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) 1.083103 0.200103 1.96103 C22 (azimuthal dep.) 0.16105 2.4105 5105 S22 (azimuthal dep.) 0.09105 0.5105 3105
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