The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context).
Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.They play a central role in describing observables (measurable quantities like energy, momentum, etc.).
and its conjugate momenta: If either L or H is independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry).
Operators in classical mechanics are related to these symmetries.
is the rotation matrix about an axis defined by the unit vector
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.
is the generator of the translation group, which in this case happens to be the derivative operator.
The whole group may be recovered, under normal circumstances, from the generators, via the exponential map.
may be obtained by repeated application of the infinitesimal translation: with the
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
The mathematical properties of physical operators are a topic of great importance in itself.
The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.
Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator.
The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment.
In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators.
In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can be represented as matrices.
Any other symmetry, mapping a physical state into another, should keep this restriction.
The wavefunction must be square-integrable (see Lp spaces), meaning: and normalizable, so that: Two cases of eigenstates (and eigenvalues) are: Let ψ be the wavefunction for a quantum system, and
, and the observable does not have a single definite value in that case.
Acting the commutator on ψ gives: If ψ is an eigenfunction with eigenvalues a and b for observables A and B respectively, and if the operators commute: then the observables A and B can be measured simultaneously with infinite precision, i.e., uncertainties
Clearly the state (ψ) of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.
If the operators do not commute: they cannot be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables even if ψ is an eigenfunction the above relation holds.
Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as Lx and Ly, or sy and sz, etc.).
can be connected to another,[3] by the expression: which is a matrix element: A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.
[1] In matrix form, operators allow real eigenvalues to be found, corresponding to measurements.
Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system.
, then the momentum eigenvalue p is the value of the particle's momentum, found by: For three dimensions the momentum operator uses the nabla operator to become: In Cartesian coordinates (using the standard Cartesian basis vectors ex, ey, ez) this can be written; that is: The process of finding eigenvalues is the same.
Since this is a vector and operator equation, if ψ is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum.