the wavefunction Ψ 10 shown in figure d on p. 923, for`= 1 and
`z= 0. (The odd value of`is possible for a rotor that doesn’t have
mirror symmetry, e.g., the carbon monoxide molecule CO.) The
ratio of the correct quantum mechanical energy to the semiclassical
one is`(`+ 1)/`^2 = 2, and the factor of two makes sense because
at the poles, the wave has equal contributions to its kinetic energy
due to oscillations in the two perpendicular directions that occur in
the Laplacian∂^2 /∂x^2 +∂^2 /∂y^2.
Discussion Question
A The correction of the semiclassical proportionality for the energy of
a rotor from`^2 to`(`+ 1) is effectively the addition of a correction equal
to`. What if someone tells you that there is an additional correction term
that depends only on`z(for a fixed`)? Is this plausible?
B Can the correction`^2 →`(`+ 1) be tested experimentally by mea-
suring the energy of a spinning steel ring in the laboratory? Can the
correctionn→n+ 1/2 be tested using a cart on an air track that vibrates
back and forth between two springs?14.3 ?A tiny bit of linear algebra
This optional section is a self-contained presentation of a very small
amount of linear algebra. None of the later physics requires this
material, but reading it may be helpful as review for the reader who
has already had an entire linear algebra course, or to help make
connections for the one who is taking such a course concurrently or
will take it in the future.
Avector space is a set of objects, which we refer to as vectors,
along with operations of addition and scalar multiplication defined
on the vectors. The scalars may be the real numbers or the com-
plex numbers. We require that the addition and scalar multiplica-
tion operations have the properties that addition is commutative
(u+v=v+u), that we have an additive identity 0 and additive
inverses (v+ (−v) = 0), and that both operations are associative
and distributive in the ways that we would expect from the no-
tation. The prototypical example of a vector space is vectors in
three-dimensional space, with the scalars being the real numbers.
The vector space of polynomials example 1
Consider the set of all polynomials. If we define addition of poly-
nomials and multiplication of a polynomial by a real number in the
obvious ways, then these functions are a vector space. Note that
there is no well-defined division operation, since dividing a poly-
nomial by a polynomial typically does not give a polynomial.
In quantum mechanics, we are interested in the vector space of
wavefunctions, with the scalars being the complex numbers.
A set of vectors is said to belinearly independentif it is not
possible to form the zero vector as a linear combination of them.964 Chapter 14 Additional Topics in Quantum Physics