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.