Note that only even multiples of~are observed in collective
rotation in figure d. This is because the nucleus’s shape has an
additional mirror symmetry, so that it is unaffected by a 180-degree
rotation. This means that the wavefunction describing the collective
rotation must oscillate twice as we pass through a full rotation.14.2.4 Corrections to semiclassical energies
So far we’ve been using the correspondence principle to make
educated guesses about quantum-mechanical expressions for the en-
ergies of vibrators and rotors. This style of reasoning is called
semiclassical, because it combines ideas from classical and quantum
physics. These expressions are guaranteed to be good approxima-
tions in the classical limit obtained when the quantum numbers are
large, but figure e shows that the approximations can be poorer
when the quantum numbers are small.e/Quantum-mechanical correc-
tions to the semiclassical results
for the energy of a vibrator and
a rotor. The rotational levels are
shown for the case of a rotor
with mirror symmetry, so that only
even values of`occur.
In the case of thenth excited state of a vibrator, the energy is
(n+ 1/2)~ω, where the +1/2 term represents a quantum correction
to the semiclassical approximation. This shifts the entire ladder up-
ward in energy by half a step. In particular, the energy of the ground
state is not zero but rather (1/2)~ω. This can be verified quantita-
tively by calculating the energy for the solution to the Schr ̈odinger
discussed using the guess-and-check method in problem 23, p. 944.
It is easy to see why the answer cannot be zero, for if it were, then
the particle in the ground state would have zero kinetic energy and
zero potential energy. To have zero kinetic energy, it would have
to have a momentum of exactly zero, so ∆p= 0, but to have zero
potential energy it would also have to sit still at exactly the equi-962 Chapter 14 Additional Topics in Quantum Physics