Page 79 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

70 General principles of quantum theory

For illustration, we consider some examples involving only one variable,
namely, the cartesian coordinate x, for which w(x) ˆ 1. An operator that results
in multiplying by a real function f (x) is hermitian, since in this case

f (x) ˆ f (x) and equation (3.8) is an identity. Likewise, the momentum
operator ^ p ˆ ("=i)(d=dx), which was introduced in Section 2.3, is hermitian
since

1 1 " 1 " 1 dø
" dø i j
ø ^ pø i dx ˆ ø j dx ˆ ø ø i ÿ ø i dx
j
j
ÿ1 ÿ1 i dx i ÿ1 i ÿ1 dx
The integrated part is zero if the functions ø i vanish at in®nity, which they
„

must in order to be well-behaved. The remaining integral is ø i ^ p ø dx,so
j
that we have

1 1

ø ^ pø i dx ˆ ø i (^ pø j ) dx
j
ÿ1 ÿ1
The imaginary unit i contained in the operator ^ p is essential for the
^
hermitian character of that operator. The operator D x ˆ d=dx is not hermitian
because

1 dø i 1 dø
ø j dx ˆÿ ø i j dx (3:10)
ÿ1 dx ÿ1 dx
where again the integrated part vanishes. The negative sign on the right-hand
side of equation (3.10) indicates that the operator is not hermitian. The operator
^ 2 x
D , however, is hermitian.
The hermitian character of an operator depends not only on the operator
itself, but also on the functions on which it acts and on the range of integration.
An operator may be hermitian with respect to one set of functions, but not with
respect to another set. It may be hermitian with respect to a set of functions
de®ned over one range of variables, but not with respect to the same set over a
different range. For example, the hermiticity of the momentum operator ^ p is
dependent on the vanishing of the functions ø i at in®nity.
The product of two hermitian operators may or may not be hermitian.
^
^
^^
Consider the product AB where A and B are separately hermitian with respect
to a set of functions ø i , so that
^^ ^ ^ ^ ^
hø j j ABø i iˆhAø j j Bø i iˆhBAø j j ø i i (3:11)
^
^
where we have assumed that the functions Aø i and Bø i also lie in the hermitian
^
^
^^
^
^
domain of A and B. The product AB is hermitian if, and only if, A and B
^
commute. Using the same procedure, one can easily demonstrate that if A and
^ ^
^
^ ^
^^
B do not commute, then the operators (AB ‡ BA) and i[A, B] are hermitian.
^
^
^^
By setting B equal to A in the product AB in equation (3.11), we see that the
square of a hermitian operator is hermitian. This result can be generalized to