Consider the function , the derivative of Ψ with respect to time; one can say that the operator acting on the function Ψ yields the function . More generally, if a certain operation allows us to bring into correspondence with each function Ψ of a certain function space, one and only one well-defined function Ψ^′ of that same space, one says the Ψ^′ is obtained through the action of a given operator A on the function Ψ, and one writes

By definition A is a linear operator if its action on the function λ₁Ψ₁ + λ₂Ψ₂, 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:

1. the differential operators ∂∕∂x,∂∕∂y,∂∕∂z,∂∕∂t, such as the one which was considered above;
2. the operators of the form f(r,t) whose action consists in multiplying the function Ψ by the function f(r,t)

Starting from certain linear operators, one can form new linear operators by the following algebraic operations:

1. multiplication of an operator A by a constant c:
   
2. the sum S = A + B of two operators A and B:
   
3. the product P = AB of an operator B by the operator A:

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 AB is not necessarily identical to the product BA; in the first case, B first acts on the function Ψ, then A acts upon the function (BΨ) to give the final result; in the second case, the roles of A and B are inverted. The difference AB − BA of these two quantities is called the commutator of A and B; it is represented by the symbol [A,B]:

(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 f(x), multiplication by function f(x), and the differential operator ∂∕∂x. Indeed we have, for any Ψ,

In other words

(2)

and, in particular

(3)

However, any pair of derivative operators such as ∂∕∂x,∂∕∂y,∂∕∂z,∂∕∂t, 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.

0.1 References

[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].