orthonormal basis
An orthonormal basis is a basis whose members are not only complete but also tidy in two specific ways: each basis state has length one (normalised), and any two different basis states are perpendicular, meaning their inner product is zero (orthogonal). In symbols, ⟨m|n⟩ equals one when m and n are the same state and zero otherwise — the cleanest possible relationship a set of states can have.
This neatness pays off enormously in calculation. To find the coefficient of a state |ψ⟩ along the basis direction |n⟩, you simply take the inner product ⟨n|ψ⟩ and read it straight off — no solving simultaneous equations, no untangling overlaps between the axes. Lengths and probabilities also become easy: the squared coefficients of a normalised state add up to one, so they can be read directly as the probabilities of the corresponding measurement outcomes.
Most familiar bases in quantum mechanics are orthonormal, and for good reason. The energy eigenstates of a system, the spin-up and spin-down states, the position states — all can be arranged to be orthonormal because they correspond to distinct, mutually exclusive measurement results. Distinct outcomes of a single sharp measurement are always perfectly distinguishable, which is exactly what orthogonality expresses.
Each basis state has unit length and is perpendicular to every other; the Kronecker delta says it crisply.
Orthonormality is convenient but not strictly required for a basis. A set of independent, complete states can serve as a basis even if they overlap; we orthonormalise them mainly because it makes coefficients and probabilities trivial to read off.