Download Angle Matrix Elements

Physics 601, Angle Matrix elements
It has been brought to my attention that the basic equation
k
4π X ∗
Pk (cos θ12 ) =
Y (r̂1 )Ykm (r̂2 )
2k + 1 m=−k km
(1)
where r̂ denotes the set of coordinates θ, φ giving the direction of the unit vector r̂ = r/r is
not well known. The version given here is from Jackson, Classical Electrodynamics, 2nd Ed.,
Eq. (3.70). The whole point of this formula is to separate the variables of particle 1 from
the variables of particle 2. Once that has been done it is easy to evaluate angular matrix
elements. That is, we need to find
< `1 m1 `2 m2 |Pk (cos θ12 )|`01 m01 `02 m02 >
(2)
Our author did this for the special case when two of the `’s were zero. The general case
is given by
<
`1 m1 `2 m2 |Pk (cos θ12 |`01 m01 `02 m02
k
4π X
∗
< `1 m1 |Ykm
(r̂1 )|`01 m01 >
> =
2k + 1 m=−k
× < `2 m2 |Ykm (r̂2 )|`02 m02 >
(3)
Using Eq. (4.6.3) of the book by Edmonds, Angular Momentum in Quantum Mechanics,
namely,
< `1 m1 |Ykm (r̂1 )|`01 m01 >= (−1)m1
0
(2`1 + 1)(2`1 + 1)(2k + 1)
4π
1/2


`1 k
0
0
`01
0


`1
k
−m1 m m01
one can write
< `1 m1 `2 m2 | Pk (cos θ12 )|`01 m01 `02 m02 >= [(2`1 + 1)(2`2 + 1)(2`01 + 1)(2`02 + 1)]1/2 (−1)m1 +m2 +m





0
0
0
0
`1 k `1
`
k
`1
` k `2
`
k `2
 1
 2
 2

× 
(5)
0 0 0
−m1 −m m01
0 0 0
−m2 m m02
where the


j1
j2
j3
m1 m2 m3
1


`01
(6)

(4)
are called three-j symbols. They are available in Mathematica, but one can also get them
by a search of the internet. In any case, just treat them as another integral that you look
up in tables. The symbol vanishes unless m1 + m2 − m03 = 0 and |j1 − j3 | ≤ j2 ≤ j1 + j3
Some special cases that occur frequently are


j−m
j
j 0

 = (−1)
√
2j + 1
m −m 0
(7)
It follows that
< `1 m1 |Ykm (r̂1 )|00 >= δk`1 δmm1
(8)
The three-j symbols are unchanged under an even permutation of the columns, and are
multiplied by (−1)j1 +j2 +j3 under an odd permutation of the columns.
2