# Temperature-Dependent Rotational Energy Spectrum

Temperature-Dependent Rotational Energy Spectrum

This Demonstration studies the pure rotational spectrum of the quantum rigid rotor problem (neglecting centrifugal distortion), described by the Hamiltonian =, where is the angular momentum operator and is the moment of inertia. The energy levels are given by =J(J+1), where ≡ is the rotational constant and the transition frequency between two adjacent levels is =ΔE=2(J+1) (selection rules only allow transitions that satisfy ). 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, , where is the Boltzmann constant and is the temperature. The left and right graphics show the spectrum for a given ratio T/hc and the energy level diagram, and the red highlighting shows the correspondence between a pair of energy levels and a specific transition frequency.

H

2

J

2I

J

I

E

J

B

B

h

8Ic

2

π

′

ν

B

ΔJ=±1

P()=N(2+1)exp-(+1)T

J

i

J

i

J

i

J

i

hc

B

k

B

k

B

T

k

B

B