Elements of Quantum Theory
91
while its square is given by
L
2
= (r × p) ⋅ (r × p)
(3.132)
Here, since r and p are operators, their order must be maintained. The
expression for L
2
simplifies to
L
2
= r. (p × (r × p))
(3.133)
=
,
[
]
i j i j
i j j i
i j
r p r p ri r p
−
∑
and with
r.p = i r r
∂
− ∂
Eq. (3.133) becomes
L
2
= r
2
p
2
+
2
2
22
2
2
∂
∂
+
∂
∂
r
r
r
r
(3.134)
Using this relation the kinetic energy T can be written as
T =
2
1 p
2m
=
2
2
2
2
2
L
2
2
∂
∂
−
+
∂
∂
r
r
r
mr
mr
(3.135)
which shows that the angular momentum is an important term in the kinetic
energy.
The expressions for the angular momentum operators in terms of spherical
coordinates are obtained from Eq. (3.131) as
L x =
sin
cos cot
i
∂
∂


φ
+
φ
θ


∂θ
∂φ


L y =
cos
sin cot
i
∂
∂


−
φ
−
φ
θ


∂θ
∂φ


(3.136)
L z = i
∂
− ∂φ
and
L
2
=
2
2
2
1
1
sin
sin
sin
i


∂
∂
∂


−
θ
+




θ ∂θ
∂θ


θ ∂φ


(3.137)
The wave functions corresponding to well-defined values of L
2
, satisfy the
equation
2
2
2
1
1
sin
( , )
sin
sin
i
Y


∂
∂
∂


−
θ
+
θφ




θ ∂θ
∂θ


θ ∂φ


= λY (θ, φ) (3.138)