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Robert B. Laughlin proposed correlated many-electron wave functions explaining important states of the fractional quantum Hall effect. The variational construction strongly suppresses configurations in which electrons approach one another while keeping them within the lowest Landau level. Its excited states behave as localized disturbances carrying a fraction of the electron electric charge. The theory connects a microscopic correlated state with an incompressible quantum fluid, rather than interpreting the fractional plateau as an ordinary filled electron band.^[ref]

A physicist.^[ref]

Robert Hofstadter investigated nuclear charge distributions through energetic electron scattering. His 1956 review compares $400\mathrm{MeV}$ electrons scattered from helium gas with predictions for point charges and several extended charge profiles. Beyond about $70^{\circ }$, the elastic scattering was over a hundred times smaller than the point-charge prediction. The extracted nuclear charge form factor distinguishes candidate spatial profiles, connecting measured angular counts with a model of nuclear size and shape.^[ref]

A physicist.^[ref]

Roberto B. Faria coauthored research on molecular orbital.^[ref]

A molecular orbital is a one-electron spatial wavefunction for the combined atomic nuclei of a molecule. For a normalized molecular orbital, the Born rule gives the position probability density $|\psi (\mathbf{r}){|}^2$, with ${\int }_{{\mathbb{R}}^3}|\psi {|}^2{d}^3\mathbf{r}=1$. The molecular orbital is the amplitude, rather than its squared modulus or a classical trajectory.

Robert S. Paton coauthored research on radical.^[ref]

A radical has one or more unpaired electrons. In a simple atomic orbital or molecular orbital occupancy picture, an unpaired electron occupies a spatial orbital without an opposite-spin partner in that orbital. A dot such as ${\mathrm{H}}^{\bullet }$ marks this electronic property; it does not mean a missing electron or positive electric charge. Free radicals can be neutral or charged. Terminology also has a boundary: metal ions exhibiting paramagnetism are not normally called free radicals just because they have unpaired electrons.

The Rutherford model places a concentrated central electric charge inside a much larger chemical atom. In 1911, Ernest Rutherford argued that rare large deflections were predominantly single close encounters, rather than accumulated small deviations in a diffuse-charge atom. He treated the central electric charge as positive for calculation while noting that the scattering deductions did not establish its sign. He explicitly deferred atomic stability.^[ref]

The physical inference is a compact region generating a strong electric field, not a miniature hard wall. Most impact parameters miss that region, while a small subset give large deflections. The model supplies a way to infer interior structure from outgoing angles, a central use of scattering throughout atomic physics and nuclear physics. This inference does not by itself specify electron quantum states or explain spectral lines.

The repulsive model also sets a probing scale. At closest approach, conservation of energy with zero radial velocity givesFor the numerical model above, even a head-on projectile turns at $r_{\min }=2a=45.504\mathrm{fm}$. This is a calculated closest-approach distance, not a measured atomic nucleus radius. A point-charge fit can constrain how compact the scattering centre must be, but cannot directly resolve structure at distances the projectiles never reach.

The Rutherford differential cross-section must not be extrapolated to zero angle: the unscreened point model has a forward divergence, while real neutral chemical atoms exhibit the screening effect, finite beam acceptance, and accumulated small deflections. At very short separation, finite nuclear size and the nuclear force can change the interaction. Nuclear recoil changes the relation between laboratory and centre-of-mass angles for lighter targets; appreciable energy loss or repeated large deflections invalidates the simple thin-foil scattering counts. These limits delimit the calculation rather than supplying evidence for a nuclear contact collision in the early experiment.

Rutherford scattering is elastic scattering by a Coulomb potential. In the repulsive classical case, a positively charged projectile bends around a positively charged atomic nucleus. A close approach can turn it backwards without a contact collision. The calculation connects an invisible impact parameter with a measurable outgoing angle, then with a differential scattering cross-section and a detector count.

Assume a projectile of mass $m$ and electric charge $ze$, a stationary point atomic nucleus of electric charge $Ze$, and positive $z,Z$. Here $e$ is the elementary charge. Neglect nuclear recoil, electron screening effect, energy loss, and the nuclear force. With vacuum permittivity ${\epsilon }_0$, defineHere $E$ and $v$ are the incident kinetic energy and speed far from the atomic nucleus, $b\ge 0$ is the impact parameter, and $L$ is the magnitude of angular momentum. The nonrelativistic limit requires $v\ll c$, with $c$ the speed of light. The fixed-target approximation requires the target mass to be much greater than $m$. In SI, $\kappa$ has units $\mathrm{J}\mathrm{m}$, $E$ has units $\mathrm{J}$, and $b$ has units $\mathrm{m}$.

A Rydberg atom has an electron excited to a large principal quantum number $n$. In a hydrogen-like model, the binding energy scales as $1/n^2$ and the orbital size as $n^2$. The outer electron is therefore weakly bound and its motion extends far beyond the compact core.

These properties make Rydberg atoms sensitive to applied electric fields and useful for studying controllable interactions such as Rydberg blockade. A field ramp can ionize different excited states at different thresholds, enabling state-selective field ionization. Real multi-electron atoms depart from the simple hydrogen-like spectrum, especially when the outer electron penetrates the core.

A salt bridge is an ionic connection between electrolyte compartments that limits bulk mixing while permitting electric charge transport. Its mobile ions compensate changes caused by the electrode half-reactions, keeping the bulk chemical solutions approximately electrically neutral during sustained electric current. It does not conduct electrons like the external wire, and its ions need not be consumed at either electrode. The ionic return path and compensating ion flows are explained in Purdue University, Electrochemical Reactions: salt bridge and charge balance.

In a discharging Daniell cell, anions moving toward the anode compartment and cations moving toward the cathode compartment both contribute to the internal conventional current from anode toward cathode. This charge-balance role is compatible with a negative anode: terminal signs do not imply that the whole anode chemical solution is negatively charged. The simple compensating-inventory example in Daniell cell does not fix all individual ion fluxes; diffusion and the cell's other mobile ions can contribute. A real bridge also introduces electrical resistance and may introduce a junction electric potential difference, so its presence does not guarantee an exact thermodynamic cell reading. See IUPAC Gold Book: salt bridge and liquid-junction potential.

Samuel C. C. Ting coauthored the 1974 observation of the narrow particle signal called J. A precise pair spectrometer measured the invariant mass of electron-Positron pairs produced when protons struck beryllium. A sharp feature near 3.1 GeV stood above the smooth background, indicating a new state rather than an arbitrary individual collision. This illustrates how a short-lived particle can be identified through reconstructed decay products.^[ref]

A physicist.^[ref]

A particle physicist who wrote a historical analysis of the Stern-Gerlach experiment. It distinguishes the experiment's original proposal and interpretation from its later connection with electron spin, and discusses the different problem of performing an analogous measurement on free electrons. This helps prevent later quantum concepts from being read back into the original experimental argument.^[ref]

The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

Saul Dushman investigated electron emission from metals as a function of temperature. Thermal emission connects the population of sufficiently energetic electrons with the energy required to escape a surface.

The strong temperature dependence makes the phenomenon useful in electron sources, while the surface state and work function affect quantitative interpretation.^[ref]

Physical chemistry.^[ref]

S. A. VanSlyke coauthored the 1987 demonstration of an efficient organic light-emitting diode with C. W. Tang at Eastman Kodak. Their device used two deposited organic layers between an indium-tin-oxide anode and a magnesium-silver cathode.

This architecture brought electrons and holes together near the organic-layer interface, producing visible electroluminescence at an applied electric potential difference below ten volts. It connected control of carrier injection and recombination to practical light output.^[ref]

The National Inventors Hall of Fame profile identifies the OLED co-inventor as Steven Van Slyke. The header preserves the original publication byline.

S. Bendjaballah coauthored research on density functional theory.^[ref]

A framework describing electronic ground-state properties using the electron density. Practical approximations require an exchange-correlation functional; this does not mean that explicit electrons have been replaced by a classical atom-only force field.

A scanning electron microscope scans a focused electron probe across a specimen and assigns a detector signal to each probe position. Secondary electrons leaving the irradiated region often reveal surface relief; backscattered electrons can provide composition-dependent contrast. Image brightness records the selected signal, rather than a transmitted projection enlarged by two lenses. Display enlargement is therefore set by the ratio of display width to scanned specimen width. Surface geometry and composition both affect brightness.^[ref]

A conducting scanning-probe tip acts as a movable electrostatic gate. Mapping conductance changes as the tip perturbs a quantum point contact or surrounding electron gas probes quantum transport. The image is a response to the tip, not automatically a noninvasive map of the unperturbed local current density.

Scanning transmission electron microscopy forms an image by scanning a focused electron probe across a thin specimen and recording transmitted or scattered electrons at each probe position. The detector's angular range determines which interaction contributes to the recorded brightness.

A detector including the direct beam gives a different image contrast mechanism from an annular detector collecting scattered electrons. High-angle annular dark-field imaging can emphasize composition and mass-thickness differences, but specimen thickness and channeling still affect interpretation. The probe size limits spatial detail, whereas the displayed-to-scanned length ratio sets magnification; increasing the latter cannot sharpen the probe.

The Schrödinger equation governs the evolution of a quantum state between instances of measurement in quantum mechanics. For a particle of mass $m>0$ moving in a prescribed real potential energy $V(\mathbf{r},t)$, its position wavefunction satisfiesHere $h$ is the Planck constant, ${\nabla }^2$ is the Laplace operator, and the Hamiltonian operator combines kinetic energy with potential energy. Specify an initial wavefunction, an operator domain and any boundary conditions: the differential expression alone does not completely specify a physical model. The Born rule then turns the evolving probability amplitude into position probabilities through $\rho =|\psi {|}^2$ and integration over a region.

This scalar Schrödinger operator describes nonrelativistic quantum mechanics, with relevant momenta much smaller than $mc$, where $c$ is the speed of light. It omits the constant rest energy $m{c}^2$, assumes fixed particle number, and suppresses active spin dynamics and coupling to a magnetic vector potential. A free electron with a fixed, uncoupled spin state can still use this scalar model; the Pauli equation supplies the additional spin components and magnetic coupling when needed. A prescribed real $V$ describes a closed evolution model or an approximation in which the source of that potential energy is treated as external. Dissipation, particle creation and a general measurement in quantum mechanics need further modelling.

For a time-independent self-adjoint Hamiltonian operator, the spectral theorem defines unitary time evolution byInitial states in the operator domain satisfy the differential equation in the Hilbert space sense; the unitary operator also evolves all other states in that Hilbert space. Time-dependent Hamiltonian operators need suitable regularity and domain hypotheses; their evolution is not generally the exponential of $-itH(t)/\hslash$.

The local reason for conservation is especially tangible. Multiply the Schrödinger equation by the complex conjugate ${\psi }^{\ast }$, subtract its conjugate multiplied by $\psi$, and use the fact that the real potential energy terms cancel. For smooth wavefunctions this gives the probability continuity equationThe divergence theorem identifies the change in probability inside a region with minus the outward probability current. Decay at infinity, or appropriate probability-conserving boundary conditions, preserves wavefunction normalization. Conservation of total probability does not require a stationary probability density.

Its usefulness comes from being a linear partial differential equation: quantum superpositions of solutions are solutions, while different quantum phases can produce interference. For a fixed Hamiltonian operator, the time-independent Schrödinger equation finds the energy eigenstates; each then acquires just its own time-dependent quantum phase. One eigenvalue problem therefore organizes both allowed energies and the evolution of arbitrary quantum superpositions, with a continuous spectrum included when required. This is the attractive bridge between a spatial wave problem and a clock: energy differences determine the rate at which relative quantum phases change. It makes atomic spectral lines, quantum tunnelling and quantum scattering calculable from a specified model.

For an explicit example without confinement, take a free particle on the whole line, $V=0$, initially in a Gaussian wave packet with position standard deviation ${\sigma }_0$ and mean momentum $\hslash k_0$:Unlike a single plane wave, this wavefunction satisfies wavefunction normalization. A Fourier transform writes it as a quantum superposition of plane waves. Their dispersion relation is $\omega (k)=\hslash k^2/(2m)$, so each Fourier component gains the factor $e^{-i\hslash k^{2}t/(2m)}$. Completing the square in the resulting Gaussian integral giveswith the square root chosen continuously from $t=0$. Its position probability density and standard deviation areThe centre moves with the group velocity ${\omega }^{\prime }(k_0)=v_0$ and the wave packet spreads because different wavenumbers have different group velocities. This is unitary wave dispersion, without a random force or loss of probability. Independently, the initial momentum standard deviation is $\Delta p=\hslash /(2{\sigma }_0)$ and the initial position-momentum covariance vanishes. Free evolution gives $x(t)=x(0)+p(0)t/m$ in the Heisenberg picture, henceThis independently checks the spreading law. The Gaussian integral of $\rho$ is one at every time; the decreasing peak is exactly offset by increasing width.

As hypothetical preparation parameters, choose a free electron with ${\sigma }_0=1\mathrm{nm}$ and $k_0=2{\mathrm{nm}}^{-1}$. Using $\hslash \simeq 1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$ and $m\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ givesThe rounded electron mass uses the 2022 adjustment.^[ref] At $t=t_s$, the centre is $4\mathrm{nm}$ and ${\sigma }_t=\sqrt{2}\mathrm{nm}$; at $t=2t_s$, they are $8\mathrm{nm}$ and $\sqrt{5}\mathrm{nm}$. The mean kinetic energy staysIt includes both mean motion and the momentum spread. The mean speed is about $7.72\times 10^{-4}c$ and the momentum spread corresponds to $1.93\times 10^{-4}c$, making the nonrelativistic model accurate for the appreciably occupied part of this ideal Gaussian distribution. Its mathematically unbounded tails are an idealization.

The complex width also explains where the probability goes. Write $\psi =\sqrt{\rho }{e}^{i\theta }$, where $\theta$ is the dimensionless quantum phase. Differentiating the exact solution givesRelative to the moving centre, the probability current $j_{\mathrm{rel}}=j-v_0\rho$ points outward on both sides for $t>0$. This is a flux statement, not an additional assumption about individual particle trajectories. At $t=t_s$ and $x-v_{0}t=\pm {\sigma }_0$, the relative flow speeds $j_{\mathrm{rel}}/\rho$ are $\pm {\sigma }_0/(2t_s)$, about $\pm 2.8942\times 10^4\mathrm{m}{\mathrm{s}}^{-1}$. The probability within a moving interval of half-width ${\sigma }_0$ falls from about $0.6827$ initially to $0.5205$ at $t_s$, exactly because of this outward probability current.

Historically, the 1926 Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger began with the stationary hydrogen atom problem. It distinguishes discrete negative energies from a positive continuous spectrum; it is not yet the general evolution equation displayed above.^[ref] The opening of quantum mechanics discusses its original vibrating-string analogy.

In the 12 December 1933 lecture The fundamental idea of wave mechanics by Erwin Schrödinger, Erwin Schrödinger explains the move from independent rays to waves through diffraction, then applies that analogy to atomic physics. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern Born rule from classical mechanics.^[ref]

The dates separate proposal from measurement: the de Broglie wavelength belongs to the 1924 thesis, the Schrödinger equation to 1926, and the Davisson–Germer experiment's electron diffraction beams to 1927.^[ref] That linked experiment describes the prepared nickel crystal, evacuated electron gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector electric current tests wave-dependent scattering probabilities; it neither measures a wavefunction directly nor establishes this equation uniquely.

Electrons partially shield an atomic nucleus electric charge. At large distances a neutral chemical atom therefore does not generate the unscreened Coulomb potential of its bare atomic nucleus.

A secondary electron is emitted from a material after excitation by incident radiation or particles, rather than simply being a high-energy primary electron redirected back out. Many detected secondaries have low enough kinetic energy that escape strongly favors regions near a surface.

This makes secondary-electron collection useful for secondary-electron topographic contrast in a scanning electron microscope. Surface angle, local electric fields and collector bias affect the signal as well as emission itself. Backscattered primaries can also excite secondary electrons away from the probe's immediate landing position, so every detected secondary need not originate directly beneath the nominal probe.^[ref]