Page 287 - Essentials of physical chemistry

The Schrödinger Wave Equation 249

Theorem 3 A single set of eigenfunctions can exist for two different Hermitian operators if
the operators commute.

Proof: Assume there really are two operators, A and B, which have the same eigenfunction c so
that Ac ¼ ac and Bc ¼ bc and that [A, B] ¼ 0. Again we set up the integral <cj[A, B]jc>.
ð ð ð ð
0 ¼ c*(0)cdt ¼ c*[A, B]cdt ¼ c*(AB BA)cdt ¼ (a*b* ba) c*cdt ¼ 0, Q.E.D.

This is an ‘‘existence’’ theorem, it means that there can be such a c or {c n } but even if we show the
two operators commute it does not help find the set {c n }. However, the most important result of this
theorem is that if two operators commute (not all do!) then we can ‘‘know’’ their eigenvalues or
expectation values simultaneously. For two operators that do not commute we may be able to find
the observable for one, while the other is not completely defined or vice versa. We can mention
that the angular momentum operators (Lx, Ly, Lz) do not commute and a given set of eigenfunctions
for the Lz operator may not give a clear interpretation of the eigenvalues for (Lx, Ly) using the
eigenfunctions of Lz. We delay further discussion of this problem until the chapter on the H atom.
The main use of this third theorem is to use the eigenfunctions of some particular operator, which
commutes with the operator we are interested in to evaluate the expectation value in the known set
of eigenfunctions. We will see that there are some blind alleys in quantum mechanics and we often
have to use any trick we can think of to evaluate what we want to know, even if it is a roundabout
approach and for those cases this third theorem can be useful.

SUMMARY

In this chapter we explored the question of how to use the De Broglie matter waves to describe
molecular phenomena.

1. The (time-independent) Schrödinger wave equation Hc ¼ Ec was derived from the second
h h
into the second
mv p
derivative of an arbitrary wave function by incorporating l ¼ ¼
derivative expression of the wave function. The characteristics of an eigenvalue equation
were noted.
2. It was noted that there is a total energy operator, H, called the Hamiltonian operator, which
can be set up by writing the kinetic energy operator in terms of momentum and the
potential energy without change except to regard it as a multiplicative operator. The key
to converting classical physics formulas into quantum mechanical operators is merely to

h d h
h .
use p op ¼ where q is the coordinate corresponding to the momentum and ¼
i dq 2p
Then all one has to do is to solve the differential equation Hc ¼ Ec to find the wave
function c and the energy E. In principle, the wave function c contains all the available
information about a given system and c*c can be interpreted as a probability. Physical
quantities are described by ‘‘operators’’ as calculus functions, which can be applied to the
wave function to obtain the quantized values of physical observables.
3. The problem of noninteracting particles trapped in a one-dimensional ‘‘box’’ with infinitely
r ffiffiffi
2 2
n h 2 npx
n
high walls was solved to find the results E n ¼ 2 and c ¼ sin . The n ¼ 0
8mL L L
level is a trivial solution for ‘‘no particle’’ because sin(0) ¼ 0 and the energy levels are non-
degenerate with quadratically increasing spacing in n. However, the actual spacing
between the levels can be quite small as for the quantized translational energy of a gas
molecule.