Linear Operators

Almost all operators encountered in quantum mechanics are linear operators. A linear operator is an operator which satisfies the following two conditions:

$$\hat{A} (f + g) = \hat{A} f + \hat{A} g$$ (43)

$$\hat{A} (c f) = c \hat{A} f$$ (44)

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)

$$(d/dx)[c f(x)] = c \; (d/dx) f(x)$$ (46)

However, $()^2$ is not a linear operator because

$$(f(x) + g(x))^2 \neq (f(x))^2 + (g(x))^2$$ (47)

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

$$\hat{A} (\lambda f + \mu g) = \lambda^{*} \hat{A} f + \mu^{*} \hat{A} g$$ (48)

Time-reversal operators are antilinear (cf. Merzbacher [2], section 16-11).