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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 $p(a)=\langle\psi|P_a|\psi\rangle$ 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 $i\hbar\partial_t\psi=H\psi$. 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>, $eV_s=h\nu-\Phi$, 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.<A Direct Photoelectric Determination of Planck's h by Robert Andrews Millikan>[1916, apparatus and stopping-potential measurements, §§2–5]{ref}
The <Stern-Gerlach experiment> sends a collimated neutral <silver> beam through an inhomogeneous <magnetic field> onto a detector. The force is approximately $F_z=\mu_z\partial_zB$ 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.<Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld by Walther Gerlach and Otto Stern>[1922; translated experimental setup and field-on/off deposition profiles]{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.<Diffraction of Electrons by a Crystal of Nickel by C. Davisson and L. H. Germer>[1927, abstract; apparatus and diffraction patterns]{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.
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