= Matrix mechanics
{wiki}
<Matrix mechanics> represents an <observable> by an array of <matrix elements> between chosen <quantum states>. In an <orthonormal basis> of <energy eigenstates>, each a <stationary state>, the diagonal entries of the <Hamiltonian operator> are the <energy levels>; off-diagonal entries of other <observables> describe couplings between those states. The same array can therefore answer two different questions: which values can a measurement return, and which pairs of <energy eigenstates> can a specified interaction connect? The <eigenvalues> answer the first question; the relevant <matrix elements>, together with the <energy> differences, answer the second.
This makes <linear algebra> a practical language for <atomic physics>. One can calculate <energy levels>, <electric-dipole selection rules> and <probability amplitudes> for transitions without assigning an <Electron> an unobserved classical <orbit>. The three-state example below separates a missing optical coupling from a missing <energy level>; the <finite oscillator commutator boundary defect> explains why a finite array requires care when approximating a <quantum harmonic oscillator>.
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