For two liquid half-spaces, integrate the magnitude of a dispersion attraction per area $A_H/(6\pi h^3)$ from a molecular contact cutoff $h_0$ to infinity. The work per area is $A_H/(12\pi h_0^2)$. Because cleavage creates two interfaces, each has the displayed surface tension in this approximation. The cutoff is essential: a continuum attraction cannot describe atomic overlap and short-range repulsion. This estimate neglects interfacial entropy, detailed density profiles and nonadditive interactions, and it cannot replace the metallic bonding or hydrogen bonds that dominate some liquids' cohesion.

A force caused by an interface through its contact-line surface tension and the capillary pressure it produces. Both contributions must be counted with consistent geometry. For a narrow, completely wetting liquid bridge between flat plates of wetted radius $a$ and gap $d\ll a$, the leading attractive force is $2\pi a^2\gamma /d$.

Drying through a fluid's thermodynamic critical point to avoid traversing an ordinary liquid-vapor coexistence interface. Removing the receding meniscus can protect compliant microstructures from capillary force during release. Solvent exchange and compatibility with the structure remain practical constraints.

A repulsive optical obstacle probes dissipation by being moved through a condensate. Heating as a function of speed tests low-speed flow stability; stationary-obstacle and thermal-cloud controls help separate other heating mechanisms. The measured onset need not equal the homogeneous critical velocity of a homogeneous condensate because density gradients, obstacle shape and vortex nucleation matter.

Two weakly linked Bose-Einstein condensates form a bosonic Josephson junction. Their relative phase controls a particle current through the weak link, analogous to the pair current of a superconducting Josephson junction. For neutral reservoirs the phase evolution follows a chemical-potential difference rather than an electric voltage. Interactions can make the tunneling dynamics nonlinear, including self-trapping at sufficiently large imbalance.

A superconducting material supports a superconducting phase under an appropriate range of temperature, magnetic field and current. Those conditions, along with its gap structure and other microscopic properties, determine how it can be used.

Critical temperature alone is insufficient to compare applications. A wire may need a high critical current in a strong magnetic field, while a sensitive device may require low dissipation and well-controlled interfaces. Structural defects can pin vortices and alter current-carrying performance, making sample preparation and measurement conditions part of a useful material specification.

Coupling a charged condensate's phase to electromagnetism changes the neutral Goldstone response into electromagnetic collective dynamics. In a bulk long-wavelength regime, the longitudinal phase-density mode has a plasma gap. Its dispersion and damping require a dynamical model; a static Ginzburg-Landau functional alone does not fix the entire excitation spectrum.

The condensation energy of a superconductor is the free-energy lowering on entering the superconducting state, with sign convention specified. For a local Landau density $a|\Delta {|}^2+b|\Delta {|}^4/2$ and $a<0$, its minimum is $-a^2/(2b)$ relative to the normal-state value. The positive energy required to destroy the condensate is the opposite sign.

An electromagnet whose winding uses a superconductor, allowing large currents without the normal winding's steady resistive loss in its superconducting state. Refrigeration, strain, field-dependent critical current and superconducting quench protection remain practical constraints. Magnetic resonance imaging, nuclear magnetic resonance and particle accelerators are applications.

Superconducting pairing outside the simple conventional isotropic spin-singlet scenario, with the precise classification depending on the context and evidence. Gap symmetry, pairing mechanism, spin structure and time-reversal behavior are separate questions; a nonstandard gap does not establish every other proposed broken symmetry.

The identity $(a^2+b^2)(c^2+d^2)=(ac-bd{)}^2+(ad+bc{)}^2$ shows that a product of two sums of two squares is another sum of two squares. It follows by multiplying Gaussian integers and taking their squared absolute values.

${\mathrm{S}}_4^{2+}$, a square-planar sulfur ring with a useful six-pi-electron aromatic description. Its 22 valence electrons are partitioned differently from a five-vertex Zintl cluster.

An ionic compound containing divalent magnesium and sulfate ions. Its nominal formula concentration $c$ contributes ionic strength $4c$ when fully dissociated into these two ion species.

A normalized ground state initially supported on $(0,a)$ is $\sqrt{2/a}\sin (\pi x/a)$. Expanding the infinite square well to $(0,2a)$ leaves this wavefunction unchanged during an ideal sudden switch. Its overlap with the new ground state $a^{-1/2}\sin (\pi x/(2a))$ is $4\sqrt{2}/(3\pi )$, so the Born rule gives $32/(9{\pi }^2)$. This differs from ground-state overlap after sudden expansion of a square well with a fixed midpoint, because translation of the new well changes the overlap. Under a fixed final Hamiltonian operator, all final energy eigenstate probabilities remain constant during subsequent unitary time evolution.

A second-order differential equation in the displayed divergence form. Under the usual regular spectral assumptions, $p>0$, $w>0$, the coefficients have suitable regularity, and separated self-adjoint boundary conditions specify a Sturm-Liouville problem. Multiplying by $y$ and using zero endpoint values gives the Rayleigh quotient $\lambda =\int (p{y}^{\prime 2}+q{y}^2)dx/\int w{y}^{2}dx$ for a nonzero real eigenfunction. Sign conventions for the potential and eigenvalue vary; they must be read from the stated equation.

Sequential reaction at the two alpha positions of a cyclic enamine, often making a bridged ring.

Stabilization by a surface-bound polymer layer that supplies both conformational/osmotic repulsion and electrostatic repulsion. Chain coverage, solvent quality and screening determine the resulting interaction barrier.

Repulsion associated with overlap of surface-bound layers, often through chain confinement and osmotic free-energy costs. Layer thickness, solvent quality and grafting/adsorption state matter; it is not simply a bare hard-sphere collision.