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Clinton Davisson and Lester Germer measured strong angular structure in electron scattering from a nickel crystal. Its dependence on incident electron energy agreed with a wavelength set by the electron momentum through the de Broglie wavelength. The crystal therefore supplied a periodic structure against which electron wave behavior could be tested. The Davisson–Germer experiment linked a tunable electron source and angularly resolved detector with the theoretical idea of matter waves.^[ref]

A physicist whose route into research included a telephone-company job when he could not afford to continue at Chicago. He later worked at Bell Telephone Laboratories, retiring in 1946 after 29 years, and taught physics at Virginia in 1947–1949.^[ref] The Davisson–Germer experiment connects that industrial-laboratory work to electron diffraction and matter waves.

The Davisson–Germer experiment turned elastic scattering from nickel into evidence for a matter wave. The search became a deliberate wave test in 1926, with diffraction beams found early in 1927. The earlier crystal investigation had begun in 1925; the de Broglie wavelength proposal was already present in the 1924 thesis.^[ref]

A heated tungsten ribbon supplied electrons; gun apertures selected a narrow beam. Two $1\mathrm{mm}$ apertures were $8\mathrm{mm}$ apart; the gun ended $7\mathrm{mm}$ from a nickel crystal cut parallel to its $\{111\}$ planes. An adjustable electric potential difference set the speed. The beam struck normally inside an evacuated, field-free enclosure with target and outer electrodes at one potential. A double-box collector, $11\mathrm{mm}$ from the target with a $1\mathrm{mm}$ entrance, selected directions. A retarding electric potential difference rejected slower electrons; the inner collector fed a sensitive galvanometer. Moving the collector scanned colatitude; rotating the crystal scanned azimuth.^[ref]

The readout was collected electric current, not a photograph of electron paths. Voltage scans held the gun-plate current constant because incident-beam current could not be held constant; the overall current trend therefore needed caution. A strong feature developed at $54\mathrm{V}$ near $50^{\circ }$ colatitude. Surface heating distinguished crystal beams from adsorbed-gas beams; finite apertures and background shifted or broadened apparent peaks.^[ref]

At the bright beam's colatitude, an azimuth scan revealed three companion directions, matching crystal symmetry. The test compared wavelengths inferred from diffraction angles and known crystal spacings with wavelengths inferred from accelerating voltage and momentum; the 1937 lecture displayed this comparison against $U^{-1/2}$. Agreement across beam directions supplied more evidence than one angular peak.^[ref]

The inference is about coherent quantum phases in scattering probabilities. It does not establish a wavy particle trajectory, and a collector-current experiment does not resolve individual detection events. Predicting the full intensity additionally requires the material's scattering amplitude, its surface condition, and instrumental angular acceptance; the de Broglie wavelength alone predicts none of those strengths. The following finite-row electron diffraction model isolates phase addition without attempting to reproduce the nickel measurements.

Lester Germer and Clinton Davisson observed electron diffraction from a nickel crystal. The intensity of scattered electrons varied strongly with direction and incident electron energy, as expected when a crystal's periodic structure acts on a wave. Agreement between the inferred wavelength and the de Broglie wavelength connected a particle's momentum with observable interference. The experiment made electron wave behavior a measured property of a material-scattering apparatus.^[ref]

A pure quantum state is a unit ray in a complex Hilbert space: normalization fixes total probability, a common phase is unobservable, and superpositions retain relative phases. Observables are self-adjoint operators. For an ideal projective measurement with spectral projector $P_a$, the Born rule gives $p(a)=⟨\psi |P_a|\psi ⟩$ and the Lüders rule gives conditional state $P_a\psi /\sqrt{p(a)}$. This preserves an eigenspace, not necessarily a unique vector when the outcome is degenerate. Between measurements, a closed system evolves unitarily according to $i\hslash {\partial }_t\psi =H\psi$. Composite systems use tensor products; quantum statistics additionally restrict identical-particle states. These are a framework supported by many experiments, not separate logical deductions from any one experiment.

In the photoelectric effect, monochromatic light illuminates a clean metal and a retarding potential measures the largest emitted electron kinetic energy. frequency controls the stopping energy, $e{V}_s=h\nu -\Phi$, while in the one-photon regime intensity mainly controls the emission rate. A threshold frequency and a stopping-energy slope $h/e$ support quantized energy exchange rather than gradual absorption of arbitrary classical-wave energy by each electron. Contact potentials, surface contamination, dark current and spectral impurities require controls. Robert Andrews Millikan tested the frequency–stopping-potential relation with careful surface and electrical control.^[ref]

The Stern-Gerlach experiment sends a collimated neutral silver beam through an inhomogeneous magnetic field onto a detector. The force is approximately $F_z={\mu }_z{\partial }_{z}B$ in an appropriate strong-field electronic description. Two resolved components rather than a continuous classical orientation spread support discrete angular-momentum projections. Field-off beam profiles and magnetic-gradient/beam-width controls distinguish splitting from ordinary broadening. In the modern description the silver ground electronic state has $J=1/2$; the 1922 experiment preceded the electron-spin interpretation. Ideal sequential analyzers along different axes illustrate incompatible observables and state preparation, but were not the original apparatus.^[ref]

In the Davisson–Germer experiment, accelerated electrons scatter from a nickel single crystal and an angularly scanned collector measures the scattered intensity. Changing voltage and crystal orientation changes narrow diffraction maxima. The inferred wavelength agrees with $\lambda =h/p$, with nonrelativistic $p=\sqrt{2meV}$. vacuum, crystal cleaning, energy selection and background measurements matter: adsorbed surface gas can produce additional diffraction features. This supports matter-wave amplitudes and interference, with measured intensities related to their squared magnitude; it does not mean an electron must follow a classical orbit through the apparatus.^[ref] Together these experiments motivate quantized outcomes, wave amplitudes and probabilistic measurement while leaving a distinction between observed evidence and the general postulates used to organize it.

The postulates of quantum mechanics give a framework for predicting observations from a prepared state. The experiments below motivate different features of that framework; no one experiment proves the whole set of postulates or specifies every Hamiltonian operator.

A pure quantum state is a normalized vector in a complex Hilbert space, with vectors differing by an overall quantum phase representing the same physical preparation. In position representation, $\int |\psi (\mathbf{r}){|}^2{d}^{3}r=1$. The linear state space permits quantum superposition, while the Born rule assigns probability ${\int }_{\Omega }|\psi {|}^2{d}^{3}r$ to a position outcome in a region $\Omega$. The wavefunction is a probability amplitude; it is not a classical distribution of definite particle positions and momenta.

An observable is represented by a self-adjoint operator. If $Q$ has spectral projectors $P_q$, its possible measurement outcomes are its spectral values and $\mathrm{Pr}(q)=⟨\psi |P_q|\psi ⟩$. The mean is $⟨Q⟩=⟨\psi |Q|\psi ⟩$. For an ideal repeatable projective measurement, obtaining $q$ leaves the normalized conditional state $P_q\psi /\sqrt{\mathrm{Pr}(q)}$. This projects onto an entire eigenspace if the outcome is degenerate. General detectors need more general measurement in quantum mechanics models; the projective update is an idealization. Noncommuting observables need not have simultaneous sharp values, as the canonical commutation relation and Heisenberg uncertainty relation illustrate.

Between measurements, an isolated system has unitary time evolution generated by its Hamiltonian operator:This preserves normalization and superpositions. Composite systems use a tensor product of state spaces, allowing entangled states as well as product states; identical particles additionally require the bosonic or fermionic exchange symmetry appropriate to their species. Statistical mixtures are described by a density matrix, with $\mathrm{Pr}(q)=\mathrm{tr}(\rho P_q)$. These rules separate preparation, coherent evolution and measurement.

The photoelectric effect exposes discrete energy transfer by light. Illuminate a clean metal cathode in an evacuated tube and measure the collected electron current while varying the light frequency, intensity and a retarding voltage. The stopping potential $V_s$ probes the largest emitted electron kinetic energy, whereas the photocurrent measures the rate at which charge is collected. In the ordinary single-photon regime there is a threshold frequency; above it the maximum energy grows with frequency, rather than with intensity. Changing intensity at fixed frequency mainly changes the number of emitted electrons. These observations are summarized by the Einstein photoelectric equationHere $\Phi$ is the surface work function, and the stopping voltage is understood after any photoelectric contact-potential correction. A photon delivers $h\nu$ in an absorption event; emission is allowed only when that energy can pay the escape cost. This accounts for the frequency threshold and the lack of the classical waiting time associated with slowly accumulating energy in a single electron. The experiment supports energy quantization of light; it does not imply that every incident photon emits an electron or that every emitted electron has the maximum energy. Surface losses and collection geometry affect the full current-voltage curve.

The Davisson–Germer experiment demonstrates a complementary wave property of matter. An electron gun uses a heated emitter, accelerating voltage and apertures to send a narrow beam onto a nickel single crystal in vacuum. A movable collector measures scattered electric current versus direction; rotating the crystal also tests its azimuthal symmetry. The crystal gives sharp angular beams whose positions depend on voltage and crystal orientation. A prominent feature occurs near $54\mathrm{V}$ accelerating voltage and roughly $50^{\circ }$ detector colatitude. The nonrelativistic de Broglie wavelength predicted at that voltage isDiffraction wavelengths inferred using the known crystal structure agree with this voltage dependence. These angular peaks arise from coherent addition of scattering amplitudes from the crystal, not from classical specular reflection of independent point projectiles. Bragg's law is $2d\sin {\theta }_B=r\lambda$ for lattice-plane spacing $d$ and angle to those planes; a detector colatitude is a different angle and must not be substituted without its geometry. The historical readout was current, not a photograph of individual electron trajectories.^[ref]

The double-slit experiment makes the role of amplitudes especially direct. A coherent source illuminates two narrow slits; an observation screen a distance $D$ away records the pattern. For slit spacing $d$ and small screen displacement $y$, the path difference is approximately $dy/D$. The two amplitudes add before probabilities are formed:For equal coherent illumination the interference term varies as $\cos (2\pi dy/(\lambda D))$, giving fringe spacing $\lambda D/d$ beneath the single-slit diffraction envelope. Blocking one slit removes that cross term. Young's optical experiment established wave interference; later particle-resolved realizations with photons or electrons show localized detection events whose accumulated distribution has the interference pattern. Low arrival rates demonstrate that this pattern does not require different incident particles to collide with one another.

If the paths become correlated with distinguishable detector states $|D_1⟩$ and $|D_2⟩$, the cross term acquires their overlap $⟨D_2|D_1⟩$. Perfectly distinguishable path records make that overlap zero, suppressing interference even if the spatial beams still overlap. This connects quantum entanglement, measurement and loss of observable coherence. Without a path measurement, the two alternatives are amplitudes for one preparation, rather than classical mutually exclusive events whose probabilities may simply be added.

Together these experiments motivate a description in which energy transfer and detector outcomes can be discrete, yet propagation retains complex quantum phases and interference. The Born rule connects those amplitudes to repeatable frequencies of outcomes. The Schrödinger equation supplies the dynamics once a Hamiltonian operator is specified, while observable operators, state preparation and the measurement model specify what the apparatus can read.

The photoelectric effect, black-body radiation and the Davisson–Germer experiment challenge different parts of a fully classical picture. Classical electromagnetic waves successfully describe propagation, interference and diffraction, while classical particles follow trajectories determined by forces. The difficulty is combining that picture with the observed exchange of energy between radiation and matter, the equilibrium distribution of radiation, and electron scattering. The three experiments expose the roles of discrete energy exchange and coherent matter waves.

In a photoelectric effect experiment, monochromatic light illuminates a metal cathode in vacuum and a second electrode collects emitted electrons. Varying the collecting voltage measures a photocurrent; reversing it until collection ceases determines the stopping potential. The observations distinguish two controls: at fixed frequency above threshold, increasing light intensity increases the emission rate, whereas the largest emitted kinetic energy is determined mainly by frequency. There is a material-dependent photoelectric threshold frequency, and emission above threshold does not require the long charging-up delay predicted by a simple picture of gradual classical energy accumulation at low intensity.

A classical wave carries continuously distributed energy, with intensity proportional to squared field amplitude. On a fully classical absorption picture, increasing intensity would be expected to increase the energy delivered to an electron, and sufficiently long exposure could accumulate the escape energy even below the observed threshold. It does not naturally produce the measured universal straight-line relation between stopping potential and light frequency. In 1905, Albert Einstein instead treated light-energy transfer as localized quanta. His argument distinguished the fastest escaping electrons from electrons losing energy before escape.^[ref]

In the single-photon regime, conservation of energy gives the Einstein photoelectric equationwhere $\Phi$ is the surface work function. The photon supplies energy $h\nu$; the escape cost is paid first, and the remainder sets the upper electron-energy edge. Increasing intensity at fixed frequency increases the number of available photons, so it raises current without raising this edge. Below ${\nu }_0$, one photon cannot pay the escape cost. In an actual measurement, stable surface preparation, suppression of unwanted higher-frequency light, and the photoelectric contact-potential correction are needed to test the slope $h/e$ reliably. Robert Andrews Millikan's experiment used selected mercury spectral lines, freshly exposed metal surfaces and an electrometer to make such a quantitative test.^[ref] The interpretation is for weak single-photon illumination; multiphoton photoemission at sufficiently intense fields is a different regime.

Black-body radiation tests equilibrium physics rather than the escape of individual electrons. A small aperture in an isothermal absorbing cavity approximates an ideal blackbody: radiation entering the hole undergoes repeated interactions with the walls, and the equilibrium spectrum depends on temperature rather than wall material. A spectrally resolving detector outside the aperture measures how emitted power varies with frequency or wavelength. Experimentally the spectrum has a finite peak, is strongly suppressed at short wavelengths, shifts toward shorter wavelengths when the cavity is heated, and has integrated emission proportional to $T^4$.

For a large cavity, the electromagnetic mode density in a box, including two independent transverse polarizations, is $g(\nu )=8\pi {\nu }^2/c^3$ per volume per unit frequency. The classical equipartition theorem assigns mean energy $k_{B}T$ to each mode, leading to the Rayleigh-Jeans lawThis agrees at low frequencies but diverges as frequency increases; integrating it over all frequencies gives infinite energy density at any nonzero temperature. This is the ultraviolet catastrophe, a failure of classical equipartition applied to the electromagnetic modes.

Max Planck's essential modification was that energy at a frequency is exchanged in units $h\nu$. In a modern oscillator description, the thermal excitation levels are $nh\nu$, $n=0,1,\dots$. Using the Boltzmann distribution and summing a geometric series givesMultiplying by the mode density yields Planck's law for the spectral energy density,When $h\nu \ll k_{B}T$, expansion of the denominator recovers the classical law. When $h\nu \gg k_{B}T$, the large excitation cost suppresses occupancy exponentially, removing the divergence. Its integral gives the Stefan–Boltzmann law, while its peak gives Wien's displacement law. Thus quantization repairs the spectral shape, total emission and temperature dependence together. The displayed thermal energy excludes the oscillator's temperature-independent zero-point energy.

The Davisson–Germer experiment supplies the complementary evidence that matter cannot always be treated as classical particles with definite scattering trajectories. In 1927, Clinton J. Davisson and Lester Halbert Germer sent a narrow, accelerated electron beam normally onto a single nickel crystal in vacuum. A movable collector measured the current of scattered electrons as direction and accelerating voltage were varied; a retarding potential selected electrons retaining approximately the incident energy. Sharp angular beams depended on crystal orientation and beam energy, with a prominent feature near $54\mathrm{V}$ and detector colatitude $50^{\circ }$. These are scattering directions and currents, not photographs of electron paths.^[ref]

The de Broglie wavelength predictsAt $U=54\mathrm{V}$, this gives $\lambda \simeq 0.167\mathrm{nm}$, comparable to atomic spacings. A crystalline arrangement supplies regularly spaced scattering sites, so scattered probability amplitudes can add constructively in selected directions. For a suitable family of crystal planes, Bragg's law expresses the condition as $2d\sin {\theta }_B=n\lambda$, with ${\theta }_B$ measured relative to those planes. The detector colatitude is not automatically this angle. The measured angular structure and its variation with electron momentum are described by electron diffraction with the de Broglie wavelength. A classical point-particle deflection model has no phase wavelength $h/p$ and does not supply that interference condition.

The common quantum account preserves the successes of wave propagation and particle detection while changing what is quantized. Photons exchange energy in frequency-dependent units; a thermal distribution of those excitations gives the blackbody spectrum. Electrons have a wavefunction whose amplitudes interfere on scattering from a crystal, while detection transfers localized charge. The Born rule turns squared amplitudes into event probabilities. These results invalidate a fully classical description of matter and radiation; the photoelectric energy law alone does not establish every property of a quantized electromagnetic field, but the three phenomena together provide complementary evidence for the quantum description.