# PlanetPhysics/Quantum Operator Concept

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

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 :

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

and, in particular

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.

### References

[edit | edit source][1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.