Legendre Special Function (Part - 4) - 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 - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The document Legendre Special Function (Part - 4) - 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.
All you need of Physics at this link: Physics

ASSOCIATED LEGENDRE FUNCTIONS

When Helmholtz’s equation is separated in spherical polar coordinates, one of the separated ODEs is the associated Legendre equation

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

With x = cos θ , this becomes

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

If the azimuthal separation constant m= 0, we have Legendre’s equation, Eq. (12.28).
The regular solutions Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  (with m not necessarily zero, but an integer) are

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev          (12.73a)

with m ≥ 0 an integer

One way of developing the solution of the associated Legendre equation is to start with the regular Legendre equation and convert it into the associated Legendre equation by using multiple differentiation. These multiple differentiations are suggested by Eq. (12.73a). 
We take Legendre’s equation

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

(1 − x 2)u′′ − 2x(m + 1)u′ + (n − m)(n + m + 1)u = 0,       (12.75)

where

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(12.76)

Equation (12.74) is not self-adjoint. To put it into self-adjoint form and convert the weighting function to 1, we replace u(x ) by

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev      (12.73b)

Solving for u and differentiating, we obtain

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Substituting into Eq. (12.74), we find that the new function v satisfies the self-adjoint ODE

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

which is the associated Legendre equation; it reduces to Legendre’s equation when m is set equal to zero. Expressed in spherical polar coordinates, the associated Legendre equation is

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Associated Legendre Polynomials 

The regular solutions, relabeledLegendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev       (12.73c)

These are the associated Legendre functions. Since the highest power of x in Pn (x ) is x n ,wemusthave m ≤ n (or the m-fold differentiation will drive our function to zero).
In quantum mechanics the requirement that m ≤ n has the physical interpretation that the expectation value of the square of the z component of the angular momentum is less than or equal to the expectation value of the square of the angular momentum vector L,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

From our definition of the associated Legendre functions Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev           (12.82)

A generating function for the associated Legendre functions is obtained, via Eq. (12.71), from that of the ordinary Legendre polynomials:

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        (12.83)

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Comparing coefficients of powers of t in these power series, we obtain the recurrence relation

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

so upon inserting Eq. (12.84) we get the recursion

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Table 12.2 Associated Legendre Functions

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Example 12.5.1 LOWEST ASSOCIATED LEGENDRE POLYNOMIALS

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Example 12.5.2 SPECIAL VALUES

For x = 1we use

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

in Eq. (12.84) and find

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev(12.90)

where
Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

for s = 0 and

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Recurrence Relations

As expected and already seen, the associated Legendre functions satisfy recurrence relations. Because of the existence of two indices instead of just one, we have a wide variety of recurrence relations:

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

These relations, and many other similar ones, may be verified by use of the generating function (Eq. (12.4)), by substitution of the series solution of the associated Legendre equation (12.79) or reduction to the Legendre polynomial recurrence relations, using Eq. (12.73c). As an example of the last method, consider Eq. (12.93). It is similar to Eq. (12.23):

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev         (12.95)

Let us differentiate this Legendre polynomial recurrence relation m times to obtain

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Now multiplying by (1 − x 2 )(m+1)/2 and using the definition of Pn (x ), we obtain the first part of Eq. (12.93).

 

Parity

The parity relation satisfied by the associated Legendre functions may be determined by examination of the defining equation (12.73c). As x →−x , we already know that Pn (x ) contributes a (−1)n .The m-fold differentiation yields a factor of (−1)m . Hence we have

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  (12.97)

A glance at Table 12.2 verifies this for 1 ≤ m ≤ n ≤ 4.
Also, from the definition in Eq. (12.73c),

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev        (12.98)

Orthogonality 

The orthogonality of the Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev follows from the ODE, just as for the Pn (x ) (Section 12.3), if m is the same for both functions. However, it is instructive to demonstrate the orthogonality by another method, a method that will also provide the normalization constant.
Using the definition in Eq. (12.73c) and Rodrigues’ formula (Eq. (12.65)) for Pn (x ),we find

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The integrand on the right-hand side is now expanded by Leibniz’ formula to give

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Since the term Xcontains no power of x greater than x2m ,wemusthave

q + m − i ≤ 2m                    (12.102)

or the derivative will vanish. Similarly,

p + m + i ≤ 2p.               (12.103)

Adding both inequalities yields

q ≤ p,                                 (12.104)

which contradicts our assumption that p< q . Hence, there is no solution for i and the integral vanishes. The same result obviously will follow if p> q .
For the remaining case, p = q , we have the single term corresponding to i = q − m.
Putting Eq. (12.101) into Eq. (12.100), we have

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Since

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Eq. (12.105) reduces to

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The integral on the right is just

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

(compare Exercise 8.4.9). Combining Eqs. (12.108) and (12.109), we have the orthogonality integral,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

or, in spherical polar coordinates,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The orthogonality of the Legendre polynomials is a special case of this result, obtained by setting m equal to zero; that is, for m = 0, Eq. (12.110) reduces to Eqs. (12.47) and (12.48). In both Eqs. (12.110) and (12.111), our Sturm–Liouville theory of Chapter 10 could provide the Kronecker delta. A special calculation, such as the analysis here, is required for the normalization constant.
The orthogonality of the associated Legendre functions over the same interval and with the same weighting factor as the Legendre polynomials does not contradict the uniqueness of the Gram–Schmidt construction of the Legendre polynomials.
Table 12.2 suggests thatLegendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev dx may be written as

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

where we defined earlier

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The functions Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev may be constructed by the Gram–Schmidt procedure with the weighting function w(x ) = (1 − x 2 )m .
It is possible to develop an orthogonality relation for associated Legendre functions of the same lower index but different upper index. We find

Legendre Special Function (Part - 4) - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note that a new weighting factor, (1 − x 2 )−1 , has been introduced. This relation is a mathematical curiosity. In physical problems with spherical symmetry solutions of Eqs. (12.80) and (9.64) appear in conjunction with those of Eq. (9.61), and orthogonality of the azimuthal dependence makes the two upper indices equal and always leads to Eq. (12.111).

 

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Dynamic Test

Content Category

Related Searches

Objective type Questions

,

UGC - NET Physics Physics Notes | EduRev

,

Semester Notes

,

Sample Paper

,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics

,

MCQs

,

Summary

,

past year papers

,

Extra Questions

,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics

,

Important questions

,

Free

,

Previous Year Questions with Solutions

,

shortcuts and tricks

,

video lectures

,

Legendre Special Function (Part - 4) - Mathematical Methods of Physics

,

ppt

,

practice quizzes

,

UGC - NET Physics Physics Notes | EduRev

,

Viva Questions

,

Exam

,

study material

,

mock tests for examination

,

pdf

,

UGC - NET Physics Physics Notes | EduRev

;