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 $A\vec v$:

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

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 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:

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

Squaring is not a linear operator:

$$\Big(c_1 f(x) + c_2 g(x)\Big)^2 = c_1^2 f^2(x) + 2 c_1 c_2 f(x) g(x) + c_2^2 g^2(x) \ne c_1 f^2(x) + c_2 g^2(x)$$

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

Matrices turn vectors into other vectors.

Operators turn functions into other functions.

2.4 Review Questions

1

So what is the result if the operator ${\rm d}/{\rm d}x$ 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 $\pi /2$: ${\cal T}_{\pi /2}f(x)=f\left(x+\frac 12\pi\right)$. (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 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 $-\sin(x)$.
Solution mathops-d