Lecture Four

Properties of the wave Function:
To be an acceptable solution,must be satisfying the following the restrictions for all values of x:

1- must be finite.
2- must be continuous.


Operator:

An operator, O (say), is a mathematical entity which transforms one function
into another: i.e.,
O (f(x)) ? g(x).

For instance, x is an operator, since x f (x) is a different function to f (x), and is

Fully speci?ed once f (x) is given. Furthermore, d/dx is also an operator, since is a different function to f(x), and is fully speci?ed once f (x) is given. Now, This can also be written
Where the operators are assumed to act on everything to their right, and a ?nal
f (x) is understood [where f (x) is a general function]. The above expression
illustrates an important point: i.e., in general, operators do not commute. Of
course, some operators do commute: e.g.,
x x2 = x2 x
Finally, an operator, O, is termed linear if





(




)

Where f (x) is a general function, and c a general complex number. All of the
operators employed in quantum mechanics are linear.