5 Spherical Harmonics Laplace Helmholtz or central force

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

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

Fig. 15. 12. Shapes of [ Re Ylm ( , ) ]2 Surfaces are given by Y 00 Y 10 Y 20 Y 11 Y 22 Y 21 Mathematica Y 30 Y 31 Y 32 Y 33

Cartesian Representations f is a polynomial Using one gets

Table 15. 4. Spherical Harmonics (with Condon-Shortley Phase ( ) ) m Mathematica Spherical. Harmonic. Y[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 r 0 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”, Mc. Graw-Hill (53) ] See Ex. 15. 5. 6 for normalization Measured Earth Moon Mars C 20 (equatorial bulge) 1. 083 10 3 0. 200 10 3 1. 96 10 3 C 22 (azimuthal dep. ) 0. 16 10 5 2. 4 10 5 5 10 5 S 22 (azimuthal dep. ) 0. 09 10 5 0. 5 10 5 3 10 5

Symmetry of Solutions have less symmetry than the Hamiltonian due to the initial conditions. L 2 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) has degeneracy = 2 l + 1. Same pt. in different coord. systems or different pts in same coord. system m degeneracy also occurs for the Laplace, Helmholtz, & central force Schrodinger eqs. see Chap. 16 for more

Example 15. 5. 3. Y 1 m Solutions for l = 1 at Arbitray Orientaion 1 0 1 Spherical 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 : 2 nd solution ( § 7. 6 ) : where the Wronskian is

Mathematica Ql obeys the same recurrence relations as Pl.

for If we define Ql (x) to be real for real arguments, Replace Note: Legendre. Q in Mathematica retains the i term. 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 : x = 0 is a regular point Special values : 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 series converges at x=1 s = 0 for odd l s = 1 for even l Ql even Ql odd Pl even Pl odd

s = 0 for odd l s = 1 for even l s = 1 , l = even s = 0 , l = odd Ql even Ql odd Ql even j = even a 1 = 0 bl = a 0 for Ql

Lowest order in x :

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