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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.

The QCD scale is the renormalization-group invariant energy at which the perturbative running coupling becomes strong. It sets the characteristic scale of confinement and most hadron masses.

It is a characteristic energy scale of quantum chromodynamics.

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.

A resistive velocity term proportional to the speed squared removes energy at rate proportional to $|\omega {|}^3$. Its derivative at zero velocity vanishes, so it does not appear in the linearization of a dynamical system at rest. Nonlinear energy arguments can still prove asymptotic stability.

For a positive self-adjoint Hamiltonian operator, finite energy means that a normalized state lies in $D(H^{1/2})$, its form domain, and has expectation value $q_H(\psi )$. This domain can be larger than $D(H)$. With a complete orthonormal basis of energy eigenstates, $\psi ={\sum }_n{c}_n{e}_n$ and $H{e}_n=E_n{e}_n$ give $q_H(\psi )={\sum }_n{E}_n|c_n{|}^2$ by applying $H^{1/2}$ and taking its squared norm. Membership in $D(H)$ instead requires ${\sum }_n{E}_n^2|c_n{|}^2<\infty$. For a Dirichlet free-particle Hamiltonian operator on an interval, the energy form is ${\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m)$ on functions with square-integrable weak derivative and zero endpoint values.

For constant density of states, weighting the difference between the Fermi-Dirac distribution and the zero-temperature step by $E-E_F$ produces a convergent integral that scales as ${\beta }^{-2}$. Hence the leading low-temperature energy correction is proportional to $T^2$.

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.

Two oriented linear filters in quadrature have responses $A\cos \phi$ and $A\sin \phi$ to a suitably matched narrowband grating. Their squared sum is $A^2$, independent of phase $\phi$, while orientation and spatial-frequency selectivity remain. A nonlinear output function or normalization can transform this energy into firing rate. Ideal quadrature gives an exact phase identity for the matched stimulus; finite approximate filter pairs need not have perfect invariance for arbitrary images.

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.

The translational concentration scale for a nonrelativistic particle in three dimensions, per internal state. It is the inverse cube of the thermal de Broglie wavelength. A classical component with number density $n$ and external one-particle energy $U$ has chemical potential $\mu =k_{B}T\ln (n/n_Q)+U$. An additional internal degeneracy must be included explicitly; it is not hidden in this definition. The dilute classical condition is $n/n_Q\ll 1$ for each occupied internal component.

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.

The quantum-critical regime is the finite-temperature region in which temperature exceeds the energy scale generated by detuning from a quantum critical point. Temperature then controls leading masses, relaxation rates, and correlation lengths.

Quantum decoherence is the suppression of observable quantum phase relations through interaction with unobserved degrees of freedom. A simple model starts with $(a|0⟩+b|1⟩)|E⟩$ and correlates the alternatives with environmental states:Taking the partial trace multiplies the off-diagonal entry of the reduced density matrix by $⟨E_1|E_0⟩$. If the environmental records become nearly orthogonal, interference between the two alternatives becomes locally inaccessible even though the joint state can remain pure.

Preserving diagonal populations is a property of dephasing, not a general requirement for quantum decoherence. An amplitude damping channel, for example, changes an excited-state population as ${\rho }_{11}\mapsto (1-p){\rho }_{11}$ while suppressing coherence as ${\rho }_{01}\mapsto \sqrt{1-p}{\rho }_{01}$. Thus energy relaxation and quantum decoherence can occur together, with different rates. Pure dephasing instead preserves populations in its selected pointer basis.^[ref]