Page 273 - Essentials of physical chemistry

The Schrödinger Wave Equation 235

We can carry the analysis of the energy units further. Classically (sophomore physics), the
2 2 2 2 2 2
mv m v p d
h
kinetic energy would be ¼ ¼ so maybe, just maybe (?)
2 2m 2m 2m dx 2
op
d 2 d p ffiffiffiffiffiffiffi
h
2
2
h
p op and there is also a momentum operator p op where i ¼ 1? In this case
dx 2 i dx
the spatial variable is x but it is known from the physics subfield called mechanics that for every
momentum in a given coordinate system there is a corresponding coordinate; for every (p, q) pair
there is a q for each momentum in that coordinate as p q . Not every coordinate system will be (x, y, z)

h d
. Note that there is no special need to rewrite the potential
so the general condition is p q ¼
i dq
energy V as anything other than a multiplicative ‘‘operator.’’ The main action of the energy operator
is in the momentum operator. Maybe it is too soon to oversimplify wave mechanics, but the main
principle is indeed simple, just follow a few direct steps.
1. Write the total energy expression in terms of classical momenta and coordinates.

h d
2. Insert the equivalent operator wherever momentum occurs.
i dq
3. Consider any function of the coordinates such as the potential energy to be a simple
multiplicative operator.
4. Form the total energy operator, H op ¼ T þ V and write the Schrödinger equation as
2
p 2 op d 2
h
H op c ¼ E tot c where H op ¼ þ V(q) ¼ þ V(q):
2m 2m dq 2
5. Solve the differential equation by whatever means to find c and E tot , noting that there may
be a set of functions {c n } with corresponding eigenvalues {E n }.

It would all be so simple if step 5 really is easy to do. We need to simultaneously introduce you to
some simple techniques in solving certain types of differential equations while at the same time
solving an easy problem that has sufficient application to laboratory measurements to be realistic.
It is traditional to use the ‘‘particle-in-a-box’’ (PIB) problem for this purpose. To provide motivation,
let us consider the ultraviolet spectrum of all-trans polyenes, trans-butadiene will suffice. We
should know from organic chemistry that there is a principle of ‘‘sigma-pi-separability,’’ which
alerts us to the idea that the four electrons in 2P p orbitals are oriented in a plane perpendicular to
the plane of the atoms H 2 C ¼ CH–CH ¼ CH 2 , which contains all the 2P s bonds and the C1s
orbitals (Figure 11.2). Sigma-Pi separability is a good approximation because the 2P p orbitals are
odd functions with a node (sign change) in the plane of the atoms while the 2P s orbitals are even
functions with respect to reflection in the molecular plane and the product of an even and odd
function integrates over all space to zero. The sigma and pi orbitals are ‘‘orthogonal’’ in a first
approximation, although certainly there is some coulomb repulsion between the electrons and even a
strange phenomenon called ‘‘exchange’’ due to the fact that electrons are indistinguishable and can
occasionally trade places! Further, the 2P s orbitals are spatially more compact than the larger,
diffuse 2P p orbitals. Thus for several reasons, we consider the sigma bond skeleton of the molecule
to be ‘‘frozen’’ and screening all but þ4 nuclear charges with the electronically soft 2P p orbitals
forming linear combinations to hold the four pi electrons. Thus we consider a path along the trans
structure as a square box in which there are four electrons.
Another simplifying assumption is that these electrons suffer no mutual repulsion but do tend to
pair into two spin pairs (a, b) p1 and (a, b) p2 . We rely on previous explanations of spin pairing in