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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
is a constant and
and
are functions.
As an example, consider the operators
and
.
We can see that
is a linear operator because
However,
is not a linear operator because
![\begin{displaymath}
(f(x) + g(x))^2 \neq (f(x))^2 + (g(x))^2
\end{displaymath}](img103.png) |
(47) |
The only other category of operators relevant to quantum mechanics is the
set of antilinear operators, for which
![\begin{displaymath}
\hat{A} (\lambda f + \mu g) = \lambda^{*} \hat{A} f + \mu^{*} \hat{A} g
\end{displaymath}](img104.png) |
(48) |
Time-reversal operators are antilinear (cf. Merzbacher
[2], section 16-11).
Next: Eigenfunctions and Eigenvalues
Up: Operators
Previous: Basic Properties of Operators
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David Sherrill
2006-08-15