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:
Starting from certain linear operators, one can form new linear operators by the following algebraic operations:
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
:
![]() |
(1) |
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
![]() |
(2) |
and, in particular
![]() |
(3) |
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.
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
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