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With bare nuclear masses, the proton separation energy is the energy needed to remove a proton. A negative value means the daughter plus proton has lower rest energy. Proton emission can still have a finite lifetime because of a Coulomb or centrifugal barrier.

In a uniform isotropic material, these scalar P-SV displacement potentials separate longitudinal P waves from vertically polarized SV-waves in an $x,z$ plane. They satisfy ${\varphi }_{tt}={\alpha }^2\Delta \varphi$ and ${\psi }_{tt}={\beta }^2\Delta \psi$, with compressional and shear speeds $\alpha ,\beta$. Reversing the sign convention for $\psi$ changes its polarization column but not the physical P-SV directional impedance matrix or the separate modal energy fluxes.

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

A circular closed string contracts and re-expands with $X^0=Rt$ and $X^1+i{X}^2=R\cos t{e}^{i\sigma }$. Its proper length is $2\pi R|\cos t|$, while its energy remains $2\pi TR$. At collapse the induced worldsheet metric degenerates, but the Nambu-Goto phase-space action supplies a regular continuation.

Water is pumped to an upper reservoir and later discharged through a turbine. For approximately constant head, useful discharge energy is $\eta \rho Vgh$ while power is $\eta \rho Qgh$; reservoir volume and flow rating set different properties.

A four-force preserving the particle's rest mass. For a massive particle, $P=mU$ and $U^2=c^2$ in the mostly-minus convention imply $F\cdot U=c^{2}dm/d\tau$. Thus constant mass is equivalent to orthogonality of the force and four-velocity. Energy transfer in a moving frame is allowed even though rest mass does not change.

For the field-free, spin-free linear rigid rotor in one fixed internal quantum state, the electric-dipole interaction is $-\mathbf{d}\cdot \mathbf{E}(t)$. A nonzero permanent electric dipole moment along the molecular axis allows pure rotational photon absorption with $\Delta J=+1$ and photon emission with $\Delta J=-1$. The matrix element between initial and final quantum states must be nonzero: the existence of an energy gap alone is insufficient. Relative to a chosen $z$ axis, polarization of an electromagnetic wave along $z$ selects $\Delta M=0$; transverse polarization of an electromagnetic wave permits $\Delta M=\pm 1$.^[ref]

Hydrogen chloride and carbon monoxide are examples with a permanent electric dipole moment. By symmetry, molecular hydrogen and carbon dioxide have no permanent electric dipole moment in the ideal symmetric internal state. They still rotate and have rotational energy eigenvalues, but the simple permanent-electric dipole spectrum described here is absent. This restriction concerns this interaction and these assumptions, not every possible way of detecting molecular rotation.

Quantum chromodynamics makes parton distribution functions depend on resolution scale through gluon radiation and parton splitting. Thus Bjorken scaling and the linear-energy neutrino cross section are approximate leading-order patterns rather than exact laws.

For a resonator, the quality factor is $2\pi$ times the stored energy divided by the energy dissipated per cycle, using a specified steady-state stored-energy convention. A series RLC circuit at resonance has $Q=L{\omega }_0/R$, because its maximum magnetic energy is $L{I}_M^2/2$ and one cycle of Joule heating dissipates $\pi R{I}_M^2/{\omega }_0$.

For an ideal pulse with negligible stored seed photons and useful output accounting for all cavity photon loss, the extracted energy is $(N_i^{\ast }-N_f^{\ast })V_g\hslash {\omega }_L/\beta$. The factor $\beta =1+g_2/g_1$ converts inversion depletion into stimulated transitions. With over-pumping $r=N_i^{\ast }/N_{\mathrm{th}}^{\ast }$, the nonzero extraction branch satisfies $r\beta \eta =-\ln (1-\beta \eta )$. A fraction $0.95$ of the maximum energy requires $r=-\ln (0.05)/0.95$.

An axially symmetric quadratic potential energy perturbation preserves parity and the $z$ component of orbital angular momentum. For $1s,2s,2p_0,2p_{\pm 1}$, its first-order energy shifts are respectively $(2\alpha +\beta )a_0^2$, $14(2\alpha +\beta )a_0^2$, $(12\alpha +18\beta )a_0^2$, and $(24\alpha +6\beta )a_0^2$. They follow by multiplying the hydrogen radial second moment by the angular average of $\alpha +(\beta -\alpha ){\cos }^2\theta$, whose ${\cos }^2\theta$ averages are $1/3,1/3,3/5,1/5$. Negative $\beta$ makes the globally extended potential unbounded below; the shifts then describe local perturbative resonance poles or a potential stabilized at large distance.

One isotropic three-dimensional bosonic branch with dispersion $ϵ=\alpha p^2$ and zero chemical potential has density of states per volume $\sqrt{ϵ}/(4{\pi }^2{\hslash }^3{\alpha }^{3/2})$. Integrating its Bose-Einstein energy and differentiating temperature gives the title coefficient. Identical independent branches multiply it by their number. Thus a measured $T^{3/2}$ coefficient fixes branch multiplicity divided by ${\alpha }^{3/2}$, not a dispersion coefficient independent of that counting convention. A gap or low-dimensional spectrum changes the asymptotic law.

For a normalized quantum state $\psi$ and a self-adjoint operator $H$ on a finite-dimensional Hilbert space, expand its unitary time evolution:Multiplying by its complex conjugate gives the survival probabilityThe function is even in $t$, so its odd coefficients vanish. The coefficient $(\Delta H{)}^2=⟨H^2⟩-⟨H{⟩}^2$ is the variance of the energy. This quadratic departure yields the quantum Zeno effect when ideal projective measurements repeatedly reset the quantum state.

The dipole coupling $-pE\cos \theta$ connects only $J\pm 1$ at fixed $m$. For $J\ge 1$, second order gives $(pE{)}^2[J(J+1)-3m^2]/[2BJ(J+1)(2J-1)(2J+3)]$; the ground-state result is separately $-(pE{)}^2/(6B)$. Here $B$ has energy units and $\theta$ is the polar angle to the field.

For an isolated weakly damped resonator mode, $Q=\omega E/P_{\mathrm{loss}}$. If $E(t)=E(0)e^{-t/{\tau }_E}$, differentiating gives $P_{\mathrm{loss}}=E/{\tau }_E$, so $Q=\omega {\tau }_E$. Since energy is quadratic in wave amplitude, the wave amplitude exponential relaxation time is ${\tau }_A=2{\tau }_E$. A decay by an energy factor $D$ in a measured interval $t_d$ gives $Q=\omega t_d/\ln D$. For a conventional reverberation time, $Q=2\pi f{T}_{60}/[6\ln (10)]$ under this exponential single-rate approximation.

For a weakly damped well, quantizing the conservative part of current-biased Josephson phase dynamics gives the local Hamiltonian operatorThe tilted potential energy is unbounded downhill. Its trapped energy levels are therefore long-lived resonances with escape lifetimes, rather than globally bound stationary states. Close to ${\varphi }_m$, set $x=\varphi -{\varphi }_m$ and expandKeeping only the quadratic term gives the quantum harmonic oscillator resultIt is useful for states localized near the minimum, with energy well below $\Delta U$. It does not count all the resonances accurately near escape. The omitted terms make successive spacings unequal. For the chosen example, the three plotted reference energies divided by $h$ are $1.44804$, $4.34412$ and $7.24020\mathrm{GHz}$ above the minimum.

A small microwave bias contributes $\delta U=-(\hslash /2e)I_{\mu }(t)\varphi$. A one-photon transition between energy levels is resonant when $hf=E_{n+1}-E_n$. Exciting a higher energy level can strongly increase its escape probability. Measuring microwave absorption indirectly through escape therefore connects an invisible quantum phase state to a visible electric potential difference. A classical oscillation can also show resonance, so the mere existence of one peak is insufficient: the bias dependence and resolved unequal spacings are the discriminating predictions.

The quantum asymmetric top has different rotational-energy coefficients on its principal axes. Its Hamiltonian commutes with $L^2$ but generally not with an individual component, so diagonalization proceeds within fixed total-angular-momentum multiplets. For a mechanical rigid rotor the positive coefficients are reciprocals of twice its principal moments of inertia. The component sign convention for body-fixed generators does not affect this squared-component energy expression.

An electric field normal to a semiconductor quantum well shifts the electron and hole confinement energies and displaces their envelopes in opposite directions. The exciton optical transition usually redshifts and its oscillator strength falls as their overlap decreases. In a symmetric unbiased well the permanent dipole $p_0$ vanishes, so the leading weak-field shift is quadratic. This optical shift includes confinement and binding changes; it is not solely a change in exciton binding energy.

Quantum escape of a current-biased Josephson junction is decay of a trapped collective quantum phase state through a barrier in the tilted Josephson washboard potential. This barrier lies in the potential energy of an electrical circuit coordinate. Transfer of Cooper pairs across the physical insulating layer already supplies the Josephson coupling energy; it does not by itself demonstrate this further collective escape. Single-electron superconducting tunneling is a third, distinct process.

At higher temperature $T$, thermal activation instead lifts the coordinate over the barrier. In the rare-escape regime its characteristic form is the Kramers escape ratewhere $k_B$ is the Boltzmann constant and the prefactor depends on damping and thermal equilibration. Cooling suppresses this route. For a quasistationary energy level at energy $E$, the leading conservative WKB approximation instead haswith the two turning points bracketing the forbidden barrier. This is an exponential estimate requiring a large action compared with $\hslash$, slow escape compared with intrawell motion, and weak environmental corrections. It is not a complete rate prediction near the barrier top or with substantial damping.

The familiar cubic-barrier exponent can be checked independently. Choose $V(x)=M_{\varphi }{\omega }_p^2{x}^2(1-x/a)/2$, with $a>0$. Its top is at $2a/3$ and $\Delta U=2M_{\varphi }{\omega }_p^2{a}^2/27$. At the well-bottom reference energy $E=0$,Substitution $u=x/a$ reduces the dimensionless integral to ${\int }_0^{1}u\sqrt{1-u}du=4/15$. This leading semiclassical action uses a cubic potential and a well-bottom reference; actual ground-state escape also needs the zero-point correction, prefactor and environment. In the chosen washboard example it gives $B\simeq 74.0$ after matching the exact barrier and curvature. This matching is an illustrative approximation, not an exact action integral for the cosine potential.

Escape need not yield a sustained voltage: the coordinate can become trapped again. The running-state readout assumes sufficient underdamping and bias to avoid such retrapping. The parallel model has quality factor $Q={\omega }_{p}RC$; large $Q$ makes energy loss per oscillation small. Low-temperature rate saturation alone cannot distinguish quantum tunnelling from unwanted heating or external noise.

A classical oscillator of amplitude $A$ has energy $m{\omega }^2{A}^2/2$. Comparing with the oscillator spacing $\hslash \omega$ gives the large-occupation scale displayed. A coherent state with this classical amplitude has exactly this mean excitation occupation and a Poisson number distribution; its total mean energy also includes the zero-point term. An observed macroscopic amplitude does not, by itself, establish a pure coherent state.