3.3 Solutions
219
Apart from the factor 1/r
2 , the angular part is seen to be
∂
2
ψ
∂θ 2 + cot θ
∂ψ
∂θ
+
1
sin
2
∂
2
ψ
∂ϕ 2
3.84 (a) The (i, j)th matrix element of an operator O is defined by
O ij =< i|o| j >
(1)
For j = 1/2, m = 1/2 and −1/2. The two states are
|1 >= |
1
2
,
1
2
> and |2 >= |
1
2
, −
1
2
>
(2)
With the notation | j, m >
Now
< j
m
|J z |jm >= mδ jj
δ mm
(3)
Thus
(J z ) 11 =< 1|J z |1 >=
1
2
(4)
(J z ) 22 =< 2|J z |2 >= −
1
2
(5)
(J z ) 12 =< 1|J z |2 >=
1
2
,
1
2
|J z |
1
2
,
1
2
= 0
( 6 )
because of (3).
Similarly
(J z ) 21 = 0
( 7 )
Therefore
J z =
2
1 0
0 −1
(8)
For J x and J y , we use the relations
J x =
1
/ 2 (J + + J − ) and J y = −
1
2i
(J + − J − )
< j, m|J x | j, m >=< j, m|
1
2
(J + + J − )| j, m >
=
1
/ 2 [( j + m + 1)( j − m)]
1/2
< j, m
| j, m + 1 >
+
1
/ 2 [( j − m + 1)( j + m)]
1/2
< j, m
| j, m − 1 >
=
1
/ 2 [( j + m + 1)( j − m)]
1/2
δ m ,m+1
+
1
/ 2 [( j − m + 1)( j + m)]
1 / 2 δ m ,m−1
That is the matrix element is zero unless m
= m + 1 or m
= m − 1.
The first delta factor survives