a Hermitian operator
/ her-MISH-un /
Measurable quantities come out as real numbers, and the states of definite value come out orthogonal and complete. What guarantees both of those good properties at once? Being Hermitian. A Hermitian operator is the mathematical species that quantum mechanics reserves for physical observables, precisely because it is built to give real eigenvalues and a clean basis of eigenstates.
The defining condition is that the operator equals its own adjoint (its conjugate transpose), A = A-dagger, which unpacked means <phi|A psi> = <A phi|psi> for all states phi and psi. Three consequences follow. Its eigenvalues are all real, so measurement outcomes are real. Its eigenvectors belonging to different eigenvalues are orthogonal. And, by the spectral theorem, its eigenvectors form a complete orthonormal basis for the Hilbert space, so any state can be expanded in them. This is the structure that makes the Born rule and the measurement postulate consistent.
A graduate-level caution physicists often gloss over: for finite matrices Hermitian and self-adjoint coincide, but for unbounded operators on infinite-dimensional spaces (like momentum p = -i hbar d/dx) they differ by subtle questions of domain and boundary conditions. An operator can be symmetric on some domain yet fail to be truly self-adjoint, and self-adjointness — not mere symmetry — is what guarantees real spectrum and unitary evolution. Momentum on a half-line is the classic cautionary example.
Position x, momentum p = -i hbar d/dx, and the Hamiltonian H = p^2/2m + V(x) are all Hermitian; the factor of i in p is exactly what is needed to make the derivative operator Hermitian rather than anti-Hermitian.
Hermiticity forces real eigenvalues, so energies, positions, and momenta come out as real numbers.
For unbounded operators, physicists' loose 'Hermitian' (symmetric) is weaker than the self-adjointness that actually guarantees a real spectrum and unitary time evolution; domains matter.