228
3 Quantum Mechanics – II
(b) First we show that the wavefunction is normalized.
|ψ|
2 dr =
1
4π
∞
0
|g(r )|
2 r
2 dr
π
0
dθ
2π
0
(1 + cos ϕ sin 2θ ) sin θdϕ
=
1
2
π
0
sin θdθ = 1
The probability for the occurrence of L z = is
1
3
−
√
2
Y 11
2
dΩ = (2/3)
(3/8π) sin
2
θ.2π sin θ dθ
= (1/2)
+1
−1
(1 − cos
2
θ)d cos θ = 2/3
The probability for the occurrence of L z = 0 is
1
3
Y 10
2
dΩ =
+1
−1
3
4π
.2π cos
2
θ d cos θ = 1/3
3.96 (a) [J z , J + ] = J z J + − J + J z = J z (J x + iJ y ) − (J x + iJ y )J z
= J z J x − J x J z + i(J z J y − J y J z )
= [J z , J x ] + i[J z , J y ] = iJ y − ii J x
= iJ y + J x = (J x + iJ y ) = J +
= J + , in units of .
(b) From (a), J z J + = J + J z + J +
J z J + |j m >= J + J z |jm > +J + |jm >
= J + m|jm > +J + |jm >
= (m + 1)J + |jm >
J + |jm > is nothing but | j, m + 1 > apart from a possible normalization
constant. Thus
J + |jm >= C jm
+ | j, m + 1 >
Given a state |jm >, the state | j, m + 1 > must exist unless C jm
+ vanishes for that particular m. Since j is the maximum value of m by definition.
There can not be a state | j, j + 1 >, i.e. C jj
+ must vanish. J + is known as
the ladder operator. Similarly, J − lowers m by one unit.
(c) J + = J x + iJ y
J − = J x − iJ y
Therefore, J x =
1
2
(J + + J − ) =
⎛
⎝
0 1/
√
2 0
1/
√
2 0 1/
√
2
0 1/
√
2 0
⎞
⎠
J y =
1
2i
(J + − J − ) =
⎛
⎝
0 −i/
√
2
0
i/
√
2
0
−i/
√
2
0
i/
√
2
0
⎞
⎠
[J x , J y ] = J x J y − J y J x = i
⎛
⎝
1 0 0
0 0 0
0 0 −1
⎞
⎠ = iJ z