Calculation of reduced matrix elements - IPNL

•〈l a ‖Y (L) ‖l b 〉= ?
The spherical harmonics Y (L)
M
(ˆr) are the coordinate representation of the 2L+1 spherical
components of a tensor of rank L, the Wigner-Eckart theorem can be written (for example
for M= 0)
( )
〈l a m a |Y (L)
0 |l bm b 〉=(−1) l a−m a la L l b
〈l
-m a 0 m a ‖Y (L) ‖l b 〉. (6)
b
Furthermore using〈l b m b |ˆr〉=Y (l a)∗
m a
(ˆr) and〈ˆr|l a m a 〉=Y (l b)
m b
(ˆr) one has
∫
〈l a m a |Y (L)
0 |l bm b 〉= dˆr Y (l a)∗
m a
(ˆr)Y (L)
0 (ˆr)Y(l b)
m b
(ˆr)
where L andl b can be coupled tolusing
√
∑
Y (L)
(2L+1)(2lb + 1)(2l+1)
0 (ˆr)Y(l b)
m b
(ˆr)=
4π
so one has
∑
〈l a m a |Y (L)
0 |l bm b 〉=
lm
lm
√
(2L+1)(2lb + 1)(2l+1)
4π
The integral gives
∫ ∫
dˆr Y (l a)∗
m a
(ˆr)Y m
(l)∗ (ˆr)=(−1)m
so
〈l a m a |Y (L)
0 |l bm b 〉=(−1) −m a
Using the properties of the 3− j symbols
( )( )
L la l b L la l b
=
0 -m a m b 0 0 0
so
〈l a m a |Y (L)
0 |l bm b 〉=(−1) −m a
( )( )
L l lb L l lb
0 m m b 0 0 0
( )( )∫
L l lb L l lb
0 m m b 0 0 0
dˆr Y (l a)∗
m a
(ˆr)Y -m (l)∗ (ˆr)=(−1)m δ la lδ ma -m
√
(2L+1)(2la + 1)(2l b + 1)
4π
√
(2L+1)(2la + 1)(2l b + 1)
4π
Y m
(l)∗ (ˆr)
dˆr Y (l a)∗
m a
( )( )
L la l b L la l b
.
0 -m a m b 0 0 0
( )( )
la L l b la L l b
δ
-m a 0 m b 0 0 0
ma m b
Using eq. (6) and eq. (7) we obtain
√
〈l a ‖Y (L) ‖l b 〉=(−1) l (2L+1)(2la + 1)(2l
a
b + 1)
4π
(ˆr)Y (l)∗ (ˆr).
( )( )
la L l b la L l b
δ
-m a 0 m b 0 0 0
ma m b
.
(7)
( )
la L l b
.
0 0 0
m
3