### 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 into a different vector :

Similarly, an operator transforms a function into another function:

Some simple examples of operators:

Note that a hat is often used to indicate operators; for example, is the symbol for the operator that corresponds to multiplying by . If it is clear that something is an operator, such as , 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 by a factor, increases by that same factor. Also, for any two functions and , will be + . For example, differentiation is a linear operator:

Squaring is not a linear operator:

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 is applied to the function ?

2.

If, say, is simply the function , then what is the difference between and ?

3.

A less self-evident operator than the above examples is a translation operator like that translates the graph of a function towards the left by an amount ​2: . (Curiously enough, translation operators turn out to be responsible for the law of conservation of momentum.) Show that turns into .

4.

The inversion, or parity, operator turns into . (It plays a part in the question to what extent physics looks the same when seen in the mirror.) Show that leaves unchanged, but turns into .