HAUSAUFGABE 5
(1) Consider two angular momenta j1 = 1 and j2 = 1 to form states with
j = 2, 1, and 0. Using either the ladder operator method or the recursion
relation, express at least two of the 9 possible {j, m} eigenkets in terms of
the |j1 j2 ; m1 m2 ! states.
(2) Consider a system with j = 1. Explicitly write the matrix representation
of the Jˆy operator in 3 × 3 matrix form. Note that the eigenstates are
defined in the z-basis.
!j
= 1, m" |Jˆy |j = 1, m!
Show that only for j = 1 it is legitimate to replace e−
1 − i(
by
Jy
Jy
)sinβ − ( )2 (1 − cosβ).
!
!
Use this relation to calculate dj (β) = $j = 1, m" |e−
iJy β
!
(3) Consider a sherical tensor of rank 1 (that is a vector)
(1)
iJy β
!
V± = ∓
Vx ± iVy
(1)
√
, V0 = Vz .
2
Use the result for d(j=1) from exercise (2) evaluate
! (1)
(1)
dqq! (β) Vq! .
q!
1
|j = 1, m!.