4.3 Angular Momentum
83
2E = I a ω
2
a + I b ω
2
b + I c ω
2
c
P
2
= I
2
a ω
2
a + I
2
b ω
2
b + I
2
c ω
2
c
(4.15)
or
2E = 2H =
P
2
a
I a
+
P
2
b
I b
+
P
2
c
I c
P
2
= P
2
a + P
2
b + P
2
c
(4.16)
4.4 Rotational Hamiltonian of the Rigid Rotor
In quantum mechanics, (4.16) remains valid, but the physical magnitudes are replaced
by operators. The operators P g contain the Planck’s constant = h
2π . In spectroscopy, it is common to define the Hamiltonian in units of frequency (i.e., divided
by h), so the transformed Hamiltonian may be written
H =
h
8π 2
P
2
a
I a
+
P
2
b
I b
+
P
2
c
I c
= AP
2
a + BP
2
b + CP
2
c
(4.17)
with the rotational constants
A =
h
8π 2 I a
≥ B =
h
8π 2 I b
≥ C =
h
8π 2 I c
(4.18)
The conversion factor may be calculated
h
8π 2 = I a (uÅ
2
) · A(MHz) = 505,379.005(50)
(4.19)
The matrix elements of the angular momentum operators may be found in the
courses of quantum mechanics. For a symmetric top rotor with z as axis of symmetry,
the diagonal elements are (Gordy and Cook 1984)
J K M|P
2
z |J K M = K
2
2
(4.20a)
J K M|P
2
x |J K M = J K M|P
2
y |J K M =
2
2
J (J + 1) − K
2
(4.20b)
and, using P
2
= P
2
x + P
2
y + P
2
z
J K M|P
2
|J K M =
2 J (J + 1)
(4.21a)
and the off-diagonal elements are