Page 271 - Essentials of physical chemistry

11 The Schrödinger Wave

Equation

INTRODUCTION
This chapter is primarily about ‘‘wave mechanics’’ since that is the most convenient way to
introduce undergraduates to quantum mechanics using calculus. An equivalent form called ‘‘matrix
mechanics’’ will be discussed briefly in a later chapter. Consider again the 1923 paper by De Broglie
and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that
if there really is some ‘‘wave’’ that describes the behavior of particles, then is there an equation that
the wave obeys? Even today it is difficult to say what the ‘‘wave’’ is, but it may help to find an
equation it obeys. Step back a moment to some basic calculus:
2
d d 2
[a cos (ax)] ¼ ( a ) sin (ax):
sin (ax) ¼
dx 2 dx
Perhaps you did not notice the pattern before but we can put this into a general form as

(Operator)(Eigenfunction) ¼ (Eigenvalue)(Eigenfunction):

The word ‘‘eigen’’ in German means ‘‘characteristic, unique, peculiar, special . . . ’’ and only certain
functions satisfy this condition called an eigenfunction equation. An analogy that has been suc-
cessful in explaining this to undergraduates is to consider an apple tree with ripe apples on it. If you
hit the branches with a stout stick some apples will fall off the tree but the tree will still be there. The
operator is the act or operation of hitting the tree with the stick, the tree is the eigenfunction and the
apples are the eigenvalue(s). The eigen word comes from German because this relationship was first
linked to the De Broglie wave idea by Erwin Schrödinger in 1926 in a series of papers that are
among the most important in modern science [1]. Schrödinger (1887–1961) was an Austrian
physicist who received the Nobel Prize for his work in 1933 (Figure 11.1).
We will now present a derivation of the Schrödinger equation that may not be the way he thought
of it but that follows from limited use of calculus and simple algebra. Since De Broglie implied there
is some sort of invisible, untouchable, mathematical pilot wave accompanying the motion of
particles, we assume the general form of such a wave and relate it to its own second derivative.
We will use c since it is universally used for the wave function.
2 2 2
2px d c d 2px 4p 2px
Let c ¼ A sin . Then ¼ A sin ¼ A sin . That would be
l dx 2 dx 2 l l 2 l
h
, so we substitute l DB .
mv
true for any wave but we want to apply it to a ‘‘matter wave’’ using l ¼
" #
2 2 2 2
d 2px 4p 2px (mv) 2px h 2
h
A sin ¼ A sin ¼ A sin . Note that ¼ .
dx 2 l h 2 l h 2 l 4p 2
mv
mv 2 (mv) 2
þ V ¼ T þ V ¼ H op where H op ¼ T þ V
One more step is needed. Recall E tot ¼ þ V ¼
2 2m
is the total energy operator with T as the conventional symbol for the kinetic energy (often K in
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