c/Energy levels of the N 2
molecule.d/Excited states of the nu-
cleus^178 Hf. Black squares
represent states that are inter-
preted as end-over-end rotation,
while white diamonds show parti-
cle excitations. For each angular
momentum, the graph shows the
lowest-energy state of each type,
where known.By the correspondence principle, we expect that when the quantum
mechanical version of such a system is highly excited, it should emit
a large number of photons of this frequencyf, so that the discrete
quantum jumps are undetectable and the radiation appears as a
classical wave. We can thus infer that for a quantum vibrator, the
excited states should show anevenly spacedladder of energy levels.
Figure c shows how the series of red lines in figure a arises. For
an excitation consisting only of vibrational motion, we expect based
on the correspondence principle to see the evenly spaced ladder of
states shown in a stack built above the ground state, with all of the
photons having the same energy. These states and transitions do
exist, but the light lies in the infrared spectrum and so is not seen
in figure a. The red visible-light lines arise as shown in the second
box, from states that involve both a certain particle excitation and
some vibration. Because the spacing of the two ladders is slightly
unequal, the red lines all have slightly different wavelengths.
14.2.3 Rotation
What about rotation? An interesting thing happens here due
to the structure of quantum mechanics. Quantum mechanics can
describe motion only as a wave, with the value of the wave oscillat-
ing from one place to another. But this implies that according to
quantum mechanics, no object can rotate about one of its axes of
symmetry, for the rotated version of a state would then be the same
state. This is why rotational excitations are never seen in individ-
ual atoms, or in nuclei that have spherical shapes. In examples like
the ones in figure b, which have a single axis of symmetry, we can
therefore have end-over-end rotation, but never rotation about the
symmetry axis. Such end-over-end rotational states are observed in
N 2 , for example, but because this involves large motions by the high-
mass nuclei, the moment of inertiaIis quite large, and therefore the
rotational energies — classically,K=L^2 / 2 I— are very small, and
infrared rather than visible photons are emitted. If rotation about
the symmetry axis were possible, then the moment of inertia would
be thousands of times smaller, because in such a rotation the nu-
clei would not move. The energies involved would be thousands of
times higher, and the photons would lie approximately in the visible
region of the spectrum. No such visible lines are actually observed.
Perhaps more vivid evidence for the nonexistence of rotation
about a symmetry axis is shown in figure d. The states involving
end-over-end rotation of the nucleus as a whole (“collective” rota-
tion) are approximately a parabola on this graph, which is reason-
able given the classical relationK=L^2 / 2 I. But angular momentum
cannot be generated along the symmetry axis through collective ro-
tation. Instead, we see an irregular set of energy levels in which first
one particle (forL≤ 8 ~) and then two (14 and 16~) are excited.
Section 14.2 Rotation and vibration 961