Consider the function
, the derivative of
with respect to time; one can say that the operator
acting on the function
yields the function
. More generally, if a certain operation allows us to bring into correspondence with each function
of a certain function space, one and only one well-defined function
of that same space, one says the
is obtained through the action of a given operator
on the function
, and one writes
By definition
is a linear operator if its action on the function
, a linear combination with constant (complex) coefficients, of two functions of this function space, is given by
Among the linear operators acting on the wave functions
associated with a particle, let us mention:
- the differential operators
,
,
,
, such as the one which was considered above;
- the operators of the form
whose action consists in multiplying the function
by the function ![{\displaystyle f(\mathbf {r} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c5a09bbfad706256f3890adbff60ec852a1e2a)
Starting from certain linear operators, one can form new linear operators by the following algebraic operations:
- multiplication of an operator
by a constant
:
- the sum
of two operators
and
:
- the product
of an operator
by the operator
:
Note that in contrast to the sum, the product of two operators is not commutative . Therein lies a very important difference between the algebra of linear operators and ordinary algebra.
The product
is not necessarily identical to the product
; in the first case,
first acts on the function
, then
acts upon the function
to give the final result; in the second case, the roles of
and
are inverted. The difference
of these two quantities is called the commutator of
and
; it is represented by the symbol
:
![{\displaystyle [A,B]:=AB-BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863994f9f066fdc4ae4679dd7aad6dc461724a90)
If this difference vanishes, one says that the two operators commute:
As an example of operators which do not commute, we mention the operator
, multiplication by function
, and the differential operator
. Indeed we have, for any
,
In other words
![{\displaystyle \left[{\frac {\partial }{\partial x}},f(x)\right]={\frac {\partial f}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acb38b46440700e53ae79e77e4c514f3a0bbfc8c)
and, in particular
![{\displaystyle \left[{\frac {\partial }{\partial x}},x\right]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d88903319efa1ebab9ed4e2684c83e71e96b8be9)
However, any pair of derivative operators such as
,
,
,
, commute.
A typical example of a linear operator formed by sum and product of linear operators is the Laplacian operator
which one may consider as the scalar product of the vector operator gradient
, by itself.