130_notes.dvi

27.6 Rotational States

Molecules can rotate like classical rigid bodies subject to the constraint that angular momentum is
quantized in units of ̄h. We can estimate the energy of these rotations to be

Erot=

1

2

L^2

I

=

ℓ(ℓ+ 1) ̄h^2

2 I

≈

̄h^2

2 Ma^20

=

m

M

α^2 mc^2

2

≈

m

M

E≈

1

1000

eV

where we have useda 0 =αmc ̄h. These states are strongly excited at room temperature.

Let’s look at the energy changes between states as we might get in aradiative transition with
∆ℓ= 1..

E=

ℓ(ℓ+ 1) ̄h^2

2 I

∆E=

̄h^2

2 I

[ℓ(ℓ+ 1)−(ℓ−1)ℓ] =

̄h^2

2 I

(2ℓ) =

̄h^2 ℓ

I

These also have equal energy steps in emitted photon energy.

With identical nuclei,ℓis required to be even for (nuclear) spin singlet and odd for triplet. This
means steps will be larger.

A complex molecule will have three principle axes, and hence, three moments of inertia to use in
our quantized formula.

Counting degrees of freedom, which should be equal to the numberof quantum numbers needed
to describe the state, we have 3 coordinates to give the position ofthe center of mass, 3 for the
rotational state, and 3N-6 for vibrational. This formula should be modified if the molecule is too
simple to have three principle axes.

The graph below shows the absorption coefficient of water for light of various energies. For low
energies, rotational and vibrational states cause the absorption of light. At higher energies, electronic
excitation and photoelectric effect take over. It is only in the regionaround the visible spectrum
that water transmits light well. Can you think of a reason for that?