### Ch 12.1 Generating Function

```講者： 許永昌 老師
1
Contents
 Preface
 Generating function
 Legendre Polynomials
 Example: multipoles
 Recurrence relations
 Special values
 Upper and Lower bounds for Pn.
2
Preface
Generating Function
Special
values
Upper &
Lower Bound
Legendre Equation
Recurrence
relations
Legendre
Polynomial
Multipoles
Rodrigues’ Formula
Orthogonality
Series
Expansion
Associated Legendre functions
Spherical
Harmonics
Theorem
Green’s
function
Atomic
Orbitals
Orthogonality
Angular
momentum
Helmholtz Eq.
Group
theory
3
Generating Function (請預讀P741~P744)

rl
1
 From
  l 1 Pl  rˆ rˆ 
r  r ' l 0 r
1
r  2r r ' r '
2
,
2
 If tr</r>, we get
g  x, t  
1
1  2tx  t 2
 Series form:
1
1  2tx  t 2

  Pn  x  t n .
n 0

 2n  !
n 0
22 n n ! n !


 2n  ! n
1
  2n
x ,
1  x n 0 2 n ! n !
 2tx  t  ,
2 n
n !  1  2 x 
  2n
 p !  n  p !
n 0 2 n ! n ! p 0


 2n  !
m
t
  2m
m 0 2
m
2
 
n
m
2
 
p
n p
t
n p
,
a  b
n
n

p 0
22 p  2m  2 p  !  1  2 x 
,

p !  m  2 p !
 m  p !
p 0
p
 1  2m  2 p  !
We get Pm  x    m
x m 2 p .
p 0 2 p !  m  p  !  m  2 p  !
m 2 p
n!
a p bn  p ,
p !  n  p !
m
n  p  m, p  0 ~   ,
2
p
4
Multipoles (請預讀P744~P747)
 If
2V 


2
, we get  G  r , r '     r  r '  & V  r    G d .
0
0
 Green’s function:
1
1
G  r , r ' 

4 r  r ' 4
V(r)

rn
P rˆ rˆ ' ,

n 1 n 
r
n 0 
 If |r| > |r’|, we get
Multipole expansion:
  r '
1  1
n
V  r    G  r , r '
d ' 
r
'
Pn  rˆ rˆ '    r '  d '.

n 1 
0
4 0 n 0 r
 Remember, in physical systems we do not encounter pure
multipoles.
5
Recurrence Relations (請預讀P749~P751)
1.
From tg:

g  x, t 
3
  x  t  g  x, t    nPn  x  t n 1 ,
t
n 0

  x  t   Pn  x  t  1  2 xt  t
n

2
n 0
  nP  x  t
n 0
n
n 1
,
xPn  Pn 1   n  1 Pn 1  2nxPn   n  1 Pn 1 ,

For each t n
  2n  1 xPn  nPn 1   n  1 Pn 1 .
2. From xg : g  x, t 
x
 tg

3
 x, t    P 'n  x  t n ,
n 0

 t  Pn  x  t  1  2 xt  t
n
n 0

     1

2
  P ' xt
n 0
Pn  P 'n 1  2 xP 'n  P 'n 1 ,
n
n
,
     2
For each t n 1
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Recurrence Relations (continue)
 From Eqs. (1) & (2) we can get
1 '  n  1  2   P 'n1  xP 'n  nPn ,
1 ' n  2 
 P 'n 1  xP 'n   n  1 Pn ,
      3
     4
 Let’s try to get the Legendre Eq.:
x  3   4 nn 1  1  x 2  P 'n  nPn 1  nxPn
      5




2
2
1

x
5

n
3

1

x

n
n

1
Pn  0.












x
x
 x

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Special Values & Parity (請預讀P752~P753)
 g(1,t)=1/|1-t|
 Pn(1)=1.
 g(-1,t)=1/|1+t|
 Pn(-1)=(-1)n.
n


1  2n  ! 2 n

1
 g(0,t)=
  2n
t   Pm  0  t m ,
2
2 n!n!
1 t
n 0
m 0
 P2n+1(0)=0,
 P2n(0)= (-1/4)n C2nn.
 g(x,t)=g(-x,-t)
 Pn(x)=(-1)nPn(x). Even or odd function.
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Upper and Lower Bounds for
Pn(cosq) (請預讀P753~P754)
n
 If we can prove that Pn  cos q    am cos mq
m 0
 Pn  cos q  
 Proof:

 Pn  cos q  t n
n 0
n
a
m 0
m
1

am  0,
 Pn 1 .

1
1
1  2t cos q  t 2
1  teiq 1  te  iq
 
1
i p q q
  2 p  2 q C p2 p Cq2 q t p  q e   , if p  q  n,
p 0 q 0 2

n
1 2 p 2 n 2 p n i 2 p n q
C p Cn  p t e
2n
n 0 p 0 2
 
n
1
1
Pn  cos q    2 n C p2 p Cn2n p2 p ei 2 p n q  2 n
2
p 0 2
n
2 p 2 n 2 p
C
 p Cn p cos  n  2 p  q .
p 0
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Homework
 12.1.6 (11.1.6e)
 12.1.7 (11.1.7e)
 12.2.1 (11.2.1e)
 12.2.5 (11.2.5e)
 12.2.7 (11.2.7e)
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