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An energy flux describes transport of energy across a surface. A flux density ${\mathbf{j}}_E$ has units of power per area; the total outward power through $S$ is ${\int }_S{\mathbf{j}}_E\cdot \mathbf{n}dS$, with outward unit normal $\mathbf{n}$. A local conservation law relates the divergence of ${\mathbf{j}}_E$ to the time derivative of the energy density and any sources.

For a one-dimensional inviscid gas with velocity $u$, pressure $p$, mass density $\rho$ and internal energy density $W$, the total mechanical-plus-internal energy flux isThe first two terms advect internal energy and kinetic energy. The term $pu$ is pressure work: a moving section of area $A$ transmits force $pA$ at velocity $u$, giving power $pAu$. This formula assumes negligible viscous stress and heat conduction; those processes contribute their own energy-transfer terms when retained.

Collection of ambient light, heat flow, thermal cycling, motion or other available energy into a usable electrical supply. Available average power, conversion loss and peak-load buffering must be distinguished.

For radial and vertical Lagrangian fluid displacements of a local axisymmetric mode, integrate the momentum equations and use incompressible flow plus the free-surface pressure condition of an incompressible disk. The result isFor nonnegative ${\kappa }^2$ and positive ${\Omega }_z^2$, every nontrivial oscillatory mode has real nonnegative ${\omega }^2$. The surface term is the gravitational restoring energy of the displaced vacuum interfaces.

For a rectangular membrane mode with peak antinode displacement A, the integrals of each squared sine factor are half the corresponding side length. Integrating local kinetic energy density $\rho ({\partial }_t\psi {)}^2/2$ gives $K=\rho ab{\omega }^2{A}^2{\sin }^2(\omega t+\varphi )/8$. The total energy is $\rho ab{\omega }^2{A}^2/8$, and potential energy has the complementary cosine-squared factor. Its time-averaged kinetic energy is $\rho ab{\omega }^2{A}^2/16$. At equal peak displacement, frequency and density, such mean energies are proportional to area.

An energy landscape assigns an energy to each possible state and describes its variation between neighbouring states. Its local minima and intervening energy barriers help distinguish favourable states from states that a specified dynamics can reach quickly. In a Hopfield network, the assigned energy is an objective function and need not be a conserved physical quantity.

An energy level is an allowed energy eigenvalue of a Quantum Hamiltonian. Its corresponding energy eigenstates satisfy $H|n⟩=E_n|n⟩$. Several independent energy eigenstates may share one energy level; their number is the energy-level degeneracy. A quantum superposition of different energy eigenstates need not have a definite energy, even though an ideal energy measurement yields one of the allowed outcomes.

Quantum confinement often produces discrete energy levels: the quantum harmonic oscillator has $E_n=\hslash \omega (n+1/2)$. An unconfined free particle instead has a continuous spectrum. In spectroscopy, a transition between two energy levels exchanges approximately $|E_f-E_i|=h\nu$ with a photon; recoil and external fields can affect the measured frequency. Selection rules determine which energy differences produce observable spectral lines.

By National Institute of Standards and Technology. Handbook of Basic Atomic Spectroscopic Data, neutral helium He I table.

The table provides ground, singlet and triplet excitation energies as spectroscopic wavenumbers, and the He II ground-state ionization limit. Multiply by $hc$ with consistent inverse-length units to obtain excitation energy.

A decrease in a projectile's energy by transfer to matter or emitted radiation. Mean loss per unit path length defines stopping power; its event-to-event variation is energy-loss straggling.

If the similarity profiles account for all kinetic and internal energy inside an adiabatic strong shock, letThe injected energy $\dot{E}t$ equals $4\pi {\rho }_1{\dot{R}}^2{R}^{3}I$. Consequently conservation of energy fixesAn interior region not included in the profiles must have its energy added separately; a radiatively cooled shell changes this energy budget.

For an unbound equatorial Schwarzschild spacetime geodesic, let $E_{\infty }$ and $L_{\infty }$ be its conserved energy and angular momentum measured relative to infinity. The ratio $J=c^2{L}_{\infty }/E_{\infty }$ has dimensions of area divided by time. For a massive test particle, $E_{\infty }=m{c}^2{\gamma }_{\infty }$ and $L_{\infty }=m\ell$, so $J=\ell /{\gamma }_{\infty }=b{v}_{\infty }$, where $\ell$ is specific angular momentum and $b$ the impact parameter. For a photon, $E_{\infty }=c{p}_{\infty }$ and $L_{\infty }=b{p}_{\infty }$, giving $J=bc$ without assigning a finite Lorentz factor or a proper time to the photon. This ratio removes the arbitrary normalization of a null affine parameter.

For elevation $h_0\sin (\omega t)\sin (kx)$, depth-integrated kinetic energy per surface area is $\rho h_0^2{\omega }^2{\cos }^2(\omega t)/(4k)$. Wavelength-averaged excess gravitational energy is $\rho g{h}_0^2{\sin }^2(\omega t)/4$. Conservation gives ${\omega }^2=gk$ in the inviscid gravity-only model.

For a small-slope travelling wave $y=A\cos (kx-\omega t+\varphi )$ on a uniform string with linear mass density $\mu$ and tension $T$, the wave equation on a string requires ${\omega }^2=T{k}^2/\mu$. Its elastic energy of a small-slope string and kinetic energy density are equal, giving total energy density $u=\mu A^2{\omega }^2{\sin }^2(kx-\omega t+\varphi )$. Its spatial average is $\mu A^2{\omega }^2/2$, and integrating over a full wavelength $\lambda =2\pi /|k|$ gives the title formula. Omitting $\lambda$ produces an energy density, not an energy; units and a direct one-period integral detect the mistake independently.

Suppose a constant Hamiltonian operator has a complete discrete orthonormal basis of energy eigenstates, $H|n⟩=E_n|n⟩$. Expanding an initial pure state and applying the time-evolution operator term by term givesA projective measurement resolving these basis vectors therefore assigns the same Born rule probability $|c_n{|}^2$ to each label $n$ at every time. An energy measurement groups labels sharing the same energy eigenvalue: $P(E)={\sum }_{n:E_n=E}|c_n{|}^2$ is also constant. If $c_m,c_n\ne 0$, their probability amplitude ratio instead evolves asOnly energy differences enter this relative quantum phase. Adding a constant $\epsilon I$ to the Hamiltonian operator multiplies the entire pure state by $e^{-i\epsilon t/\hslash }$, a global phase, and leaves the density operator unchanged.

For a fixed observable $A$, the expectation value can nevertheless vary:Terms joining distinct energy eigenvalues can oscillate; diagonal terms cannot. If $[A,H]=0$, these terms between different energies vanish, so the expectation value is constant. This distinguishes conservation of energy from the stronger claim that every observable is stationary.

An energy eigenstate has only a global phase evolution and is a stationary state. A quantum superposition entirely within one degenerate energy eigenspace is also an energy eigenstate and is stationary. For a general density operator in finite dimension, stationarity under a constant Hamiltonian operator means $[H,\rho ]=0$: differentiating $U\rho U^{†}$ proves necessity, and commuting $\rho$ through the exponential proves sufficiency. Merely fixing the diagonal energy populations does not fix the off-diagonal quantum coherence.

For $\{Q_{\alpha },Q_{\beta }^{†}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$, summing diagonal components gives $4H$. The expectation of each anticommutator is $\Vert Q_{\alpha }v{\Vert }^2+\Vert Q_{\alpha }^{†}v{\Vert }^2$, so $H$ is a positive semidefinite operator. A vacuum has zero energy exactly when every supercharge annihilates it. Spontaneously broken global supersymmetry with an existing vacuum therefore gives positive vacuum energy density. This statement fixes the energy zero through the algebra and does not apply to the general supergravity F-term potential.

A rigid body rolling with effective radius $h$ has kinetic energy $(m+I/h^2)v^2/2$. If a transition changes the rolling constraint while preserving this total energy, the translational speed changes according to the title formula. Maintaining the same speed would generally require energy loss or gain. The energy-preserving condition is an assumption about the transition, not a generic property of sudden impacts.

An energy profile plots an explicitly specified energy against a reaction coordinate. Identify whether the vertical axis is potential energy, enthalpy, or Gibbs free energy; these descriptions are related but not interchangeable. A potential-energy profile describes a path through configurations, while a free-energy description also accounts for the relevant thermodynamic conditions.

For the two chosen profiles, the reaction energy barriers from the reactants are $80$ and $50\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. From the reaction products, the same maxima are $100$ and $70\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ higher. Both paths have endpoint difference $-20\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The barrier is measured from the relevant starting endpoint, not from an arbitrary zero of potential energy.

The appeal of the diagram is that it separates two useful questions: which endpoint is lower, and which path is easier to traverse? Catalysis changes the route through transition states while retaining the endpoint thermodynamics of the same overall chemical reaction. A real catalytic route may have several steps and reaction intermediates; the two simple curves do not specify such a reaction mechanism. The plotted potential-energy difference alone is not an equilibrium constant or a calculated rate constant. See Chemistry 2e, 12.7: alternative reaction routes and catalysis.

Returning energy from a used flow or beam to an upstream supply or a useful reservoir. An electron beam decelerated before collection returns much of its acceleration energy; the net input then covers unrecovered energy and auxiliary losses.

If energies are measured upward from a species reference ${\epsilon }_0$, its absolute single-particle sum is $q_{\mathrm{abs}}=e^{-\beta {\epsilon }_0}q$. The ideal-gas chemical potential is then ${\epsilon }_0-k_{B}T\ln (q/N)$. A zero-point factor can be included either in $q$ or in the reference energy, but not both.

Use antiplane energy $\frac{\mu }{2}\int |\nabla u{|}^2$ and normalize the tip singularity as $u=k\sqrt{r}\sin (\theta /2)$ plus a less singular term. The energy release rate is the small-circle flux $\int [\mathcal{W}{n}_x-\mu u_r{u}_x]rd\theta$. Here $\mathcal{W}=\mu k^2/(8r)$, $u_r=k\sin (\theta /2)/(2\sqrt{r})$ and $u_x=-k\sin (\theta /2)/(2\sqrt{r})$. Integrating over $(-\pi ,\pi )$ gives the displayed coefficient. The full three-dimensional antiplane stress has squared Frobenius norm $2{\mu }^2|\nabla u{|}^2$, so its leading norm ahead of the tip is $\mu |k|/\sqrt{2r}$. These constants depend on the normalization of the stress intensity factor.

The decrease in potential energy per increment of crack area under specified loading constraints. Its critical value $G_c$ is a crack-growth threshold, not energy stored per specimen volume.