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The scattering interpretation connects an incoming preparation to the relative frequency of outgoing directions and final atomic states. Max Born's 1926 paper treats a distant incoming electron and a chemical atom, expands the outgoing wavefunction into final atomic states, and imposes conservation of energy. Its translated correction footnote specifies the square of the coefficient; the modern complex modulus notation below is ours.^[ref]

In his 11 December 1954 lecture, Max Born recalled the difficulty of interpreting bound electrons alone. He instead described an incident electron wave generating an outgoing spherical wave, with direction-dependent squared amplitude determining relative scattering probabilities. Distinct final atomic states describe excitation with corresponding electron energy loss. This is a retrospective account of an interpretation, rather than the report of a new apparatus.^[ref]

The distinction from the modern Born rule matters. The historical question was which collision outcome occurs: a direction of motion and a final atomic state. The general projective measurement formula applies to arbitrary observables, including degenerate outcomes and mixed states; a positive operator-valued measure describes still more general detector statistics. Those formulations should not be read back as the explicit contents of the short 1926 scattering paper. Nor does the scattering interpretation itself supply a rule for the conditional post-measurement state.

Actual experimental settings are described in the Davisson–Germer experiment and the Franck-Hertz experiment. The former scans collected electric current as an electron beam scatters from prepared nickel; changing direction, accelerating voltage and surface preparation probes electron diffraction. The latter uses acceleration through mercury vapor and a retarding collector to infer discrete energy losses from current minima, with vapor density and retarding settings as controls. Angular resolution and surface backgrounds limit the first inference; initial energy spread and collision positions limit the second. These linked experiments supply the preparation, apparatus and readout; the examples below are original calculations, not Born's measured data.

In the stationary-state terminology used here, an excited state is a normalized energy eigenstate with energy eigenvalue $E>E_0$, where $E_0$ is an attained ground-state energy. Its excitation cost above the ground state is $\Delta E=E-E_0>0$. Several energy eigenstates can share one excited energy eigenvalue through degenerate energy levels. An arbitrary quantum superposition with energy expectation value above $E_0$ need not be a single stationary excited state; its energy content is described by its Born rule probabilities or, for a continuum, its scalar spectral measure.

An excited state can be a stationary state: the second infinite square well mode is an immediate example. Spontaneous photon emission is a transition caused by coupling matter to the electromagnetic field; it is absent from an isolated matter-only Hamiltonian operator. The relevant energy eigenstate of that isolated model is therefore not automatically an energy eigenstate of the full interacting system. This distinction explains how an ideal stationary excited state can approximate a state with a finite experimental lifetime. See electric-dipole interaction for the coupling used in the simplest optical model.

For a one-photon transition between stationary internal states, neglecting recoil, conservation of energy requiresHere $h$ is the Planck constant and $\nu$ is the photon frequency. Energy matching alone does not guarantee a transition: in the electric-dipole interaction approximation its strength also involves the matrix element $⟨f|\mathbf{d}\cdot \mathbf{ϵ}|i⟩$, where $\mathbf{d}$ is the electric dipole moment operator and $\mathbf{ϵ}$ specifies photon polarization. A zero matrix element gives an electric-dipole selection rule. Thus an absent optical line does not prove the corresponding energy eigenvalue is absent; a different interaction can couple different states. ^[ref]. The Franck-Hertz experiment gives a particularly instructive comparison between collision excitation and photon emission.

The first excited state terminology assumes a least energy eigenvalue above $E_0$. Such a next level is not guaranteed by quantum mechanics: a continuous spectrum can start immediately above the spectral bottom, or energy eigenvalues can accumulate there. For example, a Hamiltonian operator with energy eigenvalues $0,\epsilon ,\epsilon /2,\epsilon /3,\dots$, where $\epsilon >0$, has a ground-state energy zero but no least positive energy eigenvalue: every candidate $\epsilon /n$ is exceeded downward by $\epsilon /(n+1)$. A free particle on the whole line has neither a ground state nor a first stationary excited state. Each use of a discrete excitation ladder therefore needs a specified Hamiltonian operator and its boundary conditions.

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.^[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.^[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 relationThis relates position and momentum through an intrinsically noncommuting product.^[ref]

For the quantum harmonic oscillator, the canonical commutation relation and creation and annihilation operators give $H=\hslash \omega (a^{†}a+I/2)$, 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.

Awarded jointly to James Franck and Gustav Ludwig Hertz for electron-impact laws in chemical atoms. They received the prize in 1926; the Franck-Hertz experiment explains the evidence.^[ref]

In the Franck-Hertz experiment, electrons emitted from a heated cathode are accelerated through a dilute atomic vapour toward a collector. A small retarding potential before the collector rejects electrons that have lost too much kinetic energy. Below the first excitation threshold, mainly elastic collisions occur and the collector current rises as the accelerating voltage increases. At threshold an inelastic collision can transfer a definite excitation energy to an atom, causing a current drop because the slowed electron cannot overcome the retarding potential.

Raising the voltage further permits acceleration again, and repeated losses produce successive current minima. For mercury their spacing is about $4.9\mathrm{V}$, corresponding to an atomic excitation of about $4.9\mathrm{eV}$ and an associated ultraviolet photon near $254\mathrm{nm}$. Contact potentials shift the absolute voltages, so the spacing is more informative than the first minimum's position alone. Discrete energy losses support discrete atomic energy levels; a continuously variable excitation energy would not produce the same recurring fixed loss scale. The experiment establishes quantized excitations without requiring the classical electron orbit of the Bohr model to be literally correct.