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Put $c=m\kappa /{\hslash }^2$, $k_n=(n+1/2)\pi /a$, and ${\delta }_n=a({\lambda }_n-k_n)$. The quantization equation gives ${\delta }_n=\arctan (c/{\lambda }_n)$, so ${\lambda }_n{\delta }_n\to c$ and ${\delta }_n\to 0$. Therefore ${\lambda }_n^2-k_n^2\to 2c/a$, proving the displayed absolute energy shift. Its ratio to the unperturbed energy tends to zero. The normalized unperturbed even state has probability density $1/a$ at the origin, so its first-order delta-barrier expectation is also $\kappa /a$.

A first-order electric dipole transition connects opposite parities, but this does not prohibit all transitions between states of the same parity. In a symmetric infinite square well, use the centered position operator $X=x-L/2$. Each factor of $X$ reverses parity, so a second-order transition with two factors can connect the ground state to an excited state of the same parity through opposite-parity intermediate states.

For two-photon absorption, the second-order amplitude contains sums of products $⟨f|X|m⟩⟨m|X|i⟩$ divided by detuning-dependent energy denominators. Equal initial and final parity is therefore allowed, provided appropriate intermediate states and drive frequencies exist. “Allowed” does not guarantee a nonzero amplitude at every parameter value: different intermediate contributions can cancel. The distinction separates a symmetry-forbidden first-order matrix element from an all-orders prohibition.

The occupied molecular orbital of greatest energy in a specified reference electron configuration. A fixed-orbital model removes an electron from this level first; relaxation and electron correlation can modify the resulting energy.

A strongly stretched inextensible worm-like chain has two small transverse tangent components. Its quadratic energy is $\frac{1}{2}\int [A|{\partial }_s{\mathbf{t}}_{\perp }{|}^2+f|{\mathbf{t}}_{\perp }{|}^2]ds-fL$. For a long bulk chain, the equipartition theorem assigns each Fourier mode covariance $k_{B}T/(A{q}^2+f)$. Integrating over $q$ and summing both components gives $⟨|{\mathbf{t}}_{\perp }{|}^2⟩=k_{B}T/\sqrt{Af}$, hence the extension deficit is half that value. With persistence length $L_p=A/(k_{B}T)$, the deficit is $\sqrt{k_{B}T/(4L_{p}f)}$. It requires small slopes and negligible stretching of the backbone.

For a fast ion traversing a fixed material thickness with only a small fractional energy loss, the dipole excitation probability per atomic encounter scales as $Z^2/v^2$ at fixed impact parameter. Summing transferred excitation energies and integrating the encounter measure $2\pi bdb$ gives the leading electronic stopping power scaling, with slower logarithmic dependence from the allowed excitation energies and impact-parameter range.

At fixed ion kinetic energy $E=M{v}^2/2$, a heavier isotope is slower, so the leading loss scales as $Z^{2}M/E$. At fixed velocity, isotopes of the same element instead have approximately the same electronic loss; plotting against energy per nucleon removes the leading isotope mass dependence. The charge should be the effective projectile charge if its bound electrons screen the nucleus. At lower speeds, excitation thresholds, slow-passage suppression, charge exchange and nuclear energy transfer invalidate a simple extrapolation of the high-speed law.

Robert Hofstadter's Stanford installation selected electrons up to $190\mathrm{MeV}$. Magnetic spectrometer scans counted shielded Lucite Cherenkov radiation pulses per monitor charge. Carbon-12 at $187\mathrm{MeV}$ and $80^{\circ }$ resolved $185.1\mathrm{MeV}$ elastic scattering and $180.7\mathrm{MeV}$ inelastic scattering ($4.43\mathrm{MeV}$ excitation). Calibration used forward-beam energy, hydrogen nuclear recoil and carbon-12 energy levels. Closed entrance slits checked background.^[ref]

The key distinction is between an electron counter and an energy measurement. Here the counter registers particles admitted by the magnetic spectrometer; changing its magnetic field selects another outgoing momentum. A peak is therefore reconstructed from a scan, rather than read as one pulse whose height equals the scattered electron energy. Dividing counts by monitored incident electric charge compensates for changes in beam intensity. A peak's height alone is insufficient: its area and the instrument's accepted momentum interval determine the selected yield.

An inelastic scattering peak corresponds to a different final nuclear quantum state, so including it in the elastic scattering yield would answer a different structural question. Conversely, finite energy resolution can hide sufficiently nearby excitations beneath an apparent elastic scattering peak. Background subtraction, target-loss corrections, angular acceptance and counter efficiency all enter before counts become a differential scattering cross-section. A dip in uncorrected counts is therefore not automatically a zero of a nuclear charge form factor.

A holonomic constraint is a relation $f(q,t)=0$ involving generalized coordinates and possibly time, rather than a genuinely nonintegrable velocity relation. Independent smooth constraints of full row rank define a local configuration submanifold. Ideal constraint forces are normal to its tangent virtual displacements and can be represented by Lagrange multipliers. A time-dependent holonomic constraint need not conserve mechanical energy.

For a uniform three-dimensional ideal gas of spinless bosons with energy $p^2/(2m)$ and ground energy set to zero, the excited number density is $n_{\mathrm{ex}}={\lambda }_T^{-3}{\mathrm{Li}}_{3/2}(z)$, with quantum gas fugacity $z=e^{\mu /(k_{B}T)}\le 1$. Its finite maximum $\zeta (3/2)/{\lambda }_T^3$ gives the displayed critical temperature. Below it, particle conservation gives condensate fraction $N_0/N=1-(T/T_c{)}^{3/2}$. The excited-state internal energy is $U=(3/2)V{k}_{B}T{\lambda }_T^{-3}{\mathrm{Li}}_{5/2}(z)$, with $z\to 1$ below the transition. The exponent differs from that of a harmonic trap. Condensation is not by itself a demonstration of superfluidity; the ideal quadratic dispersion has vanishing Landau critical speed.

For a scalar theory with positive kinetic and gradient energy, a potential attaining a global minimum, and boundary conditions admitting a constant minimizing field, the energy is bounded below by the integral of that potential minimum. A static constant field at the minimum attains the bound. A potential unbounded below, such as $-{\varphi }^4$, invalidates the existence claim; incompatible boundary values can require an inhomogeneous minimizer.

For a Newtonian central configuration, set ${\mathbf{x}}_i(t)=a(t){\mathbf{a}}_i$ with $a>0$. The gravitational force scales as $a^{-2}$, reducing all equations to the displayed scalar ordinary differential equation. Its first integral is ${\dot{a}}^2/2-\Lambda /a=k/2$. The total energy is $kI/2$, so the scalar energy constant exactly tracks the many-body energy for the fixed shape.

For $\ddot{q}=-{\omega }^{2}q+g_2{q}^2+g_3{q}^3+(\tau +f_{1}q+f_2{q}^2)\dot{q}+\cdots$, the Hopf bifurcation amplitude equation has cubic coefficient $(f_2+f_1{g}_2/{\omega }^2)/8$ in the leading position amplitude. The quadratic restoring term produces mean shift $g_2{r}^2/(2{\omega }^2)$ and second harmonic $-g_2{r}^2\cos (2\omega t)/(6{\omega }^2)$. Consequently $⟨q{\dot{q}}^2⟩=g_2{r}^4/8$ and $⟨q^2{\dot{q}}^2⟩={\omega }^2{r}^4/8$, giving the coefficient by energy averaging. A negative value gives a supercritical stable limit cycle; a zero value requires higher-order analysis.

A recurrent network with symmetric interactions and an energy function. In the binary version, units take values $\pm 1$, self-couplings are zero, and asynchronous threshold updates can retrieve stored patterns from incomplete or corrupted cues. Fixed points are local energy minima, which need not be the intended memories or global minima.

For a symmetric matrix of couplings $W$ with $w_{ii}=0$, define the energy landscapeThe factor $1/2$ removes double counting. The local input is $h_k={\sum }_{j\ne k}{w}_{kj}{x}_j-{\theta }_k$. Choosing $x_k$ to agree with $h_k$ lowers this energy; this mathematical energy need not be a conserved physical quantity. Its pairwise form connects the model to the Ising model; competing couplings also motivate the comparison with a spin glass. Here retrieval seeks a useful nearby local minimum, so reaching the global minimum is not the goal.

John Hopfield's 1982 computer study stored random binary strings in neural networks of 30 and 100 units. Couplings were prepared from the stored patterns; asynchronous updates supplied the computational apparatus. Starting at each nominal memory, the readout counted Hamming distances after settling as storage load varied. Corrupted starting patterns separately tested recovery. Varying the load and cue damage exposed interference between memories. These simulations supported collective retrieval, not a measurement of biological memory capacity.^[ref]

Store only $\xi =(1,-1,1,-1,1)$ with $M=5$, zero thresholds, and Hebbian memory weights for a Hopfield network. If $d$ bits differ from $\xi$, their Hamming distance is $d$ and ${\sum }_i{\xi }_i{x}_i=M-2d$. Squaring this sum and subtracting the $M$ diagonal terms yields the displayed energy. Thus $d=0,1,2,3,4,5$ gives $E=-2,-2/5,2/5,2/5,-2/5,-2$.

For the cue $x=(-1,1,1,-1,1)$, the first unit has input $h_1=2/5$ and flips to $1$. The second then has $h_2=-4/5$ and flips to $-1$, completing recall. The energy drops $2/5\to -2/5\to -2$. Every cue with at most two wrong bits is corrected under a fair asynchronous schedule: a wrong bit sees the correct majority among the other four, while a correct bit either sees agreement or retains its value at a tie. The opposite pattern $-\xi$ is also stored implicitly, so cues with at least three errors retrieve it. This is a limitation of this symmetric one-pattern construction.

A network whose characteristic long-string separation grows with the horizon scale can lose energy into loops and radiation. Its density can scale as $\mu /t^2$ in natural units. This is distinct from the $a^{-2}$ dilution of an ideal frozen noninteracting string network.

If the typical wall separation grows as cosmic time, surface energy per volume is of order wall tension divided by time. This decreases more slowly than the usual radiation-era total density and can lead to a wall-domination problem.

Hot-air heating transports thermal energy with moving air and transfers it to a region as that air mixes and cools. A fan can establish a controlled flow; buoyancy can also drive natural circulation when warm air has lower mass density than surrounding air at similar pressure.

For a steady stream with approximately constant composition and specific heat capacity, its sensible-heating contribution is $\dot{Q}\simeq \dot{m}{c}_p(T_{\mathrm{supply}}-T_{\mathrm{return}})$. This is an enthalpy transport estimate, so both the flow rate and the temperature difference matter. Warmer supply air does not determine delivered power without knowing how much air moves. Spatial temperature gradients and circulation paths can make a single room thermometer an incomplete description. Comparing measurements at several heights tests whether a lumped thermal model captures the actual heating process.

A device concept that seeks to extract carriers through energy-selective contacts before their excess photon energy is lost by thermalization. Suppressing cooling while maintaining extraction is the central challenge.

Dilute shock-heated interstellar plasma with characteristic temperatures around $10^6\mathrm{K}$. supernova energy injection and subsequent cooling are important to its evolution; temperature alone does not determine its volume filling factor.

Hot-water heating carries thermal energy in circulating liquid water. The water gives up sensible heat in a radiator or pipe network and returns at a lower temperature. Under approximately constant pressure and specific heat capacity, the transferred power iswhere $\dot{m}$ is the mass flow rate.

The radiator's thermal conduction and its exchange with the room through convection and thermal radiation determine how large a temperature difference is needed. A circulation pump supplies mechanical work against fluid resistance, while the main heating source replenishes the lost enthalpy. Measuring inlet and outlet temperatures together with mass flow estimates the delivered thermal power. The circulating water can remain entirely within the liquid phase, making sensible cooling rather than a phase transition the principal transport budget.

Physical Review D 68, 103503. Horizon-scale matching depends on the inflationary energy history and subsequent reheating. Section II explicitly distinguishes instantaneous conversion with almost constant inflationary energy from cases with a substantial energy drop before the end.