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Almost all operators encountered in quantum mechanics are linear
operators. A linear operator is an operator which satisfies the following
two conditions:
where c is a constant and f and g are functions.
As an example, consider the operators d/dx and ()^{2}.
We can see that d/dx is a linear operator because
(d/dx)[f(x) + g(x)] 
= 
(d/dx)f(x) + (d/dx)g(x) 
(45) 
(d/dx)[c f(x)] 
= 

(46) 
However, ()^{2} is not a linear operator because

(47) 
The only other category of operators relevant to quantum mechanics is the
set of antilinear operators, for which

(48) 
Timereversal operators are antilinear (cf. Merzbacher
[2], section 1611).
Next: Eigenfunctions and Eigenvalues
Up: Operators
Previous: Basic Properties of Operators