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A transverse confinement eigenstate that defines a threshold for propagation along a narrow conductor. Above that threshold, its longitudinal dispersion supplies an incident channel from each reservoir. spin can label separate channels; an unresolved spin-degenerate mode contributes twice the single-spin spin-resolved conductance quantum. This physical use of mode is unrelated to the most probable value of a probability distribution.

For exchange $J>0$, the Jordan–Wigner transformation gives a band ${ϵ}_k=h-J\cos k$. At zero temperature, negative-energy states form a filled interval $|k|<\arccos (h/J)$. Since each fermion decreases total magnetization by one, the thermodynamic magnetization per site is $1/2-\arccos (h/J)/\pi$. It saturates at $1/2$ for $h\ge J$. A finite ring instead has discrete level crossings and parity-dependent momentum grids.

Suppression of tunnelling because adding or removing a discrete island charge costs more energy than the accessible transport and thermal energy window. Weak junction coupling and small island capacitance make it observable.

A biased double-barrier structure whose intermediate quasi-bound levels control transmission from reservoirs. Bias-dependent alignment can produce negative differential resistance when a resonance moves out of the available injection window.

At energy equal to a finite rectangular barrier height, the stationary interior Schrödinger equation has a linear solution $\psi =bx+c$. Its probability current can be nonzero: $J=(\hslash /m)\mathrm{Im}(c^{\ast }b)$. Setting the interior classical kinetic energy to zero does not set the transmitted quantum flux to zero.

Transport by successive real charge-state changes across weak tunnel barriers. Charge electrochemical potentials must enter the source-drain bias window; outside it sequential transport is Coulomb blocked. Higher-order virtual cotunnelling is a distinct process.

Finite lifetime of an island state due to tunnelling into leads gives an energy-width scale of order $\hslash \Gamma$. The numerical factor depends on whether $\Gamma$ denotes a population decay rate, an amplitude decay rate or a sum of partial rates.

A bosonic state with Gaussian phase-space characteristic function, specified by its first and second optical quadrature moments. Vacuum, coherent states, optical squeezed vacuum and thermal states are examples. Gaussian here describes phase-space statistics, not a Gaussian optical pulse envelope.

A particle incident below a semi-infinite step has a unit-modulus quantum reflection amplitude but a nontrivial phase. Continuity of the oscillating and evanescent solutions yields the displayed coefficient, for the incident phase origin at the interface. Its decaying exterior wave has zero net probability current. The phase depends on the chosen scattering convention; the unit reflection probability does not.

A quantum reflection amplitude is the complex ratio of the outgoing reflected wave to the incident wave in a chosen scattering convention. In one reflected channel with the same exterior speed, the reflection probability is $|r{|}^2$. Its phase affects interference even when its modulus is one; shifting the reference plane changes that phase.

When the free angular-momentum phases all recur at the driving period, successive identical kicks add coherently. For the full revival $\hslash T/(2I)=2\pi$ and an initial zero-angular-momentum state, the kinetic energy after $N$ kicks is ${\hslash }^2{\eta }^2{N}^2/(4I)$. This ballistic law is specific to the stated resonance and boundary conditions.

Pair production creates a particle and its antiparticle while conserving energy, momentum and the relevant charges. The available invariant mass must reach the sum of their rest masses.

Preservation of relative electronic phase along alternative paths so they can interfere. Loss of coherence suppresses interference structure, but semiclassical magnetic focusing can persist if momentum trajectories remain sufficiently ballistic.

For massless distinguishable quarks exchanging one gluon, summing the squared generator contraction gives two with $\mathrm{Tr}(T^a{T}^b)={\delta }^{ab}/2$. Averaging nine initial colors and four initial spins leaves the coefficient $1/18$ multiplying the two Dirac traces. Their contraction is $8(s^2+u^2)$, yielding the displayed tree-level result. Identical quarks require an additional exchange amplitude and its interference.

Within the $l=1$ multiplet, $|1,0⟩$ has energy ${\hslash }^2(a_x+a_y)$. The symmetric and antisymmetric combinations of $|1,1⟩$ and $|1,-1⟩$ have the other two pairwise coefficient sums. Principal-axis permutation relates these energies; equality requires corresponding coefficient degeneracies rather than angular-momentum conservation alone.

A quadric surface obtained by extending a parabola along a fixed direction outside its plane. For example, $x=c-y^2/(4c)$ with arbitrary $z$ is a parabolic cylinder parallel to the $z$ axis. Each cross-section of fixed $z$ has vertex $(c,0)$ and focus $(0,0)$ in its $xy$ plane. It occurs as a coordinate surface in parabolic cylindrical coordinates.