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The local energy difference between the conduction and valence bands along the boundary of the first Brillouin zone. In a nearest-neighbour model it vanishes at the corner valleys and reaches $2t$ at edge midpoints; this angular variation is compatible with a globally gapless material.

Assign a real value $x_i$ to each vertex. The Graph Laplacian quadratic form measures disagreement across edges:This makes the Graph Laplacian a positive semidefinite matrix. The energy is small when neighbours have similar values, even if distant vertices have very different values; it depends on connections rather than where the graph is drawn. In the three-vertex path graph, take $x=(2,0,-1{)}^T$. Thenand the two edges contribute $(2-0{)}^2=4$ and $(0-(-1){)}^2=1$.

For a short proof, expand $x^T(D-A)x$. Each edge contributes $x_i^2+x_j^2$ through the degree matrix and $-2x_i{x}_j$ through the adjacency matrix, yielding $(x_i-x_j{)}^2$. The sum over unordered edges has no factor $1/2$; a sum over all ordered pairs does:Equivalently, let $B$ be an oriented incidence matrix, with one row per edge. Then $Bx$ records endpoint differences and $L=B^{T}B$, independently of the chosen orientations. This translates a local comparison into a global quadratic form and makes its nonnegativity visible.

For an isolated spherical distribution, shell assembly gives gravitational potential energy $W=-{\int }_0^{\infty }GM(r)r^{-1}dM(r)$. Use $S^{\prime }=-2\pi r\rho$ and $M^{\prime }=-2r{S}^{\prime }$ to write $W=2G\int M{S}^{\prime }dr=-2G\int S{M}^{\prime }dr=4G\int rS{S}^{\prime }dr$. A second integration by parts gives the displayed negative quadratic functional. The boundary terms $MS$ and $r{S}^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.

Gravitational energy is the potential energy associated with gravity. In Newtonian gravity, separated point masses haveFor a continuous isolated distribution, the corresponding self-energy is $E_{\mathrm{grav}}=\frac{1}{2}\int \rho \Phi d^{3}x$, where $\Phi$ is its own Newtonian gravitational potential with zero at infinity. The half avoids counting every pair twice.

For a spherical body with a fixed dimensionless mass density profile, $E_{\mathrm{grav}}=-CG{M}^2/R$, with $C>0$ determined by that profile; a uniform sphere has $C=3/5$. The inverse-radius scaling follows by rescaling every separation at fixed total mass.

This competes with the internal energy of an ultrarelativistic degenerate electron gas, which scales as $\hslash c{N}_e^{4/3}/R$ at fixed electron number $N_e$. Its electron degeneracy pressure instead scales as $R^{-4}$; it is the internal energy, rather than the pressure itself, that shares the gravitational $R^{-1}$ scaling. With $N_e=M/({\mu }_e{m}_u)$, comparing the two energy coefficients gives the Chandrasekhar mass scalewhere $m_u$ is one unified atomic mass unit and ${\mu }_e$ the mean number of atomic mass units per electron. The numerical coefficient requires a stellar structure model.

Gravitational memory is a persistent change in the relative configuration of freely falling bodies after a gravitational-wave burst has passed. In the weak-field transverse-traceless description, an initially separated pair obeys $\Delta {\xi }^i\simeq \frac{1}{2}\Delta h_{ij}^{\mathrm{TT}}{\xi }^j$. A nonzero net strain change leaves a displacement rather than only a temporary oscillation.

Memory can arise from changes in the motion of material sources and from the energy carried away by radiation itself. The latter contribution reflects the nonlinear gravitational role of wave energy. Distinguishing this slow or step-like response from detector drift requires more than identifying an oscillatory waveform.^[ref]

For identical point masses of total mass $M$, let $W=-G{M}^2/R_g$ exclude self-interactions. The virial theorem gives $⟨v_{\mathrm{eq}}^2⟩=GM/R_g$. Since the sum of individual potential energies counts each pair twice, the mass-weighted mean frozen-potential escape velocity squared is $-4W/M$. A dissipation-free collapse from zero total energy instead has kinetic energy $-W$, hence mean squared speed $-2W/M$. These are three-dimensional speeds relative to the centre of mass; a coherent collapse speed is not a local random velocity dispersion.

A gravitational wave is a propagating disturbance of spacetime curvature produced by changing mass-energy multipole moments. In general relativity, its two transverse gravitational wave polarizations alternately stretch and squeeze freely falling separations. The leading radiation from a slowly moving source is governed by its mass quadrupole moment, rather than by its total mass alone.

The experimental breakthrough was GW150914: observed on 14 September 2015 and published on 11 February 2016. The LIGO Scientific Collaboration and Virgo Collaboration reported a merging pair of black holes, using the two LIGO detectors at Hanford and Livingston; Virgo was being upgraded and did not observe this event.^[ref]

The 2017 Nobel Prize in Physics recognized Rainer Weiss, Barry C. Barish and Kip S. Thorne for contributions to LIGO and gravitational wave observation. Rainer Weiss developed interferometric detection and its noise analysis; Barry C. Barish led the project into construction and operation; Kip S. Thorne developed its theoretical and astrophysical foundations. The experiment depended on a large collaboration.^[ref]

A gravity assist is a spacecraft flyby that uses the gravity of a moving planet or other massive body to change the spacecraft's trajectory and its orbital energy relative to another body, such as the Sun. In the ideal unpowered two-body encounter, the incoming and outgoing speeds far from the planet are equal in the planet's reference frame. Their directions differ. Adding the planet's motion converts that directional change into a possible speed change in the Sun's reference frame.^[ref]

Write the planet's nearly constant velocity during the encounter as $\mathbf{V}$, and the incoming and outgoing planet-relative velocities as ${\mathbf{u}}_i$ and ${\mathbf{u}}_o$. Then ${\mathbf{v}}_i=\mathbf{V}+{\mathbf{u}}_i$, ${\mathbf{v}}_o=\mathbf{V}+{\mathbf{u}}_o$, and $|{\mathbf{u}}_i|=|{\mathbf{u}}_o|$. Subtracting the squared speeds gives the local change in specific orbital energy:For a hypothetical $90^{\circ }$ turn, choose $\mathbf{V}=(10,0)$, ${\mathbf{u}}_i=(-5,0)$ and ${\mathbf{u}}_o=(0,5)$, all in $\mathrm{km}{\mathrm{s}}^{-1}$. Planet-relative speed stays $5$, while Sun-relative speed changes from $5$ to $\sqrt{125}\approx 11.18\mathrm{km}{\mathrm{s}}^{-1}$. The specific-energy gain is $50\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}$. Whether a given turn is physically available depends on the planet's mass, the incoming speed and the permitted closest approach.

Conservation of energy is maintained because the planet and spacecraft exchange momentum and energy. The planet's change is extremely small because of its much greater mass. A different encounter geometry can remove orbital energy or mainly change orbital inclination; Ulysses spacecraft used the latter possibility to reach high solar latitudes.^[ref]

Gross atomic structure denotes the leading nonrelativistic Coulomb binding levels, before fine and hyperfine splittings. For a hydrogenic two-body system, the reduced mass controls the binding energy and characteristic separation. Fine structure is smaller by roughly ${\alpha }^2$; nuclear magnetic hyperfine effects have an additional electron-to-nuclear mass suppression.

A ground state is a normalized energy eigenstate that attains the lowest possible energy of a quantum system. Precisely, let the Hamiltonian operator $H$ be a self-adjoint operator bounded below on a Hilbert space, and write $E_{\ast }=\inf \sigma (H)$ for the infimum of its spectrum. A ground state is a unit vector ${\psi }_0$ in the operator domain $D(H)$ satisfyingThus a ground state exists exactly when the spectral bottom $E_{\ast }$ is an energy eigenvalue. The lowest energy eigenvalue among some listed bound states is insufficient if other parts of the spectrum lie below it. The ground-state subspace collects all minimizing energy eigenstates; it can have ground-state degeneracy. The quantum variational principle characterizes $E_{\ast }$ as the infimum of normalized energy expectation values. ^[ref].

For the one-dimensional quantum harmonic oscillator, the ground state has energy $\hslash \omega /2$ above the minimum of the potential energy. This zero-point energy is compatible with being the lowest allowed energy: a localized wavefunction cannot simultaneously have zero kinetic energy. For a free particle on the whole line, the spectral bottom is zero but is unattained by any normalizable wavefunction; see ground-state energy. A lower bound therefore need not describe a physically preparable energy eigenstate.

The ground state supplies a reference for excitation costs and for the quantum variational principle. A ground state is also a stationary state under a time-independent Hamiltonian operator: its quantum phase changes, while its position probability density stays fixed. A nonzero ground-state energy does not imply motion along a classical trajectory, and ground-state degeneracy means there need not be a unique lowest quantum state.

When a ground state exists, its ground-state degeneracy is the dimension of the ground-state subspace:Here $I$ is the identity operator, and the kernel of a linear map is taken within the operator domain of the Hamiltonian operator $H$. The count concerns linearly independent energy eigenstates, rather than every normalized quantum superposition or every choice of global phase. Even a single energy eigenspace contains infinitely many representing vectors; its dimension measures independent freedom. Ground-state degeneracy is the lowest-energy instance of degenerate energy levels.

For an exact finite model with energy unit $\epsilon >0$, let the Hamiltonian operator be the diagonal matrixIts ground-state energy is zero and $g_0=2$: the first two coordinate unit vectors $e_1,e_2$ span the ground-state subspace. Every normalized $a{e}_1+b{e}_2$, with $|a{|}^2+|b{|}^2=1$, is a ground state, including $(e_1+e_2)/\sqrt{2}$. Its energy measurement is certain to return zero, even though a measurement distinguishing $e_1$ from $e_2$ need not have a definite outcome. Under the unchanged Hamiltonian operator, this quantum superposition is a stationary state because both components acquire the same global phase.

Ground-state degeneracy matters when an interaction selects or splits previously equal energy eigenvalues. In the finite model, adding $\epsilon \mathrm{diag}(\delta ,-\delta ,0,0)$ with $0<\delta <2$ gives the formerly lowest pair the distinct energies $\pm \epsilon \delta$, leaving a unique ground state. This direct calculation illustrates why degenerate perturbation theory first examines the perturbation inside the whole ground-state subspace.

A classical continuous-symmetry model can instead have a manifold of minimizing configurations, such as the sphere of orientations of a Néel state in an isotropic Heisenberg antiferromagnet. That geometric description does not by itself count the dimension of a finite quantum ground-state subspace.

The ground-state dissociation energy starts at the lowest vibrational quantum state, with the lowest allowed rotational quantum state and the fragment states explicitly specified. For a diatomic molecule whose fragments are ground-state chemical atoms, omitting any additional rotational correction,where $E_{\mathrm{ZP}}$ is the vibrational zero-point energy measured upward from the well minimum. The initial molecule already has this energy, so less additional energy is needed to reach the same final limit.^[ref]

For fragments that are themselves molecules, include their zero-point energies too. Define $D_e$ as the electronic energy difference between separately relaxed products and reactant; then, with consistent rotational conventions,Thus “subtract the parent's zero-point energy” is sufficient for atomic products but generally incomplete for molecular products. An excited final quantum state raises its channel threshold by its excitation energy; that threshold does not become the ground-product $D_0$ just because a detector selects it.

The ground-state energy $E_0$ is the energy eigenvalue attained by a ground state. For a self-adjoint operator $H$ bounded below, distinguish this attained value from the always-defined spectral bottom $E_{\ast }=\inf \sigma (H)$. When a ground state exists, $E_0=E_{\ast }$. Some usage calls $E_{\ast }$ the ground-state energy even when it is unattained; here an unattained bottom is explicitly identified as an infimum. The quantum variational principle statesApproaching this infimum with trial wavefunctions does not ensure that a minimizing wavefunction exists. ^[ref].

For the whole-line free particle with mass $m>0$,Its spectrum is continuous and contains no energy eigenvalues represented by normalizable wavefunctions. The Fourier transform makes $H$ multiplication by ${\hslash }^2{k}^2/(2m)$; a fixed energy restricts $k$ to a set of zero length, on which a nonzero square-integrable function cannot be supported. ^[ref]. An explicit family of normalized Gaussian wave packets demonstrates how the bottom is approached:The formula follows from ${\psi }_s^{\prime }=-x{\psi }_s/(2s^2)$, integration by parts, and $⟨x^2{⟩}_s=s^2$. Every finite width gives positive kinetic energy. At zero energy, $H\psi =0$ would require an affine function on the whole line, which is never a nonzero normalizable wavefunction. Spreading a Gaussian wave packet indefinitely produces an unattained infimum, rather than a zero-energy ground state.

An additive choice of energy origin changes the numerical ground-state energy without changing the ground-state subspace. For $H^{\prime }=H+CI$, every energy eigenvalue becomes $E^{\prime }=E+C$, while all differences $E_j-E_i$ stay fixed. Time evolution gains only the global phase $e^{-iCt/\hslash }$. This makes excitation costs and spectroscopic differences more intrinsic than the isolated number $E_0$.

For $V(x)=0$ on $0<x<a$ and $V(x)=V_w>0$ outside, the lowest bound state is even about $a/2$, has no nodes, and has $0<E<V_w$. Write $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_w-E)}/\hslash$. The interior wavefunction is proportional to $\cos [k(x-a/2)]$; the exterior tails decay exponentially. Wavefunction matching at a finite potential step gives the displayed condition. Its lowest root has $0<ka<\pi$, so $E<{\pi }^2{\hslash }^2/(2m{a}^2)$, the infinite square well ground state energy. As $V_w$ increases, the penetration length $1/\kappa$ tends to zero and that limiting energy is approached from below.

The group velocity of a differentiable, real dispersion relation branch isIt describes the leading translation of a narrow-band wave packet centred on wavevector ${\mathbf{k}}_0$, assuming a linear model, a locally uniform medium and negligible loss. Narrow-band means that the appreciably excited wavevectors occupy a small neighbourhood on which one smooth branch can be linearized. Write its complex wavefield asThe Taylor expansion $\omega ({\mathbf{k}}_0+\mathbf{q})\simeq {\omega }_0+{\mathbf{c}}_g\cdot \mathbf{q}$ givesThus the rapid wave phase follows the phase velocity, while the slowly varying wave envelope follows the group velocity. This Fourier analysis argument is an outline of the approximation, rather than a statement that every wave packet translates rigidly forever. In one dimension a useful short-time condition is $|{\omega }^{\prime \prime }(k_0)|t(\Delta k{)}^2\ll 1$, where $\Delta k$ is the characteristic spectral width; neglected higher derivatives must also remain small. Curvature of the dispersion relation eventually changes the wave envelope.^[ref]

Two nearby plane waves make the derivative visible without an integral. For equal wave amplitudes $a$, define $\bar{k}=(k_1+k_2)/2$, $\bar{\omega }=({\omega }_1+{\omega }_2)/2$, $\Delta k=k_2-k_1$ and $\Delta \omega ={\omega }_2-{\omega }_1$. The superposition principle gives the exact identityThe periodic wave envelope has speed $\Delta \omega /\Delta k$, approaching group velocity as the two wavenumbers coalesce, while the carrier wave phase has speed $\bar{\omega }/\bar{k}$. Two tones alone form an infinite beat train, not a localized wave packet. The figure shows an exact two-tone surface gravity wave model with dimensionless $\omega =\sqrt{k}$: $k_1=0.9$, $k_2=1.1$. Its carrier phase velocity is approximately $0.99875$, its finite-difference envelope velocity approximately $0.50063$, and ${\omega }^{\prime }(1)=0.5$ is the limiting group velocity. The marked carrier phase line need not coincide exactly with a maximum of the full modulated wave.

The deep-water surface gravity wave therefore has $c_g=c_p/2$, whereas the deep-water capillary wave limit has $c_g=3c_p/2$: crests can drift either forward or backward relative to a wave envelope. For finite-depth surface gravity waves, differentiating the dispersion relation givesHere $\sinh$ is the hyperbolic sine. The shallow-water limit has $c_g\simeq c_p\simeq \sqrt{gh}$; the deep-water limit has $c_g\simeq c_p/2$.^[ref]

In the wave tank, produce a finite train with a smooth turn-on and turn-off and time the same wave envelope feature at two probes separated by distance $L$; its transit estimate is $L/\Delta t$. Independently follow wave crests through that train. Techet's 2003 lab asks students to observe a deep-water train's foremost crest disappearing as the group moves forward, and to compare deep- and shallow-water cases. The interpretation is that a wave crest overtakes the slower wave envelope, dwindling at its front while other crests become visible at its rear. This is a qualitative expected observation in the teaching protocol, not a published set of measured curves.^[ref]

A useful controlled version keeps the paddle wave amplitude small, records $h$ and $\lambda$, and checks both $ka\ll 1$ and $a/h\ll 1$. Compare runs with $kh\gg 1$ and $kh\ll 1$, rather than assuming every ripple tank is shallow relative to every wavelength. Increase train duration at fixed central frequency to narrow its spectrum, and check that the envelope transit estimate stabilizes. Calibrate both probes, sample fast enough to resolve the carrier, and compare records before reflected waves return; an absorbing beach reduces this contamination. A current changes laboratory velocities, large wave amplitudes introduce nonlinear effects, and very short wavelengths require surface tension. Strong damping or noticeable reshaping makes the chosen threshold or peak an unreliable marker of simple group velocity. These are proposed controls and model limits, not claims that the cited lab measured every correction.

Three velocities should consequently be kept separate. The group velocity predicts narrow-band wave envelope motion. An energy-transport velocity is energy flux divided by energy density, using the appropriate time average; it agrees with group velocity in standard lossless linear wave models when all stored energy is included.^[ref] A signal-front velocity concerns the earliest response to a new disturbance and requires a causal model over its full spectrum. A strictly band-limited wave packet has no sharply bounded onset, and a mathematical two-tone envelope cannot test front arrival. In strongly absorbing or broadband propagation, a single group velocity need not describe the observed peak, energy transport, or first detectable response. A complex $\omega$ or $k$ also prevents interpreting its derivative as an ordinary real transport velocity without further analysis.

In particular, an unusually large or negative group velocity from pulse reshaping does not establish superluminal information transmission. For the causal dielectric model analysed by Fitzpatrick, the field ahead of the front vanishes for $t<x/c$, even when its phase velocity exceeds the vacuum speed of light $c$. That result concerns the electromagnetic model's causality; the ideal water-wave diagrams above contain no relativistic signal-front claim.^[ref]

Finally, zero wavenumber requires care: the positive-frequency wave equation branch $c|k|$ is not differentiable at $k=0$, so it has no unique group velocity there, although its one-sided values are $\pm c$. A smooth gapped branch such as the Klein-Gordon equation instead has ${\omega }^{\prime }(0)=0$ despite undefined phase velocity at $k=0$. The same signed branch must be used in both derivatives and ratios; a negative group velocity means an envelope moving in the negative coordinate direction, not a negative elapsed time. The practical value of group velocity is precisely this translation between a local slope on a dispersion diagram and a measurable packet transit, with its assumptions made explicit.

The Gaussian envelope in a Klein-Gordon medium supplies a quantitative next step: the first derivative translates the wave envelope, the second derivative spreads it, and higher derivatives limit that approximation. At finite spectral width, a peak's measured speed need not equal the group velocity at the central wavenumber. Neither a peak nor a fixed fraction of its height generally identifies front velocity.

For $\omega =\sigma 2\Omega k_{\parallel }/k$, differentiating the dispersion relation gives the displayed group velocity. It is perpendicular to the wavevector. If $|k_{\parallel }|\ll k_{\perp }$, energy propagates almost along the rotation axis while the phase velocity is almost horizontal. In an orthonormal basis ${\mathbf{e}}_1,{\mathbf{e}}_2$ normal to $\mathbf{k}$, with ${\mathbf{e}}_2=\hat{\mathbf{k}}\times {\mathbf{e}}_1$, the complex velocity is $A({\mathbf{e}}_1-\sigma i{\mathbf{e}}_2)$ and its real oscillation is circular. In the anisotropic limit the vertical and transverse-horizontal velocity amplitudes are comparable, while the horizontal component parallel to ${\mathbf{k}}_{\perp }$ is smaller by $|k_{\parallel }|/k_{\perp }$.

Gustav Ludwig Hertz and James Franck accelerated electrons through Mercury vapor and measured the collected electric current as the accelerating voltage changed. Repeated features in that curve indicate a preferred loss of kinetic energy in inelastic collisions. An electron must first acquire enough energy to excite the relevant atomic transition; further acceleration can enable another such collision. The Franck-Hertz experiment therefore connects an electrical current–voltage measurement with discrete atomic excitation energies.^[ref]

Physics.^[ref]

A particle calorimeter designed to contain hadronic showers and estimate incident hadron energy. Noncompensating response, invisible nuclear breakup energy and leakage complicate its calibration relative to an electromagnetic calorimeter.

Hagai Netzer develops models connecting the spectral lines of active galactic nuclei to the conditions of their surrounding gas. In his account of emission-line physics, photons from the central source heat the gas by photoionization, while several atomic processes remove or redistribute that energy.

The resulting electron temperature follows a heating–cooling balance. It need not equal either the temperature assigned to the radiation field or the excitation temperature describing atomic level populations. This distinction is essential when interpreting a bright spectral line as a diagnostic of an illuminated cloud.^[ref]

If a density of states has high-energy growth proportional to a power of energy times $e^{{\beta }_{H}E}$, its canonical Boltzmann sum fails for $\beta <{\beta }_H$. The Hagedorn temperature is the corresponding threshold. The open bosonic string mass spectrum and large-level degeneracy of an open bosonic string give ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$ in units with Boltzmann's constant, the speed of light and Planck's constant set to one. Polynomial prefactors affect behavior at the threshold itself; the growth alone does not distinguish a limiting temperature from a phase transition.