p z , y
Â
à j ψi ¼
ħ
i
∂
∂z
y À y
∂
∂z
j ψi ¼
ħ
i
y
∂jψi
∂z
À y
∂jψi
∂z
¼ 0:
Since jψi is arbitrarily chosen, this relation implies that p z and y commute. We
obtain similar relations regarding L y
2 and L z
2 as well. Thus, we have
L
2
¼ L x
2
þ L y
2
þ L z
2
¼ y
2 p z
2
þ z
2 p y
2
þ z
2 p x
2
þ x
2 p z
2
þ x
2 p y
2
þ y
2 p x
2
À
Á
þ x
2 p x
2
À x
2 p x
2
þ y
2 p y
2
À y
2 p y
2
þ z
2 p z
2
À z
2 p z
2
À
Á
À yzp z p y þ zyp y p z þ zxp x p z þ xzp z p x þ xyp y p x þ yxp x p y
À
Á
þiħ yp y þ zp z þ zp z þ xp x þ xp x þ yp y
À
Á
¼ y
2 p z
2
þ z
2 p y
2
þ z
2 p x
2
þ x
2 p z
2
þ x
2 p y
2
þ y
2 p x
2
À
þx
2 p x
2
þ y
2 p y
2
þ z
2 p z
2
Þ À x
2 p x
2
þ y
2 p y
2
þ z
2 p z
2
À
þyzp z p y þ zyp y p z þ zxp x p z þ xzp z p x þ xyp y p x þ yxp x p y Þ
þiħ yp y þ zp z þ zp z þ xp x þ xp x þ yp y
À
Á
¼ r
2
Á p
2
À r r Á p
ð
ÞÁp þ 2iħ r Á p
ð
Þ:
ð3:7Þ
In (3.7), we are able to ease the calculations by virtue of putting a term
(x
2 p x
2
À x
2 p x
2 + y
2 p y
2
À y
2 p y
2 + z
2 p z
2
À z
2 p z
2 ). As a result, for the second term
after the second to the last equality we have
À x
2 p x
2
þ y
2 p y
2
þ z
2 p z
2
À
þyzp z p y þ zyp y p z þ zxp x p z þ xzp z p x þ xyp y p x þ yxp x p y Þ
¼ À x xp x þ yp y þ zp z
À
Á
p x þ y xp x þ yp y þ zp z
À
Á
p y þ z xp x þ yp y þ zp z
À
Á
p z
Â
Ã
¼ Àr r Á p
ð
ÞÁp:
The calculations of r
2
Á p
2 [the first term of (3.7)] and r Á p (in the third term) are
straightforward.
In a spherical coordinate, momentum p is expressed as
p = p r e
r
ð Þ
þ p θ e
θ
ð Þ
þ p ϕ e
ϕ
ð Þ ,
ð3:8Þ
where p r , p θ , and p ϕ are components of p; e
(r) , e
(θ)
, and e
(ϕ) are orthonormal basis
vectors of ℝ
3 in the direction of increasing r, θ, and ϕ, respectively (see Fig. 3.1). In
60
3 Hydrogen-Like Atoms