Page 74 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3

General principles of quantum theory

3.1 Linear operators
The wave mechanics discussed in Chapter 2 is a linear theory. In order to
develop the theory in a more formal manner, we need to discuss the properties
^
of linear operators. An operator A is a mathematical entity that transforms a
function ø into another function ö
^
ö ˆ Aø (3:1)
Throughout this book a circum¯ex is used to denote operators. For example,
multiplying the function ø(x) by the variable x to give a new function ö(x)
may be regarded as operating on the function ø(x) with the operator ^ x, where ^ x
means multiply by x: ö(x) ˆ ^ xø(x) ˆ xø(x). Generally, when the operation is
simple multiplication, the circum¯ex on the operator is omitted. The operator
^
D x , de®ned as d=dx, acting on ø(x) gives the ®rst derivative of ø(x) with
respect to x, so that in this case
dø
^
ö ˆ D x ø ˆ
dx
^
The operator A may involve a more complex procedure, such as taking the
integral of ø with respect to x either implicitly or between a pair of limits.
^
The operator A is linear if it satis®es two criteria
^ ^ ^
A(ø 1 ‡ ø 2 ) ˆ Aø 1 ‡ Aø 2 (3:2a)
^
^
A(cø) ˆ cAø (3:2b)
where c is any complex constant. In the three examples given above, the
operators are linear. Some nonlinear operators are `exp' (take the exponential
2
of) and [ ] (take the square of), since

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