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Temperature-Dependent Rotational Energy Spectrum

​
k
B
T/hc
B
3
J
i
0
1
2
3
4
5
6
7
8
9
This Demonstration studies the pure rotational spectrum of the quantum rigid rotor problem (neglecting centrifugal distortion), described by the Hamiltonian
H
=
2
J
2I
, where
J
is the angular momentum operator and
I
is the moment of inertia. The energy levels are given by
E
J
=J(J+1)
B
, where
B
≡
h
8
2
π
Ic
is the rotational constant and the transition frequency between two adjacent levels is
′
ν
=ΔE=2(J+1)
B
(selection rules only allow transitions that satisfy
ΔJ=±1
). The left graphic shows the spectrum consisting of a series of equally spaced lines, where the relative intensity of each line depends on the probability to occupy the initial state in thermal equilibrium,
P(
J
i
)=N(2
J
i
+1)exp-
J
i
(
J
i
+1)
hc
B
k
B
T
, where
k
B
is the Boltzmann constant and
T
is the temperature. The left and right graphics show the spectrum for a given ratio
k
B
T/hc
B
and the energy level diagram, and the red highlighting shows the correspondence between a pair of energy levels and a specific transition frequency.
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