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The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.

Wave reflection returns part of an incident wave into the medium from which it arrived, as a consequence of an interface or spatial variation of material properties. An amplitude reflection coefficient compares the reflected and incident waves; its squared modulus gives their wave intensity ratio when both use the same medium and normalization. Reflected energy has not necessarily undergone wave absorption.

Wave scattering is redistribution of an incident disturbance by an object or spatial variation in a medium. A small obstacle in a ripple tank redirects part of an incident water wave, and a variation of refractive index can redirect light. In a linear model, the total field is the sum of the prescribed incident wave and the outgoing scattered wave.

The scattered field carries information about the object's size, shape and material response relative to the wavelength. A far-field detector measures its angular distribution; the differential cross-section for scalar wave scattering compares scattered energy flux with incident energy flux. Wave scattering can remove power from a forward beam without wave absorption, because that energy can leave in other directions. Repeated encounters lead to multiple scattering. When interpreting a transmission experiment, collecting radiation over a larger angle helps distinguish redistributed power from genuinely absorbed power.

Weak coupling means a suitably dimensionless interaction strength is small. In BCS theory, ${\nu }_{s}V\ll 1$ gives a gap exponentially small compared with the pairing-shell energy cutoff; the expansion parameter is not the dimensional quantity $V$ alone.

For the imperfect soft Duffing-van der Pol oscillator at positive small $\lambda$ and $|\epsilon |\ll \sqrt{\lambda }$, the symmetric connection of the Hamiltonian system is $v=\sqrt{\lambda }\tanh (\sqrt{\lambda /2}t)$. Its damping integral is $4{\lambda }^{3/2}(\kappa -\lambda /5)/(3\sqrt{2})$. The two saddle equilibrium barrier heights differ by $-4\epsilon {\lambda }^{3/2}/3$ to first order. Equating the energy change to this difference gives the two oppositely directed connection curves in the display. These are weak-damping asymptotic curves, not exact global formulas at order-one parameters.

// Destination: astrophysical-fluid-dynamics.bigb

Use the leading nonrelativistic magnetic coupling for an electronic level with total angular momentum operator $\mathbf{J}$, and choose the $z$ axis along a uniform external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$, $B\ge 0$. Write its magnetic quantum number as $m_J$, so $J_z|J,m_J⟩=\hslash m_J|J,m_J⟩$. There are $2J+1$ possibilities from $-J$ to $J$. With Landé g-factor $g_J$, the first-order shift isHere $e>0$ is the magnitude of the electron electric charge, $m_e$ its mass, $h$ the Planck constant, and ${\mu }_B$ the Bohr magneton. Both $g_J$ and $m_J$ are dimensionless; with $B$ measured in tesla and ${\mu }_B$ in joules per tesla, $\Delta E$ is in joules. The photon frequency shift $\Delta {\nu }_{ul}$ is in hertz, not angular frequency; indices $u,l$ denote upper and lower levels. NIST states this weak-field convention and its orbital/spin assumptions explicitly.^[ref]

The sign follows directly from magnetic dipole potential energy. Within this electronic $J$ multiplet, the projected magnetic dipole moment is $\mathbf{\mu }=-g_J{\mu }_B\mathbf{J}/\hslash$. Therefore $H_Z=-\mathbf{\mu }\cdot \mathbf{B}$ has the displayed positive coefficient. Subtracting the shifts of the two levels and using photon energy $h{\nu }_{ul}=E_u-E_l$ gives the transition formula. For the orbital case, ${\mu }_B/h=e/(4\pi m_e)$ gives the same leading frequency spacing as the classical charged oscillator. This explains the historical sensitivity to charge divided by mass; the quantum state derivation here is modern.

“Weak” compares interaction energies: mixing with neighbouring fine structure levels must be small, so $J$ remains a useful quantum number. The formula keeps the term linear in $B$ and neglects quadratic shifts. It also omits hyperfine structure; if that is resolved and controls the coupling, the appropriate total electronic-plus-nuclear angular momentum and splitting factor must replace $J,g_J$. Weak compared with fine structure does not automatically mean weak compared with hyperfine structure.

The weak Huygens principle is the domain-of-dependence statement that initial data outside a backward light cone cannot influence its tip. It permits an interior tail and is weaker than the Strong Huygens principle. For $u_{tt}-\Delta u+V(x)u=0$ with $V\ge 0$, integrating the local energy over shrinking balls proves this statement: the boundary contribution is $u_t{\partial }_{\nu }u-\frac{1}{2}(u_t^2+|\nabla u{|}^2+V{u}^2)\le 0$.

The weak interaction changes particle species through the charged current and also acts through the neutral current. Familiar examples include beta-minus decay and neutrino scattering. It is short-range and chiral: its W boson and Z boson couplings distinguish chirality. Above the electroweak scale it belongs to the unbroken $SU(2{)}_L\times U(1{)}_Y$ electroweak interaction. At much lower energies, W boson exchange is approximated by the Fermi interaction.

A particularly direct window into the weak interaction is the Wu experiment: a sample with oriented nuclear spins emits electrons preferentially toward one end of its spin axis. The comparison tests parity conservation, because spin is an axial vector and momentum is a polar vector. The polarized beta-decay angular distribution translates that geometric distinction into a measurable counting asymmetry.

At large but finite magnetic field, opposite-spin admixtures in approximate product eigenstates let nominally spin-forbidden electric-dipole channels borrow intensity. Their amplitudes scale as spin-orbit interaction energy divided by Zeeman energy; their intensities therefore vanish quadratically in that ratio.

Surface water counted as ice needs warming but has already undergone melting. Its presence therefore biases a dry-ice estimate of latent heat of fusion downward. This is why drying the added ice is a quantitative control in ice-melting calorimetry, rather than cosmetic preparation. Let $m_a$ be the total added mass at $T_m$, of which a fraction $r$ is already liquid and $(1-r)$ remains solid. The correct constant-property balance isIf an analysis incorrectly treats all $m_a$ as solid and sets $Q_{\mathrm{env}}=0$, it obtainsThus $r=0.10$ gives a $10\%$ low estimate with no heat leak. Unaccounted incoming heat also lowers the estimate; outgoing heat raises it. These signs follow directly from the displayed energy balance. Conversely, dry ice initially below $T_m$ absorbs additional sensible heat; neglecting that term overestimates $L$. These are different biases and may partially cancel, so agreement with a reference value alone does not validate the controls.

For this model with $r=0$ and $Q_{\mathrm{env}}=0$, the balance predicts $T_f-T_m\simeq 8.829\mathrm{K}$, with $16.700\mathrm{kJ}$ consumed by fusion and $1.845\mathrm{kJ}$ by warming the melted water. At $r=0.10$ with the same zero leak, it predicts $T_f-T_m\simeq 10.369\mathrm{K}$ and $L_{\mathrm{app}}=300.6\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$. A net incoming $1\mathrm{kJ}$ shifts $L_{\mathrm{app}}$ downward by a further $20\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$ because $m_a=0.050\mathrm{kg}$. The warmer final water in the wet case is evidence of a smaller fusion budget, not a larger true $L$.

W. G. Penney developed a solvable model of electrons in a periodic lattice with R. de L. Kronig. Their 1931 calculation replaced the complicated crystal environment by a repeating one-dimensional potential, exposing allowed and forbidden electron-energy intervals.^[ref]

The Kronig-Penney model makes energy bands concrete: matching wave functions across repeated cells constrains the possible energies. Its value lies in isolating the effect of periodicity, not reproducing every interaction in a real solid.

A white dwarf is a compact stellar remnant supported mainly by electron degeneracy pressure. Its ions carry most of the mass, while its electrons provide most of the supporting pressure. The Pauli exclusion principle makes compression fill quantum states with larger momenta; support therefore persists even when thermal motion becomes small. Cooling and hydrostatic equilibrium are separate questions: a white dwarf can radiate stored energy while retaining degeneracy support.

The surprising trend is that a more massive cold white dwarf is smaller. Its higher mass density raises the Fermi momentum, eventually requiring special relativity. The degenerate-electron pressure crossover explains why this response approaches a limiting stiffness and why the Chandrasekhar limit exists. Subrahmanyan Chandrasekhar's 1983 lecture develops this connection between quantum statistics, stellar structure and stellar evolution.^[ref]

White-dwarf cooling is the gradual loss of a white dwarf's stored thermal energy after sustained core fusion has ended. Dense degenerate electrons provide most of the pressure support, so luminosity can decline while the star remains mechanically supported.

Cooling depends on core heat capacity, envelope insulation, early neutrino losses and later crystallization processes. A population's luminosity distribution therefore contains age information only through a cooling model and its formation history. Crystallization can delay the decline through latent heat and compositional energy release; luminosity is not a simple universal linear clock.

A Wick rotation deforms a time or energy contour from real to imaginary values, converting suitable Minkowski integrals into Euclidean integrals while respecting pole prescriptions.

The Wiedemann-Franz law relates the electronic thermal conductivity of a normal degenerate metal to its electrical conductivity:An isotropic kinetic theory estimate gives $K_e=c_e{v}_F^2{\tau }_Q/3$. With electronic heat capacity density $c_e={\pi }^{2}n{k}_B^{2}T/(2E_F)$, $v_F^2=2E_F/m^{\ast }$ and $\sigma =n{e}^2{\tau }_{\sigma }/m^{\ast }$, the ratio is $L_0{\tau }_Q/{\tau }_{\sigma }$. Approximately elastic scattering with a smooth energy dependence gives equal heat and charge relaxation times. Inelastic electron-phonon scattering can make them different; the law also concerns the electronic contribution, not additional lattice heat conduction.^[ref]

An experimental nuclear physicist whose electron speed-and-energy measurement provides a direct test of relativistic kinematics. Bertozzi's electron speed-energy experiment measures flight time and calorimetric energy independently.

Official profile: physics.mit.edu/faculty/william-bertozzi/

Willis Lamb and Robert Curtis Retherford measured a small separation between hydrogen energy levels that the simplest relativistic quantum mechanics placed at equal energy. Their 1947 experiment used microwave transitions involving metastable atoms, rather than trying to resolve the separation in an ordinary optical line. The Lamb shift required improved quantum electrodynamics, including the interaction of an electron with the quantized electromagnetic field. A small spectral discrepancy thereby became a quantitative test of the theory.^[ref]

A physicist.^[ref][ref]

Charles Thomson Rees Wilson's 1927 lecture recalls electron cloud trails produced with X-rays in 1911, then alpha particle trails from a radium-tipped source. The improved 1912 chamber had a wide, shallow space with a suddenly falling floor to limit stirring. A roof–floor clearing electric field removed unwanted ions. Sharp photography required expansion, then the ionizing particle, then illumination.^[ref]

The piston mechanism connects the space beneath the moving floor to an evacuated vessel: opening the valve lets the floor drop. Water in a trough supplies the vapor. The Smithsonian's surviving demonstration object is explicitly a 1961 replica of the 1912 apparatus.^[ref]

After dust removal, X-rays increased droplet formation above a lower expansion threshold; larger expansions also produced dense background cloud.^[ref]

The control logic separates a localized ionization trail from fog. Delays permit ion displacement and drop growth; stirring deforms the trail. A photograph alone supplies neither an absolute event time nor a calibrated particle energy.

A binary in which a black hole captures part of its companion's stellar wind. The captured gas can dissipate energy and, with sufficient specific angular momentum, form an accretion disk. This differs from gas overflowing a Roche lobe through the inner Lagrange point. The fraction captured, its angular momentum and changes in the wind affect the accretion flow and emitted X-rays.

A device combining sensing, local processing, a radio and a power subsystem. An energy budget must account for sleep leakage, active operation, startup and peak communication current; stored energy can bridge intermittent harvesting.