Observables and operators

In a finite-dimensional Hilbert space, an observable is represented by a Hermitian operator $A=A^\dagger$. Its eigenvalues are real and its eigenspaces provide orthogonal measurement alternatives. A matrix is the representation of that operator in a chosen basis, so a nondiagonal matrix is not a defective observable. Diagonalizing it finds a basis adapted to its outcomes. The eigenvector equation says that applying the operator to an eigenstate only scales that state.

In infinite-dimensional spaces, domains and boundary conditions become part of the operator definition. The technically appropriate condition for an observable is self-adjointness; formal integration-by-parts symmetry alone can be insufficient. The momentum expression $-i\hbar\,d/dx$ is therefore not a complete physical specification until the space and allowed functions are stated. Here we first use finite matrices, where those domain subtleties do not obstruct the calculation.

The expectation value $\langle A\rangle=\langle\psi|A|\psi\rangle$ equals the probability-weighted mean of the eigenvalues. It predicts the average across many identically prepared systems measured in the same way. It does not say that any one detector must display that number. A two-outcome apparatus can have a mean between its two readings even though no single trial ever returns that intermediate value.

The variance $\langle A^2\rangle-\langle A\rangle^2$ measures the intrinsic spread of that outcome distribution for the stated preparation. Detector calibration error and technical noise can add further spread, but they are distinct ingredients. If the state is entirely in one eigenspace, the ideal variance vanishes. A zero mean, by contrast, does not imply zero spread: equally likely positive and negative readings can cancel in the average.

Operator multiplication represents composition, and generally $AB\ne BA$. The commutator $[A,B]=AB-BA$ measures this failure. For finite Hermitian matrices, commuting observables admit a common orthonormal eigenbasis. In that basis, both values can be definite together. A nonzero commutator rules out a complete common eigenbasis, although selected special states can still have additional properties that must be checked directly.

Position and momentum illustrate the physical consequences: on an appropriate common domain, $[x,p]=i\hbar I$. Replacing quantum operators by commuting ordinary numbers would erase this structure. Later, angular momentum and creation operators will let us solve problems through algebra rather than differential equations. At each step, retain the order of factors and ask what space the operators act on; a visually convenient rearrangement is not automatically valid.