2.4 Operators

This section defines operators, which are a generalization of matrices. Operators are the principal components of quantum mechanics.

In a finite number of dimensions, a matrix A can transform any arbitrary vector $v$ into a different vector $A\vec{v}$:

\begin{displaymath}
\vec v
\quad
\begin{picture}(100,0)
\put(50,15){\mak...
...{\vector(1,0){100}}
\end{picture}
\quad
\vec w = A \vec v
\end{displaymath}

Similarly, an operator transforms a function into another function:

\begin{displaymath}
f(x)
\quad
\begin{picture}(100,10)
\put(50,15){\make...
...0,2){\vector(1,0){100}}
\end{picture}
\quad
g(x) = A f(x)
\end{displaymath}

Some simple examples of operators:

\begin{displaymath}
f(x)
\quad
\begin{picture}(100,10)
\put(50,13){\make...
...0,2){\vector(1,0){100}}
\end{picture}
\quad
g(x) = x f(x)
\end{displaymath}


\begin{displaymath}
f(x)
\quad
\begin{picture}(100,23)
\put(50,21){\make...
...(0,2){\vector(1,0){100}}
\end{picture}
\quad
g(x) = f'(x)
\end{displaymath}

Note that a hat is often used to indicate operators; for example, ${\widehat x}$ is the symbol for the operator that corresponds to multiplying by $x$. If it is clear that something is an operator, such as ${\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$, no hat will be used.

It should really be noted that the operators that you are interested in in quantum mechanics are linear operators. If you increase a function $f$ by a factor, $Af$ increases by that same factor. Also, for any two functions $f$ and $g$, $A(f+g)$ will be $(Af)$ + $(Ag)$. For example, differentiation is a linear operator:

\begin{displaymath}
\frac{{\rm d}\Big( c_1 f(x) + c_2 g(x)\Big)}{{\rm d}x} =
...
...rac{{\rm d}f(x)}{{\rm d}x} + c_2 \frac{{\rm d}g(x)}{{\rm d}x}
\end{displaymath}

Squaring is not a linear operator:

\begin{displaymath}
\Big(c_1 f(x) + c_2 g(x)\Big)^2 =
c_1^2 f^2(x) + 2 c_1 c_2 f(x) g(x) + c_2^2 g^2(x) \ne
c_1 f^2(x) + c_2 g^2(x)
\end{displaymath}

However, it is not something to really worry about. You will not find a single nonlinear operator in the rest of this entire book.


Key Points
$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
Matrices turn vectors into other vectors.

$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
Operators turn functions into other functions.

2.4 Review Questions
1.

So what is the result if the operator ${\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$ is applied to the function $\sin(x)$?

Solution mathops-a

2.

If, say, $\widehat{x^2}\sin(x)$ is simply the function $x^2\sin(x)$, then what is the difference between $\widehat{x^2}$ and $x^2$?

Solution mathops-b

3.

A less self-evident operator than the above examples is a translation operator like ${\cal T}_{\pi /2}$ that translates the graph of a function towards the left by an amount $\pi$$\raisebox{.5pt}{$/$}$​2: ${\cal T}_{\pi /2}f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $f\left(x+\frac 12\pi\right)$. (Curiously enough, translation operators turn out to be responsible for the law of conservation of momentum.) Show that ${\cal T}_{\pi /2}$ turns $\sin(x)$ into $\cos(x)$.

Solution mathops-c

4.

The inversion, or parity, operator ${\mit\Pi}$ turns $f(x)$ into $f(-x)$. (It plays a part in the question to what extent physics looks the same when seen in the mirror.) Show that ${\mit\Pi}$ leaves $\cos(x)$ unchanged, but turns $\sin(x)$ into $\vphantom0\raisebox{1.5pt}{$-$}$$\sin(x)$.

Solution mathops-d