - 6)- 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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Polar Angle Dependence

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev −n ≤ m ≤ n,              (12.152)

which are orthonormal with respect to the polar angle θ .

 

Spherical Harmonics

The function Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevm (ϕ ) (Eq. (12.150)) is orthonormal with respect to the azimuthal angle ϕ . We take the product of Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevm (ϕ ) and the orthonormal function in polar angle from Eq. (12.152) and define

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

to obtain functions of two angles (and two indices) that are orthonormal over the spherical surface. These Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (θ , ϕ ) are spherical harmonics, of which the first few are plotted in Fig. 12.15. The complete orthogonality integral becomes

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The extra (−1)m included in the defining equation of Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (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 Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (cos θ) is to introduce an alternation of sign among the positive m spherical harmonics. This is shown in Table 12.3.
The functions Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (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 Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (cos,ϕ ) is the angular part of such a solution.

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Table 12.3 Spherical Harmonics (Condon–Shortley Phase)

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

In the framework of quantum mechanics Eq. (12.145) becomes an orbital angular momentum equation and the solution Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (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):

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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)

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


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:

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

and we want to show that

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

From L+ ψLL = 0,L being the largest M , using the form of Lgiven in Exercises 2.5.14 and 12.6.7, we have

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (12.158)

and thus

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (12.159)

Normalizing, we obtain

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev            (12.160)

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev              (12.161)

This completes our first step.

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Again, note that the relative phases are set by the ladder operators. Land L operating

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Repeating these operations n times yields

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

From Eq. (12.163),

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

and for M =−L:

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note the characteristic (−1)L phase of ψL,−L relative to ψL,L .This (−1)L enters from

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev           (12.167)

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Equations (12.165) and (12.168) agree if

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (12.169)

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (12.170)

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev          (12.171)

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

in complete agreement with Section 12.6. Then by Eq. (12.73c), Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev for negative superscript is given by

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev                    (12.174)

  • Our angular momentum eigenfunctions ψLM (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, (θ11 ) and (θ22 ) denote two different directions in our spherical coordinate system (x1 ,y1 ,z1 ), separated by an angle γ (Fig. 12.16). The polar angles θ1are measured from the z1 -axis. These angles satisfy the trigonometric identity

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (12.175)

which is perhaps most easily proved by vector methods 
The addition theorem, then, asserts that

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

or equivalently,

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

 

Derivation of Addition Theorem

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

We need for our proof only the coefficient  Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevwhich we get by multiplying Eq. (12.179) by  Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  and integrating over the sphere:

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev (12.180)

Similarly, we expand P(cos γ) in terms of spherical harmonics Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (12.181)

 

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

In terms of spherical harmonics Eq. (12.182) becomes

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev              (12.184)

Now we evaluate Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  using the expansion of Eq. (12.179) and noting that the values of (γ , ξ ) corresponding to (θ1) = (θ22 ) are (0, 0). The result is

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        (12.185)

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev             (12.186)

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),

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(12.187)

Rotating our coordinate system to put the source at (a , θ22) and the point of observation at (r, θ11), we obtain

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Exercises

1. Two protons are uniformly distributed within the same spherical volume. If the coordinates of one element of charge are (r111 ) and the coordinates of the other are (r222 ) and r12 is the distance between them, the element of energy of repulsion will be given by

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Here

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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 .

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev
Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Find the corresponding electrostatic potential. Calculate the potential from

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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.

Legendre Special Function (Part - 6)- Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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