228 3 Quantum Mechanics – II
(b) First we show that the wavefunction is normalized.
∫
|ψ|^2 dr=1
4 π∫∞
0|g(r)|^2 r^2 dr∫π0dθ∫ 2 π0(1+cosφsin 2θ)sinθdφ=
1
2
∫π0sinθdθ= 1The probability for the occurrence ofLz=is
∫ [√
1
3(
−
√
2
)
Y 11
] 2
dΩ=(2/3)∫
(3/ 8 π)sin^2 θ. 2 πsinθdθ=(1/2)
∫+ 1
− 1(1−cos^2 θ)dcosθ= 2 / 3The probability for the occurrence ofLz=0is
∫ [√
1
3Y 10
] 2
dΩ=∫+ 1
− 1(
3
4 π)
. 2 πcos^2 θdcosθ= 1 / 3
3.96 (a)[Jz,J+]=JzJ+−J+Jz=Jz(Jx+iJy)−(Jx+iJy)Jz
=JzJx−JxJz+i(JzJy−JyJz)
=[Jz,Jx]+i[Jz,Jy]=iJy−iiJx
=iJy+Jx=(Jx+iJy)=J+
=J+,in units of.
(b) From (a),JzJ+=J+Jz+J+
JzJ+|jm>=J+Jz|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>=Cjm+|j,m+ 1 >
Given a state|jm>, the state|j,m+ 1 >must exist unlessCjm+van-
ishes for that particularm. Sincejis the maximum value of m by definition.
There can not be a state|j,j+ 1 >, i.e.Cjj+must vanish.J+is known as
the ladder operator. Similarly,J−lowers m by one unit.
(c)J+=Jx+iJy
J−=Jx−iJy
Therefore,Jx=1
2
(J++J−)=
⎛
⎝
01 /
√
20
1 /
√
201 /
√
2
01 /
√
20
⎞
⎠
Jy=1
2 i(J+−J−)=
⎛
⎝
0 −i/√
20
i/√
20 −i/√
2
0 i/√
20
⎞
⎠
[Jx,Jy]=JxJy−JyJx=i⎛
⎝
10 0
00 0
00 − 1
⎞
⎠=iJz