Hermitian operator

A Hermitian operator is one that equals its own adjoint — the operator you get by taking its complex conjugate and swapping rows with columns. This balanced symmetry sounds like a technicality, but it is exactly the property that makes an operator fit to represent something you can measure. Every genuine observable in quantum mechanics is built from a Hermitian operator.

Hermitian operators have two precious features that flow from their symmetry. First, all their eigenvalues are real numbers, which is essential because a measurement result has to be an ordinary real reading, not an imaginary one. Second, their eigenstates corresponding to different eigenvalues are orthogonal and can be chosen to span the whole space, so any state can be written as a superposition of them — the basis in which that observable has definite values.

These two facts together explain why Hermitian operators sit at the centre of the theory. They guarantee that the possible outcomes of a measurement are real, that those outcomes are mutually exclusive, and that the probabilities of all outcomes add up to one. When physicists say an observable is represented by an operator, they always mean a Hermitian one.