15. Legendre Functions Legendre Polynomials Orthogonality Physical Interpretation of Generating Function Associated Legendre Equation Spherical Harmonics Legendre Functions of the Second Kind
Schrodinger eq. for a central potential Associated Legendre eq. Legendre eq. with
1. Legendre Polynomials Legendre eq. x = 1 & x = are regular singular points.
Frobenius Series See § 8.3 & Mathematica Indicial eq. series diverges for x2 1 unless terminated Set a1 = 0 a2 j + 1 = 0 s = 0 even order : s = 1 odd order :
Generating Function
& highest power of x in coeff. of tn is n. Coeff. of xn in Pn(x) = Coeff. of xn tn Coeff. of highest power of x in Pn(x)
Summary Power Expansion : Ex.15.1.2
Recurrence Relations
Table 15.1. Legendre Polynomials Mathematica
& Eliminate x Pn1 term
More Recurrence Relations Any set of functions satisfying these recurrence relations also satisfy the Legendre ODE. Ex.15.1.1
Upper & Lower Bounds for Pn (cos )
Coeff. invariant under j j Coeff. invariant under j ( j+1)
For P2n , x = 1 are global max. For P2n+1 , x = +1 is a gloabal max, while x = 1 is a gloabal min. Mathematica
Rodrigues Formula From § 12.1 : If has self-adjoint form then Legendre eq. : Self-adjoint form :
Coefficient of xn in Pn(x) Coefficient of xn is :
2. Orthogonality is self-adjoint [ w(x) = 1 ] Pn(x) are orthogonal polynomials in [ 1, 1 ].
Normalization Let via Rodrigues formula : Ex.15.2.1
Legendre Series Eigenfunctions of an ODE are complete { Pn (x) } is completeness over [1,1] . For any function f (x) in [1,1] : unique
Solutions to Laplace Eq. in Spherical Coordinates General solution : finite l = 0,1,2, … Solution with no azimuthal dependence ( m = 0 ) : Solution that is finite inside & outside a boundary sphere :
Example 15.2.1 Earth’s Gravitational Field Gravitational potential U in mass-free region : Neglect azimuthal dependence : Earth’s radius at equator g includes rotational effect Note: Let al dimensionless Earth is a sphere
Slightly distorted Earth with axial symmetry : CM located at origin See Mathematica for proof. Experimental data : Data including longitudinal dependence is described by a Laplace series (§15.5).
Example 15.2.2 Sphere in a Uniform Field Grounded conducting sphere (radius r0 ) in uniform applied electric field everywhere For :
Surface charge density : SI units induced dipole moment Ex.15.2.11 Mathematica
Example 15.2.3 Electrostatic Potential for a Ring of Charge Thin, conducting ring of radius a, centered at origin & lying in x-y plane, has total charge q. Outside the ring, Axial symmetry no dependence Mathematica For r > a : On z-axis, Coulomb’s law gives :
See Eg.15.4.2 for magnetic analog.
3. Physical Interpretation of Generating Function Leading term : (point charge) for r > a. for r < a.
Expansion of 1 / | r r | Let : either r or r on z-axis
Electric Multipoles Electric dipole : point dipole Leading term :
(Linear) Multipoles Let = 2l -pole potential with center of charge at z = r. Mono ( 20 ) -pole : Di ( 21 ) -pole : Quadru ( 22 ) –pole : ( 2l ) –pole : Quadrupole Mathematica
Multipole Expansion If all charges are on the z-axis & within the interval [zm , zm ] : for r > zm where is the (linear) 2l –pole moment. For a discrete set of charges qi at z = ai .
If one shifts the coord origin to Z. l is independent of coord, i.e., Z iff Multipole expansion for a general (r) are done in terms of the spherical harmonics.