4. Associated Legendre Equation Let   Set  Mathematica

Frobenius Series  with indicial eqs. By definition,  or Mathematica

Series diverges at x = 1 unless terminated.  For s = 0 & a1= 0 (even series) : ( l,m both even or both odd )  Mathematica For s = 1 & a1=0 (odd series) : ( l,m one even & one odd )  Plm = Associated Legendre function

Relation to the Legendre Functions Generalized Leibniz’s rule :   

Associated Legendre function : Set Associated Legendre function : ()m is called the Condon-Shortley phase. Including it in Plm means Ylm has it too.  Rodrigues formula :    Mathematica

Generating Function & Recurrence     ( Redundant since Plm is defined only for l  |m| 0. ) &

  as before    

 ( Redundant since Plm is defined only for l  |m| 0. ) & 

Recurrence Relations for Plm (1) = (15.88) (2) (1) : (3) (3)  (2) : (15.89)

Table 15.3 Associated Legendre Functions Using one can generate all Plm (x) s from the Pl (x) s. Mathematica

Example 15.4.1. Recurrence Starting from Pmm (x) no negative powers of (x1)      

l = m  l = m+k1  E.g., m = 2 :

Parity & Special Values Rodrigues formula :  Parity Special Values : Ex.15.4-5

Orthogonality Plm is the eigenfunction for eigenvalue of the Sturm-Liouville problem where Lm is hermitian  ( w = 1 ) Alternatively : 

No negative powers allowed For p  q , let  & only j = q ( x = + 1) or j = kq ( x =  1 ) terms can survive

p  q : For j > m :  For j < m + 1 : 

p  q :   Only j = 2m term survives  

Ex.13.3.3 B(p,q)  

 For fixed m, polynomials { Ppm (x) } are orthogonal with weight ( 1  x2 )m . Similarly

Example 15.4.2. Current Loop – Magnetic Dipole Biot-Savart law (for A , SI units) : By symmetry : Outside loop :  E.g. 3.10.4 Mathematica  

 For r > a :  

 For r > a :

on z-axis : or   (odd in z)

Biot-Savart law (SI units) : Cartesian coord:  

 For r > a :

s 1 2 s 3/2 15/8 s 1 2 s 1 3/4 5/8

Electric dipole :  

5. Spherical Harmonics Laplace, Helmholtz, or central force Schrodinger eq. Set  Set  Orthonormal solutions

 Spherical harmonics Orthonormality :  Real valued form of : ( with Condon-Shortley phase via Plm )  Real valued form of :

Fig.15.12. Shapes of [ Re Ylm (,  ) ]2 Surfaces are given by Y00 Y10 Y11 Y22 Y20 Y21 Mathematica Y30 Y31 Y32 Y33

Cartesian Representations  f is a polynomial   Using one gets

Table 15.4. Spherical Harmonics (with Condon-Shortley Phase ()m ) Mathematica SphericalHarmonicY[l,m,,] Mathematica

Mathematica

Overall Solutions Laplace eq.:  Helmholtz eq.: 

Laplace Expansion = eigenstates of the Sturm-Liouville problem  S is a complete set of orthogonal functions on the unit sphere.  Laplace series

Example 15.5.1. Spherical Harmonic Expansion Problem : Let the potential on the surface of a charge-free spherical region of radius r0 be . Find the potential inside the region.   regular at r = 0 

Example 15.5.2. Laplace Series – Gravity Fields Gravity fields of the Earth, Moon, & Mars had been described as where [ see Morse & Feshbach, “Methods of Theoretical Physics”, McGraw-Hill (53) ] See Ex.15.5.6 for normalization Measured Earth Moon Mars C20 (equatorial bulge) 1.083103 0.200103 1.96103 C22 (azimuthal dep.) 0.16105 2.4105 5105 S22 (azimuthal dep.) 0.09105 0.5105 3105

Symmetry of Solutions Solutions have less symmetry than the Hamiltonian due to the initial conditions. L2 has spherical symmetry but none of Yl m ( l  0) does. { Yl m ; m = l, …, l } are eigenfunctions with the same eigenvalue l ( l + 1).  { Yl m ; m = l, …, l } spans the eigen-space for eigenvalue l ( l + 1).  eigenvalue l ( l + 1) has degeneracy = 2l + 1. Same pt. in different coord. systems or different pts in same coord. system see Chap.16 for more m degeneracy also occurs for the Laplace, Helmholtz, & central force Schrodinger eqs.

Example 15.5.3. Solutions for l = 1 at Arbitray Orientaion Y1m 1 1 Spherical Cartesian Cartesian coordinates : Unit vector with directional cosine angles { , ,  } :  Same pt. r, different coord. system.

Further Properties Special values:   Recurrence ( straight from those for Plm ) :



6. Legendre Functions of the Second Kind Alternate form : 2nd solution ( § 7.6 ) : where the Wronskian is   

Ql obeys the same recurrence relations as Pl .    Mathematica Ql obeys the same recurrence relations as Pl . 

for Note: LegendreQ in Mathematica retains the i  term. If we define Ql (x) to be real for real arguments, Replace for |x| > 1. For complex arguments, place the branch cut from z = 1 to z = +1. Values for arguments on the branch cut are given by the average of those on both sides of the cut.

Fig.15.13-4. Ql (x) Mathematica

Properties Parity :  Special values : x = 0 is a regular point See next page Ex.15.6.3

Alternate Formulations Singular points of the Legendre ODE are at ( Singularity at x =  is removable )  Ql has power series in x that converges for |x| < 1. & power series in 1/x that converges for |x| > 1. Frobenius series : s = 0 for even l s = 1 for odd l Pl even Pl odd  series converges at x=1  s = 0 for odd l s = 1 for even l Ql even Ql odd  

s = 0 for odd l s = 1 for even l Ql even Ql odd  s = 1 , l = even  Ql odd  j = even  a1 = 0 bl = a0 for Ql s = 0 , l = odd  Ql even  j = even  a1 = 0

Lowest order in x :      

For series expansion in x for Ql , see Ex.15.6.2 Similarly, one gets Mathematica which can be fitted as For series expansion in x for Ql , see Ex.15.6.2 For series expansion in 1/x for Ql , see Ex.15.6.3