Atomic Structure Theory

26 1 Angular Momentum
The following three relations may also be proven without difficulty:
Y (−1)
JM (θ, φ) =ˆr YJM(θ, φ), (1.140)
Y (0)
JM (θ, φ) =
Y (1)
JM (θ, φ) =
1
J(J +1) L YJM(θ, φ), (1.141)
r
J(J +1) ∇ YJM(θ, φ) . (1.142)
These equations are essential in evaluating matrix elements of the vector operators
r, L and p between atomic states.
Problems
1.1. Derive the relations
J 2 = J+J− + J 2 z − Jz ,
J 2 = J−J+ + J 2 z + Jz .
1.2. Show that the normalization factor c in the equation Θl,−l(θ) =c sin l θ
is
c = 1
2l
(2l + 1)!
,
l! 2
and, thereby, verify that (1.30) is correct.
1.3. Write a maple or mathematica program to obtain the first 10 Legendre
polynomials using Rodrigues’ formula.
1.4. Legendre polynomials satisfy the recurrence relation
lPl(x) =(2l−1)xPl−1(x) − (l − 1)Pl−2(x).
Write a maple or mathematica program to determine P2(x) through P10(x)
(starting with P0(x) =1andP1(x) =x) using the above recurrence relation.
1.5. Write a maple or mathematica program to generate the associated
Legendre functions and P m
m
l (x). Determine all Pl (x) withl≤4and1≤m≤ l.
1.6. The first two spherical Bessel functions are:
sin x
j0(x) =
x ,
sin x cos x
j1(x) = −
x2 x .
Spherical Bessel functions satisfy the recurrence relation
(2n +1)
jn+1(x)+jn−1(x) = jn(x).
x
Use maple or mathematica to obtain an expression for j6(x).