1.4 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 $A\vec v$:

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

Similarly, an operator transforms a function into another function:

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

Some simple examples of operators:

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

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

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 ${\rm d}/{\rm d}x$, 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 $f$ by a number, $Af$ increases by that same number; also, if you sum $f$ and $g$, $A(f+g)$ will be $Af$ plus $Ag$.

Key Points

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Matrices turn vectors into other vectors.

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Operators turn functions into other functions.

1.4 Review Questions

1

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

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 $x^2$? Answer

3

A less self-evident operator than the above examples is a shift operator like $E^{\pi /2}$ that shifts the graph of a function towards the left by an amount $\pi /2$: $E^{\pi /2}f(x)=f\left(x+\frac 12\pi\right)$. (Curiously enough, shift operators turn out to be responsible for the law of conservation of momentum.) Show that $E^{\pi /2}$ turns $\sin(x)$ into $\cos(x)$. Answer

4

The inversion operator ${\rm Inv}$ 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 ${\rm Inv}$ leaves $\cos(x)$ unchanged, but turns $\sin(x)$ into $-\sin(x)$. Answer