This section defines operators, which are a generalization of
matrices. Operators are the principal components of quantum
In a finite number of dimensions, a matrix A can transform any arbitrary vector into a different
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
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.
- Matrices turn vectors into other vectors.
- Operators turn functions into other functions.
2.4 Review Questions
So what is the result if the operator is applied to the function ?
If, say, is simply the function , then what is the difference between and ?
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 .
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 .