Quantum Mechanics for Engineers 

© Leon van Dommelen 

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 ?
Solution mathopsa

2.

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

3.

A less selfevident 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 .
Solution mathopsc

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 .
Solution mathopsd