Operators, Commutators and Expectation Values

Observables and operators

Each measurable quantity is represented by an operator that acts on the wavefunction. Position multiplies by the coordinate; momentum differentiates; the Hamiltonian gives the total energy.

Operators must be Hermitian, which guarantees two things that matter physically: their eigenvalues are real, so a measurement gives a real number, and eigenfunctions belonging to different eigenvalues are orthogonal.

Eigenfunctions and definite values

If acting with the operator returns the same function multiplied by a constant, the wavefunction is an eigenfunction and the observable has the definite value a. Every measurement gives that value.

Expectation values

for a normalised wavefunction. It is the average of many measurements on identically prepared systems, not the result of any single one.

Standard exercises include the average position of a particle in a box, which is the centre by symmetry, and the average momentum, which is zero because the particle is equally likely to be moving either way. Both can be argued from symmetry before any integration, which is worth doing as a check on the algebra.

Commutators

If the commutator is zero the operators commute, and the two observables can have definite values simultaneously — a common set of eigenfunctions exists. If it is non-zero they cannot.

Position and momentum do not commute, and their commutator is what produces the uncertainty principle. The principle is therefore not an additional postulate but a consequence of the operator algebra.

The last row explains why an angular momentum state is labelled by its magnitude and one component only — conventionally the z component. Being asked why all three cannot be specified is answered by the commutator.

Why this matters chemically

Quantum numbers are exactly the eigenvalues of commuting operators. The set of quantum numbers used to label a state is the set of observables that can be specified simultaneously, which is why some combinations appear together and others never do.

Selection rules also come from operators: a transition is allowed when a particular integral involving the two states and the relevant operator is non-zero, and symmetry determines when it vanishes. So the formal machinery connects directly to which spectral lines are observed.

Frequently asked questions

Why must operators be Hermitian?

To guarantee real eigenvalues, since a measured quantity must be a real number, and to guarantee orthogonality of eigenfunctions, which the expansion of arbitrary states requires.

What does it mean physically that two operators commute?

That both observables can have definite values at the same time, because a common set of eigenfunctions exists.

Is the expectation value ever actually measured?

Not in a single measurement. It is the average over many measurements on identically prepared systems.

Why can only one component of angular momentum be specified?

Because the components do not commute with one another. They each commute with the total, so the magnitude and one component form the largest specifiable set.