Download presentation
Presentation is loading. Please wait.
1
4. Associated Legendre Equation
Let Set Mathematica
2
Frobenius Series with indicial eqs. By definition, or Mathematica
3
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
4
Relation to the Legendre Functions
Generalized Leibniz’s rule :
5
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
6
Generating Function & Recurrence
( Redundant since Plm is defined only for l |m| 0. ) &
7
as before
8
( Redundant since Plm is defined only for l |m| 0. ) &
9
Recurrence Relations for Plm
(1) = (15.88) (2) (1) : (3) (3) (2) : (15.89)
10
Table 15.3 Associated Legendre Functions
Using one can generate all Plm (x) s from the Pl (x) s. Mathematica
11
Example 15.4.1. Recurrence Starting from Pmm (x)
no negative powers of (x1)
12
l = m l = m+k1 E.g., m = 2 :
13
Parity & Special Values
Rodrigues formula : Parity Special Values : Ex
14
Orthogonality Plm is the eigenfunction for eigenvalue of the Sturm-Liouville problem where Lm is hermitian ( w = 1 ) Alternatively :
15
No negative powers allowed
For p q , let & only j = q ( x = + 1) or j = kq ( x = 1 ) terms can survive
16
p q : For j > m : For j < m + 1 :
17
p q : Only j = 2m term survives
18
Ex B(p,q)
19
For fixed m, polynomials { Ppm (x) } are orthogonal with weight ( 1 x2 )m .
Similarly
20
Example 15.4.2. Current Loop – Magnetic Dipole
Biot-Savart law (for A , SI units) : By symmetry : Outside loop : E.g Mathematica
21
For r > a :
22
For r > a :
23
on z-axis : or (odd in z)
24
Biot-Savart law (SI units) :
Cartesian coord:
25
For r > a :
26
s 1 2 s 3/2 15/8 s 1 2 s 1 3/4 5/8
27
Electric dipole :
28
5. Spherical Harmonics Laplace, Helmholtz, or central force Schrodinger eq. Set Set Orthonormal solutions
29
Spherical harmonics Orthonormality : Real valued form of :
( with Condon-Shortley phase via Plm ) Real valued form of :
30
Fig.15.12. Shapes of [ Re Ylm (, ) ]2
Surfaces are given by Y00 Y10 Y11 Y22 Y20 Y21 Mathematica Y30 Y31 Y32 Y33
![Fig Shapes of [ Re Ylm (, ) ]2](http://slideplayer.com./slide/16602726/96/images/30/Fig+Shapes+of+%5B+Re+Ylm+%28%EF%81%B1%2C+%EF%81%A6+%29+%5D2.jpg "Surfaces are given by. Y00. Y10. Y11. Y22. Y20. Y21. Mathematica. Y30. Y31. Y32. Y33.")
31
Cartesian Representations
f is a polynomial Using one gets
32
Table 15.4. Spherical Harmonics (with Condon-Shortley Phase ()m )
Mathematica SphericalHarmonicY[l,m,,] Mathematica
33
Mathematica
34
Overall Solutions Laplace eq.: Helmholtz eq.:
35
Laplace Expansion = eigenstates of the Sturm-Liouville problem
S is a complete set of orthogonal functions on the unit sphere. Laplace series
36
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
37
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) 1.083103 0.200103 1.96103 C22 (azimuthal dep.) 0.16105 2.4105 5105 S22 (azimuthal dep.) 0.09105 0.5105 3105
38
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.
39
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.
40
Further Properties Special values:
Recurrence ( straight from those for Plm ) :
41
42
6. Legendre Functions of the Second Kind
Alternate form : 2nd solution ( § 7.6 ) : where the Wronskian is
43
Ql obeys the same recurrence relations as Pl .
Mathematica Ql obeys the same recurrence relations as Pl .
44
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.
45
Fig Ql (x) Mathematica
46
Properties Parity : Special values : x = 0 is a regular point
See next page Ex
47
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
48
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
49
Lowest order in x :
50
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
Similar presentations
