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An electron annihilation operator restricted to the no-double-occupancy sector of the t-J model. It removes a spin-$\sigma$ electron only when the opposite-spin site occupation vanishes. The corresponding creation operator cannot create a doubly occupied site. These projected operators do not obey the unrestricted canonical on-site fermion algebra on all states.

The proton is a positively charged spin-$1/2$ baryon with ordinary valence quark content $uud$. It carries electric charge $+e$ and baryon number one. A hydrogen atomic nucleus is a single proton, while other ordinary atomic nuclei contain protons together with neutrons.

A proton is composite even when an atomic calculation treats it as a pointlike source. Electron scattering at increasing momentum transfer probes its spatial and magnetic response, whereas low-energy atomic behavior often needs only its charge, mass and magnetic moment. Its mass is about $1836$ electron masses; that ratio does not make it a bound collection of electrons.

The proton–proton chain first produces deuterium by a weak reaction and then helium-3 by proton capture. pp I ends with ${}^3\mathrm{He}+{}^3\mathrm{He}\to {}^4\mathrm{He}+2p$. pp II passes through beryllium-7 electron capture to lithium-7 and then proton capture. pp III passes through boron-8 beta-plus decay and beryllium-8 breakup. Each branch produces one net helium-4 nucleus from four protons, but the neutrino spectrum and deposited heat differ. In pp II the net weak reactions involve one emitted positron and one captured electron.

Unintended exposure near a written feature caused by electron scattering in the resist and substrate. The deposited-energy distribution is broader than the incident probe, requiring dose or geometry correction for dense and isolated patterns.

A p-type semiconductor has more mobile electron holes than electrons; it can nevertheless be electrically neutral overall.

A radio wave traversing a dilute cold plasma has a group velocity that depends on its frequency. In the ideal unmagnetized limit, the dispersion relation is ${\omega }^2=c^2{k}^2+{\omega }_p^2$, where ${\omega }_p$ is the angular plasma frequency. For $\omega \gg {\omega }_p$, differentiate to obtain $v_g=c\sqrt{1-{\omega }_p^2/{\omega }^2}$, then expand $1/v_g$ to first order. The excess travel time over a vacuum path isHere $f=\omega /(2\pi )$ is ordinary frequency, $e$ is the magnitude of the elementary charge, $m_e$ is the electron mass, ${ϵ}_0$ is the vacuum permittivity, $n_e$ is electron number density, and $c$ is the speed of light. This approximation neglects magnetic corrections, collisions and multipath scattering; its positive sign means lower frequencies arrive later.

A planar six-membered aromatic ring with nitrogen at opposite ring sites. Its molecular point group is the D2h point group. The six ring $p_z$ atomic orbitals form a six-electron conjugated system; the in-plane nitrogen lone pairs are outside this $\pi$ basis.

The electronic contribution to differential capacitance arising from a change of chemical potential as particles are added. For noninteracting electrons at zero temperature, it is $e^2$ times the density of states of the whole sample at the Fermi energy. The quantum capacitance of a graphene cone gives a worked example.

An electric field normal to a semiconductor quantum well shifts the electron and hole confinement energies and displaces their envelopes in opposite directions. The exciton optical transition usually redshifts and its oscillator strength falls as their overlap decreases. In a symmetric unbiased well the permanent dipole $p_0$ vanishes, so the leading weak-field shift is quadratic. This optical shift includes confinement and binding changes; it is not solely a change in exciton binding energy.

A quantum dot confines electrons and electron holes in all three spatial directions, producing discrete energy levels. A semiconductor island surrounded by a larger-band gap material is one realization. Its optical excitation and electron-hole recombination depend on confinement, Coulomb interactions and electric-dipole selection rules.

Filling a degenerate orbital shell produces a larger addition energy for the next electron. In a circular two-dimensional harmonic potential, shell $q=n_x+n_y$ has $q+1$ orbitals and capacity $2(q+1)$ including spin. Cumulative closed-shell occupancies are $(q+1)(q+2)=2,6,12,20,\dots$. Charging and orbital contributions must be separated, and anisotropy or exchange can split the ideal shells.

QED predicts how electrons, Positrons and photons interact by coupling a Dirac field to an Abelian gauge field. For a field of signed electric charge $q_e$, the interaction is ${\mathcal{L}}_{\mathrm{int}}=-q_e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$; an electron has $q_e=-e$, $e>0$. Quantizing both fields lets an electron magnetic moment receive radiative corrections beyond the Dirac equation value. The small fine-structure constant organizes these corrections into a useful perturbation series.

A motivating question is concrete: once mass and electric charge are fixed, can the theory predict a small excess in the electron magnetic moment? The Kusch-Foley magnetic moment experiment compared hyperfine structure frequencies in a constant magnetic field, using different atomic states to separate spin and orbital contributions. Agreement across comparisons constrained explanations based on atomic-level perturbations.^[ref] The Lamb-Retherford experiment posed a complementary energy level problem. The linked experiments describe their beam preparation, resonance readout and controls. Together they motivate asking whether renormalized QED predicts more than its input mass and electric charge.

Choose conventional surface current density along $+x$, magnetic field along $+z$, and negative electron charge. An isotropic in-plane response has the following sheet conductivity and sheet resistivity tensors for $s\ge 0$, $k>0$:These are matrix inverses, rather than componentwise reciprocals. The sheet conductivity has units siemens, the sheet resistivity ohms. With $K_y=0$, the sign is $E_y/K_x<0$, agreeing with the negative Hall coefficient of a single electron species. Contact polarity can reverse the measured voltage sign, so quote the Hall magnitude. For a homogeneous strip of width $W$ and longitudinal probe separation $L$, $I=W{K}_x$ givesAt $s=0$, $k=N$, these reduce to $R_K/N$ and zero. A nonzero transverse sheet resistivity creates no local Joule heating: the antisymmetric contribution cancels in $\mathbf{E}\cdot \mathbf{K}=(e^2/h)s|\mathbf{E}{|}^2$.

For an original nonideal example, choose $k=2$, $s=0.01$ and $L/W=2$. Then $V_{xx}/I\simeq 129.061\Omega$, while the Hall magnitude is below $R_K/2$ by the relative amount $s^2/(k^2+s^2)\simeq 25.0$ parts per million. An unwanted admixture $0.02V_{xx}$ in the Hall readout would instead introduce approximately $2.58\Omega$. The chosen calculation shows why a small intrinsic tensor correction does not by itself guarantee an accurate contact measurement.

The quantum Hall effect connects an electrical measurement to a quantum counting rule: a two-dimensional electron gas in a perpendicular magnetic field can exhibit plateaus of transverse electrical resistance. The integer case supplies a resistance scale built from the Planck constant and elementary charge, rather than the sample's dimensions. The fractional quantum Hall effect involves additional many-electron physics.

A quantum particle is a quantum system whose possible measurement outcomes are described by a quantum state rather than a definite classical trajectory. For a single spinless particle, a wavefunction gives a probability amplitude for each position; the Born rule turns its squared modulus into a probability density. A localized detection does not imply that the particle followed a known path before detection.

An electron sent through an interferometer is a useful example: individual arrivals are localized, while many arrivals reveal matter-wave interference. Changing the relative quantum phase of the available paths changes the detection distribution. For several identical quantum particles, bosonic or fermionic exchange symmetry further restricts the joint quantum state.

A short electrostatically defined constriction connecting two electron reservoirs. Transverse confinement produces transport modes, and sufficiently narrow contacts can collimate carriers for electron-focusing experiments. Its quantum conductance depends on mode transmission; an opening described only as narrower than a micrometre is not automatically in a single-mode limit.

Quantum transport studies the flow of particles, electric charge or energy when quantum states, their occupation and their transmission probabilities determine the flow. In a short quantum wire, transverse quantum confinement permits only discrete transverse electronic modes, even though motion along the wire remains possible.

The Landauer formula treats two contacts as reservoirs supplying incident electrons. Their different occupations drive an electric current; a potential barrier changes the transmission probability of each incident mode. At low temperature, opening one perfectly transmitted spin-resolved mode increases the electrical conductance by the spin-resolved conductance quantum $e^2/h$. Quantum tunnelling, interference and Coulomb blockade explain departures from a simple classical resistor picture.

A quasi-Fermi level is a local electron or hole population parameter in a semiconductor driven out of equilibrium, assuming each population is internally close enough to thermal equilibrium to use a Fermi-Dirac distribution. Separate electron and hole parameters replace one common equilibrium chemical potential. Their spatial profiles depend on transport, generation and recombination as well as electrostatics. For ideal ohmic contacts with the p voltage $V$ relative to n, the contact electron electrochemical potentials obey ${\mu }_p-{\mu }_n=-eV$.

The difference between electron and hole quasi-Fermi levels under nonequilibrium conditions. For ideal selective contacts their terminal separation sets $qV$; poor contacts can lose part of the internal splitting.

The Q value of a reaction or particle decay is the initial total rest energy minus the final total rest energy. Positive $Q$ releases kinetic energy; a negative $Q$ requires energy supplied by incident motion or another source. Atomic and bare nuclear mass conventions must be distinguished when counting electrons in beta decay.