The simplest example of the utility of operators is the study of symmetry. Because of this, they are a very useful tool in classical mechanics. In quantum mechanics, on the other hand, they are an intrinsic part of the formulation of the theory.
Let us consider a classical mechanics system led by a certain hamiltonian , function of the generalized coordinates and its conjugate momenta. Let us consider this function to be invariant under the action of a certain group of transformations , i.e., if , . The elements of are physical operators, which map physical states among themselves.
An easy example is given by space translations. The hamiltonian of a translationally invariant problem does not change under the transformation . Other straightforward symmetry operators are the ones implementing rotations.
If the physical system is described by a function, as in classical field theories, the translation operator is generalized in a straightforward way:
Notice that the transformation inside the parenthesis should be the inverse of the transformation done on the coordinates.
If the transformation is infinitesimal, the operator action should be of the form
where is the identity operator, is a small parameter, and will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.
As it was stated, . If is infinitesimal, then we may write
This formula may be rewritten as
where is the generator of the translation group, which happens to be just the derivative operator. Thus, it is said that the generator of translations is the derivative.
The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.
The translation for a finite value of may be obtained by repeated application of the infinitesimal translation:
with the standing for the application times. If is large, each of the factors may be considered to be infinitesimal:
But this limit may be rewritten as an exponential:
To be convinced of the validity of this formal expression, we may expand the exponential in a power series:
The right-hand side may be rewritten as
which is just the Taylor expansion of , which was our original value for .
Once the interest of the operators in classical mechanics has been exposed, it has to be said that it is in quantum mechanics where they reach their full interest. The mathematical description of quantum mechanics is built upon the concept of operator.
Physical pure states in quantum mechanics are unit-norm vectors in a certain vector space (a Hilbert space). Time evolution in this vector space is given by the application of a certain operator, called the evolution operator. Since the norm of the physical state should stay fixed, the evolution operator should be unitary. Any other symmetry, mapping a physical state into another, should keep this restriction.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The values which may come up as the result of the experiment are the eigenvalues of the operator. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue.