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

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

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

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

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

6. Mathematica

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

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

9. 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 

10. 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 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

11. 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.

12. 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.

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

14. 

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

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

17. 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.

18. Fig Ql (x)
Mathematica

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

20. 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  

21. 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

22. Lowest order in x :      

23. 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
For series expansion in 1/x for Ql , see Ex