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Low-temperature solid helium-3 can have greater entropy than the liquid through disordered nuclear spins, enabling compression cooling. ^[ref]

The sign follows from the Clapeyron equation. Let $S_l,S_s$ be molar entropies and $V_l,V_s$ molar volumes of liquid and solid, respectively, on their phase coexistence curve. Equality of their chemical potentials, with $d\mu =-SdT+VdP$, givesWith $S_s>S_l$ and $V_l>V_s$, the slope is negative. Reversibly freezing one mole at fixed temperature requires heat into the sample,Here $L_{s\to l}$ uses the melting convention for latent heat of fusion. This is the opposite sign from ordinary ice melting. The inequalities specify the regime; freezing does not universally cool a substance. In an approximately isolated sample, the growing higher-entropy phase draws on the sample's thermal energy instead of an external heater. Irreversible losses and incoming heat oppose the cooling.

A population inversion puts more emitters in the upper than in the lower quantum state of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing photon absorption rate proportional to $N_1$, so net optical gain is proportional to $N_2-N_1$. For $N_2=6\times 10^{20}{\mathrm{m}}^{-3}$ and $N_1=4\times 10^{20}{\mathrm{m}}^{-3}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per statistical weight of an atomic level, rather than simply $N_2>N_1$.

Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both photon absorption and stimulated emission. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation isat a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain population inversion in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible laser gain mechanisms. ^[ref].

In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the energy: extra levels provide a route to population inversion, not free energy. ^[ref].

For a free field, its positive-frequency part consists of the annihilation terms multiplying $e^{-i\omega t}$ with $\omega >0$; its negative-frequency part consists of the creation terms. The sign refers to positive energy, even though the exponential has a minus sign. Imposing the Lorenz gauge only through $(\partial \cdot A{)}^{(+)}|\mathrm{phys}⟩=0$ is the subsidiary condition in Gupta-Bleuler quantization, not an operator identity for the whole divergence.

Positronium is the electromagnetic bound state of an electron and a Positron. Their equal masses give reduced mass $m_e/2$, half the hydrogenic Rydberg binding scale and approximately twice the electron-proton separation of hydrogen. Its spin singlet and triplet differ in hyperfine energy; annihilation also affects their physical lifetimes and splitting.

After the quantum quench, hold $H_f$ fixed and evolve without an intervening energy measurement. The unitary time evolution isProjection onto the original $|0⟩$ gives probability amplitude $c^2{e}^{+i{\Delta }_{f}t/(2\hslash )}+s^2{e}^{-i{\Delta }_{f}t/(2\hslash )}$. Squaring its modulus yields the two-level quantum return probabilityTheir sum is one. For $\theta =\pi /3$, the first minimum of $P_0$ is $1/4$ at $t=\pi \hslash /{\Delta }_f$, and the return is complete at $T=2\pi \hslash /{\Delta }_f$. At $\theta =0$ or $\pi$ there is no oscillation, because the initial quantum state is already a final energy eigenstate.

Changing measurement basis reveals the evolving relative quantum phase. Measuring the final energy first instead destroys this quantum coherence between final energy eigenstates in the unconditioned ensemble: its later original-basis return probability is $c^4+s^4=1-\frac{1}{2}{\sin }^2\theta$, equal to $5/8$ for this example. That constant differs from the oscillation of the unmeasured pure state. The work protocol and the oscillation protocol therefore use separate preparations.

Potential energy assigns energy to a system's arrangement, so a change of arrangement accounts for the work done by a conservative force. For a time-independent potential energy $V(\mathbf{r})$,The gradient points toward increasing potential energy; the force points toward decreasing potential energy. Only differences matter: adding a constant to $V$ changes neither the force nor its work. This endpoint rule follows by substituting $-\nabla V$ into the line integral for work. ^[ref]

Near Earth's surface, stipulate constant gravitational acceleration $g=9.81m/{\mathrm{s}}^2$ and measure height $h$ upward from a chosen zero. Then gravitational potential energy is $V=mgh$. Raising a $2\mathrm{kg}$ mass by $0.5\mathrm{m}$ increases its gravitational potential energy by $9.81\mathrm{J}$, while gravity does $-9.81\mathrm{J}$ of work. These are computed model values, not measurements of a particular lift. An ideal spring obeying Hooke's law instead has $V=k{x}^2/2$, with displacement $x$ from its relaxed state and spring constant $k$: $k=100N/m$ and $x=0.2\mathrm{m}$ give $V=2\mathrm{J}$ relative to $x=0$. ^[ref]

A region of lower potential energy surrounded by higher potential energy; a particle must gain sufficient energy to escape classically.

Power describes how quickly energy is transferred. For mechanical work, the instantaneous power supplied by a force isThe SI unit is $\mathrm{W}=J/s=\mathrm{N}m/s$, distinct from the velocity unit $m/s$. The italic symbol $W$ for work and the upright unit $\mathrm{W}$ for power mean different things. Two processes can transfer the same energy while requiring different powers: doing $12\mathrm{J}$ of work in $2\mathrm{s}$ requires an average $6\mathrm{W}$, whereas doing it in $4\mathrm{s}$ requires an average $3\mathrm{W}$. ^[ref]

The singular equation $2x{w}^{\prime \prime }-w^{\prime }+\lambda x^{2}w=0$ becomes $W^{\prime \prime }+(2\lambda /9)W=0$ under $z=x^{3/2}$, $w(x)=W(z)$. Its zero-endpoint finite-energy quotient transforms as $2{\int }_0^1{x}^{-1/2}(w^{\prime }{)}^{2}dx/{\int }_0^1{x}^{1/2}{w}^{2}dx=(9/2){\int }_0^1(W^{\prime }{)}^{2}dz/{\int }_0^1{W}^{2}dz$. The Poincaré inequality therefore proves the sharp minimum $9{\pi }^2/2$, attained by $w=\sin (\pi x^{3/2})$. This coordinate change turns an apparently singular weighted variational problem into the usual interval energy problem. The admissible eigenfunction is $O(x^{3/2})$, making the integration-by-parts boundary term $w{x}^{-1/2}{w}^{\prime }$ vanish and both energy integrals converge.

The ratio of signal power delivered to a specified output load to signal power accepted at a specified input, with $P_{\mathrm{in}}>0$. An amplifier draws additional energy from its supply, so a power gain above one does not create energy.

A radiation spectrum whose spectral flux density follows a power law over a stated frequency range. For $F_{\nu }\propto {\nu }^{-\alpha }$, the spectral energy distribution satisfies $\nu F_{\nu }\propto {\nu }^{1-\alpha }$. Thus $\alpha =1$ gives equal energy flux in equal logarithmic frequency intervals. A spectral break invalidates extrapolation with a single index.

A small longitudinal electric field drives conductor current in a lossy coaxial line. Its cross product with the azimuthal magnetic field sends energy radially into both conductors, where it becomes heat; forward power decreases with distance.

The Poynting theorem is the local energy balance of classical electromagnetism. For vacuum field laws, with current density $\mathbf{J}$, the electromagnetic energy per unit volume isandThus a decrease in local electromagnetic energy can supply outward energy flux or work on electric charge. The term $\mathbf{J}\cdot \mathbf{E}$ is power transferred to matter per unit volume. Its sign matters: positive values remove field energy, while negative values describe matter supplying the field. The magnetic part of the Lorentz force supplies no work term.

For a fixed volume $V$, the divergence theorem gives the directly measurable balanceFor example, $2.0\mathrm{W}$ entering a region with $0.50\mathrm{W}$ leaving and $1.20\mathrm{W}$ transferred to matter means stored electromagnetic energy increases at $0.30\mathrm{W}$. In source-free vacuum only the boundary flux changes the stored energy; a nonzero local Poynting vector can still have zero net closed-surface flux.

A brief proof takes the dot product of the Ampère-Maxwell equation with $\mathbf{E}/{\mu }_0$ and uses Faraday's law with $\mathbf{B}/{\mu }_0$. The identity $\nabla \cdot (\mathbf{E}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{E})-\mathbf{E}\cdot (\nabla \times \mathbf{B})$ turns the remaining terms into the divergence of the Poynting vector. ^[ref].

In vacuum the Poynting vectoris the local energy flux, measured in watts per square metre. Its surface integral $P={\int }_A\mathbf{S}\cdot \mathbf{n}dA$ gives signed power crossing a surface with unit normal vector $\mathbf{n}$. Positive flux follows the chosen normal; for a closed surface the usual convention is outward. Power depends on the normal component and area, rather than just the field magnitude.

The Poynting vector applies to steady fields as well as electromagnetic waves. In a driven resistive electrical circuit, surrounding fields can carry energy into the conductor, where electrical work becomes Joule heating. This turns a circuit-level power balance into a spatial description of energy delivery. ^[ref].

The reflected-to-incident compressional pressure or normal-stress amplitude ratio at a planar interface. For normal incidence between lossless homogeneous media, the displayed expression uses acoustic impedances $Z_j={\rho }_j{V}_{pj}$. The reflected energy fraction is $r_p^2$, while a same-coordinate particle-velocity reflection coefficient has the opposite sign.

The transmitted-to-incident compressional pressure or normal-stress amplitude ratio. At normal incidence, continuity of normal stress gives $t_p=1+r_p$, with pressure-amplitude reflection coefficient $r_p$. The transmitted energy fraction is $(Z_1/Z_2)t_p^2$, not $t_p^2$ alone.

Prey is an organism consumed by a predator in predation. The name describes the receiving end of that interaction: a small fish can be prey for a large fish while being a predator of zooplankton. In a food web drawn with resource-to-consumer arrows, a feeding arrow goes outward from the prey toward its predator. This convention makes the route of food and energy easy to follow.

Energy measured at the resource or initial-input stage before downstream conversion to final useful energy. Comparisons with electrical output must state the accounting convention and avoid treating a primary-energy thermal input as the same service as delivered electricity.

A primary producer builds organic chemical compounds from inorganic sources. Photosynthetic algae, plants and cyanobacteria use light energy for this role through photosynthesis and carbon fixation. The role supplies new living material to a food web; it does not mean that the organism creates energy.

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.