= Wave mechanics
Wave mechanics represents a <quantum state> by a <wavefunction> and <observables> by operators acting on it. For a nonrelativistic spinless particle in a <potential energy> $V$, the <Schrödinger equation> reads
$$
i\hbar\partial_t\psi=\left[-\frac{\hbar^2}{2m}\nabla^2+V\right]\psi.
$$
The spatial shape and <boundary conditions> determine possible <energy eigenfunctions>; the <Born rule> gives their experimental interpretation as <probability amplitudes>.
For the <infinite square well>, vanishing at both walls selects standing <sine> waves and hence discrete <energy levels>. This explains quantization through a boundary-value problem rather than by imposing a separate allowed-orbit rule. <Erwin Schrödinger>'s first 1926 paper developed quantization as an <eigenvalue> problem for the <hydrogen atom>. <Matrix and wave mechanics equivalence> explains how the same physical predictions arise from <matrix mechanics> in a suitable <orthonormal basis>.<Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger>[1926, pp. 361–363: quantization as an eigenvalue problem and the vibrating-string analogy]{ref}
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