= From Heisenberg arrays to Born-Jordan commutation
In 1925, <Werner Heisenberg> proposed replacing classical orbital quantities with arrays of transition amplitudes and frequencies. His paper's opening demanded relations between quantities observable in principle; section 1 built products by summing over an intermediate state and noted their dependence on order. The formulas above express that construction in modern <linear operator> notation.<On the quantum reinterpretation of kinematical and mechanical relationships by Werner Heisenberg>[Heisenberg, 1925; introduction and section 1, equations (5), (8), (15), (20), (24)–(25); Doria translation]{ref}
Pumping and a heated bath prepared <Mercury> vapor around a hot <platinum> wire, mesh and collector in the original <Franck-Hertz experiment>. A <galvanometer> read collector <electric current> while accelerating <voltage> varied at fixed retarding <voltage>. Retarding scans distinguished outgoing <Electron> energies.<Über Zusammenstöße zwischen Elektronen und den Molekülen des Quecksilberdampfes und die Ionisierungsspannung desselben by James Franck and Gustav Ludwig Hertz>[1914, pp. 459–461, Figs. 1–2]{ref} The losses constrain excitation of <chemical atoms>; they do not determine every optical coupling. The <Franck-Hertz experiment> explains the original mistaken <ionization> interpretation.
Such excitation and <spectral line> evidence motivated a mechanics of discrete <energy levels> and transitions. Replacing one representation by another was a theoretical construction; the <Franck-Hertz experiment> does not measure a <matrix product> or establish a unique representation of an <observable>.
Later in 1925, <Max Born> and <Pascual Jordan> developed the array calculus systematically. Their section 3 states $pq-qp=hI/(2\pi i)$, equivalently the modern <canonical commutation relation>
$$
[X,P]=i\hbar I.
$$
This relates <position> and <momentum> through an intrinsically noncommuting product.<Zur Quantenmechanik by Max Born and Pascual Jordan>[Born and Jordan, Zur Quantenmechanik, 1925; section 3, equation (38), translation p. 14]{ref}
For the <quantum harmonic oscillator>, the <canonical commutation relation> and <creation and annihilation operators> give $H=\hbar\omega(a^\dagger a+I/2)$, recovering both the spacing and <zero-point energy> algebraically. One must retain infinitely many <number states> for the exact relation: the finite cutoff below reveals the obstruction explicitly.
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