ABSTRACT
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:
d2
dx2
sin (ax) ¼ d
dx
[a cos (ax)] ¼ (a2) sin (ax):
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 successful 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).
