Quantum mechanics
= Quantum mechanics
{wiki}
Quantum mechanics describes a <quantum system> using a <quantum state>, with <observables> represented by <self-adjoint operators>. For a single particle in one dimension, a <pure state> can be described by a <wavefunction> $\psi(x,t)$: its squared <modulus> gives the <probability density> for position. The <Schrödinger equation> predicts how this <probability amplitude> evolves between interventions; <measurement in quantum mechanics> connects it to experimental outcomes through the <Born rule>. A <wavefunction> therefore predicts a distribution of possible positions rather than a single trajectory as in <classical mechanics>.
The <infinite square well> makes three central ideas visible with elementary <sine> functions. Confinement selects discrete <energy eigenvalues>; <wavefunction normalization> converts a shape into quantitative <probabilities>; and a <quantum superposition> can have a changing <probability density> even when each constituent <energy eigenstate> has a stationary one. This combination explains why <quantum mechanics> uses both <linear algebra> and <differential equations>: <eigenfunctions> describe the building blocks, while <quantum phases> determine how they interfere. The same tools help explain the discrete energies of <chemical atoms> and <molecules>, although their <Quantum Hamiltonians> require different potentials and often many particles.
In his first 1926 paper, Erwin Schrödinger formulated quantization as an <eigenvalue> problem for a <hydrogen atom> in <nonrelativistic quantum mechanics>. On its opening page he compared the emergence of integers with the number of <nodes of a standing wave> on a vibrating string. The <infinite square well> below realizes that appealing idea especially transparently: the allowed integers follow from the <boundary conditions>. This historical analogy concerns the origin of discrete energies; the <Born rule> supplies the statistical interpretation of the modern <wavefunction>. <Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger>[Schrödinger, Quantisierung als Eigenwertproblem, 1926, pp. 361–363]{ref}.