Page 77 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

68 General principles of quantum theory
^ 2
Another example is the operator D acting on either sin nx or cos nx, where
x
n is a positive integer (n > 1), for which we obtain
^ 2 x 2
D sin nx ˆÿn sin nx
2
2
^
D cos nx ˆÿn cos nx
x
^ 2
The functions sin nx and cos nx are eigenfunctions of D with eigenvalues
x
2
ÿn . Although there are an in®nite number of eigenfunctions in this example,
these eigenfunctions form a discrete, rather than a continuous, set.
In order that the eigenfunctions ø i have physical signi®cance in their
application to quantum theory, they are chosen from a special class of func-
tions, namely, those which are continuous, have continuous derivatives, are
single-valued, and are square integrable. We refer to functions with these
properties as well-behaved functions. Throughout this book we implicitly
assume that all functions are well-behaved.

Scalar product and orthogonality
The scalar product of two functions ø(x) and ö(x) is de®ned as

1

ö (x)ø(x)dx
ÿ1
For functions of the three cartesian coordinates x, y, z, the scalar product of
ø(x, y, z) and ö(x, y, z)is

1

ö (x, y, z)ø(x, y, z)dx dy dz
ÿ1
For the functions ø(r, è, j) and ö(r, è, j) of the spherical coordinates r, è,
j, the scalar product is

2ð ð 1
2
ö (r, è, j)ø(r, è, j)r sin è dr dè dj

0 0 0
„
In order to express equations in general terms, we adopt the notation dô to
indicate integration over the full range of all the coordinates of the system
being considered and write the scalar product in the form

ö ø dô
For further convenience we also introduce a notation devised by Dirac and
write the scalar product of ø and ö as hö j øi, so that

hö j øi ö ø dô

The signi®cance of this notation is discussed in Section 3.6. From the de®nition
of the scalar product and of the notation hö j øi, we note that