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The ratio of an evaporation cutoff to thermal energy. Keeping the cutoff fixed while lowering temperature increases this ratio and reduces population in the escaping tail.

Evaporative cooling of atoms lowers the mean motional energy of a trapped gas by preferentially removing its more energetic atoms. Subsequent elastic collisions redistribute the remaining energy, producing a cooler approximately thermal sample.

In forced evaporative cooling, a decreasing trap-depth threshold or a radiofrequency evaporation knife continuously removes the high-energy tail. Losing atoms alone is insufficient: losses that sample all energies equally do not provide the same cooling mechanism. Useful evaporation increases phase-space density of an atomic gas while retaining enough collisions for rethermalization. An expansion measurement before and after evaporation tests the velocity distribution as well as the retained atom number.

For a classical three-dimensional harmonic trap of fixed frequencies, $E=3N{k}_{B}T$. If each lost atom removes mean energy ${ϵ}_{\mathrm{ev}}=(\eta +\kappa )k_{B}T$, differential energy balance gives $d\ln T/d\ln N=(\eta +\kappa -3)/3$. The result assumes rapid elastic equilibration and excludes separate work from changing trap frequencies.

The predator-prey first integral $H=cX-d\log X+bY-a\log Y$ changes by the displayed amount when $h$ prey are removed. At a fixed prey threshold this increment is constant; sufficiently high thresholds reduce oscillation energy until the threshold is no longer crossed. At a fixed predator threshold, the increment depends on the prey abundance and on the crossing direction. A rising-predator balanced-quota cycle has negative return multiplier of magnitude greater than one; its leading small-oscillation approximation is marginal rather than a robust attractor.

A theoretical nuclear physicist who develops descriptions of nuclear forces through effective field theory. His joint review relates the allowed low-energy interactions of nucleons to symmetries of quantum chromodynamics. Organizing interactions systematically makes it possible to improve a nuclear calculation in stages and compare few-body predictions with measured nuclear properties.^[ref]

A confining sextic potential energy can have an exactly known ground state. Introduce a length scale $\ell >0$ and energy scale $ϵ={\hslash }^2/(2m{\ell }^2)$. Forthe normalizable wavefunction isThe version with $\ell =1$ uses the corresponding length unit; powers of a dimensional position cannot otherwise be added without scale factors.

To verify the result, set $y=x/\ell$. Then ${\partial }_y^2{\psi }_0=(y^6-3y^2){\psi }_0$, so the Schrödinger equation gives $H{\psi }_0=2ϵ{\psi }_0$. It is the lowest eigenstate by the nodeless theorem for a one-dimensional ground state.

There is also a direct positivity proof. DefineIntegration by parts on the bound-state domain gives $H=Q^{†}Q+E_0$, with $Q{\psi }_0=0$. Since $⟨Q^{†}Q⟩=\Vert Q\psi {\Vert }^2\ge 0$, no normalized state can have energy below $E_0$.

In a spatially flat universe dominated by a homogeneous canonical scalar field, use natural units and reduced Planck mass $M$. For $\lambda >0$, ${\varphi }_0>0$ and $t>0$, takeThis is an exact solution for a particular scalar potential, not for an arbitrary prescribed potential. The identities $\dot{H}=-{\dot{\varphi }}^2/(2M^2)$ and $V=3M^2{H}^2-{\dot{\varphi }}^2/2$ reconstruct it asTo check the reconstruction, $\dot{\varphi }=2\lambda {\varphi }_0/\sinh (2\lambda t)$ and $\sinh (\varphi /{\varphi }_0)=-1/\sinh (2\lambda t)$. The scalar equation then follows from the Friedmann equation and energy conservation where $\dot{\varphi }\ne 0$.

For $2/3<\beta <2$, the potential is positive and expansion begins as a decelerating power law $a\propto t^{\beta /2}$. At late times $H\to \lambda \beta$ and expansion becomes exponential. More precisely ${ϵ}_H=-\dot{H}/H^2=(2/\beta ){\mathrm{sech}}^2(2\lambda t)$ decreases through one. This illustrates entry into late accelerated expansion rather than a graceful exit from inflation.

For the normalized velocity-reset collision operator on the weighted Hilbert space, $L=\Pi -I$ vanishes on the equilibrium direction and equals $-I$ on its orthogonal complement. The spectral gap is exactly one. Conservation of the projected component and the displayed energy identity give exact exponential decay of the remaining component; the full solution need not decay to zero.

In a two-level system, let the Hamiltonian operator have matrix $H=\mathrm{diag}(0,2)$ in the orthonormal basis $|0⟩,|1⟩$. The normalized real trial quantum states and their Rayleigh quotient areThe excited-state probability amplitude is $\sin \theta$, so its probability is ${\sin }^2\theta$. Thus $E(\theta )\ge 0=E_0$, with equality if and only if the excited-state probability amplitude vanishes: $\theta \in \pi \mathbb{Z}$. The remaining quantum state is $|0⟩$ up to a global phase. At $\theta =\pi /4$ the trial energy is $1$; at $\theta =\pi /2$ it is $2$. These are upper bounds on the ground-state energy, even though neither is a minimizing trial.

The real trial family contains the ground state, so it attains the exact minimum despite not covering every complex quantum state. More generally, a family covering all normalized vectors in a two-dimensional Hilbert space gives the exact lowest eigenvalue of any Hermitian matrix by minimizing its Rayleigh quotient. For a complex trial $c_0|0⟩+c_1|1⟩$ with $|c_0{|}^2+|c_1{|}^2=1$, the same calculation gives $E=2|c_1{|}^2$: a relative quantum phase cannot change this diagonal Hamiltonian energy. This finite example is the weighted mean proof of the variational method with just two energy eigenstates.

The exchange interaction is an effective interaction between spins originating in the quantum mechanics of electrons. Its sign convention must accompany any formula. For the illustrative classical fixed-length spin pair used here, writeWith this plus-sign convention, $J>0$ favors antiparallel spins: $E(\pi )=-J{S}^2$, compared with $E(0)=+J{S}^2$. The energy difference is $2J{S}^2$. If instead the Hamiltonian is written with an overall minus sign, the same physical preference has a negative coupling. This prevents a sign convention in the Ising model from being mistaken for a disagreement about the ordering.

Excitation energy is an energy level's energy above a specified ground state. Emitted photon energies measure differences between levels, with a small recoil correction when the emitter has finite mass.

In the stationary-state terminology used here, an excited state is a normalized energy eigenstate with energy eigenvalue $E>E_0$, where $E_0$ is an attained ground-state energy. Its excitation cost above the ground state is $\Delta E=E-E_0>0$. Several energy eigenstates can share one excited energy eigenvalue through degenerate energy levels. An arbitrary quantum superposition with energy expectation value above $E_0$ need not be a single stationary excited state; its energy content is described by its Born rule probabilities or, for a continuum, its scalar spectral measure.

An excited state can be a stationary state: the second infinite square well mode is an immediate example. Spontaneous photon emission is a transition caused by coupling matter to the electromagnetic field; it is absent from an isolated matter-only Hamiltonian operator. The relevant energy eigenstate of that isolated model is therefore not automatically an energy eigenstate of the full interacting system. This distinction explains how an ideal stationary excited state can approximate a state with a finite experimental lifetime. See electric-dipole interaction for the coupling used in the simplest optical model.

For a one-photon transition between stationary internal states, neglecting recoil, conservation of energy requiresHere $h$ is the Planck constant and $\nu$ is the photon frequency. Energy matching alone does not guarantee a transition: in the electric-dipole interaction approximation its strength also involves the matrix element $⟨f|\mathbf{d}\cdot \mathbf{ϵ}|i⟩$, where $\mathbf{d}$ is the electric dipole moment operator and $\mathbf{ϵ}$ specifies photon polarization. A zero matrix element gives an electric-dipole selection rule. Thus an absent optical line does not prove the corresponding energy eigenvalue is absent; a different interaction can couple different states. ^[ref]. The Franck-Hertz experiment gives a particularly instructive comparison between collision excitation and photon emission.

The first excited state terminology assumes a least energy eigenvalue above $E_0$. Such a next level is not guaranteed by quantum mechanics: a continuous spectrum can start immediately above the spectral bottom, or energy eigenvalues can accumulate there. For example, a Hamiltonian operator with energy eigenvalues $0,\epsilon ,\epsilon /2,\epsilon /3,\dots$, where $\epsilon >0$, has a ground-state energy zero but no least positive energy eigenvalue: every candidate $\epsilon /n$ is exceeded downward by $\epsilon /(n+1)$. A free particle on the whole line has neither a ground state nor a first stationary excited state. Each use of a discrete excitation ladder therefore needs a specified Hamiltonian operator and its boundary conditions.

Unrestricted minimization of a Rayleigh quotient seeks the ground state. To bound the $k$th excited energy from above, restrict normalized trial wavefunctions to the orthogonal complement of the exact lower-energy eigenspaces. In an exact orthonormal eigenbasis, all remaining expansion coefficients multiply energies at least $E_k$, so their weighted average is at least $E_k$. A known symmetry sector can supply some of these exclusions automatically. Orthogonality only to inaccurate approximate lower states does not automatically retain the same rigorous excited-state upper bound.

An exciton is a bound state of an electron and an electron hole. Their opposite electric charges attract through the Coulomb potential, producing a neutral quasiparticle with internal relative motion and motion of its center of mass. In a semiconductor, promoting an electron from the valence band into the conduction band leaves an electron hole; the pair can occupy either an exciton bound state or an unbound scattering state. Merely creating the two charge carriers does not establish binding. A neutral exciton can transport excitation energy without transporting net electric charge. Coupling this material excitation strongly to a photon can form a polariton.

The striking optical consequence is that an exciton optical transition can absorb a photon with less energy than the threshold for creating unbound electrons and electron holes. The difference is the exciton binding energy, not an extra one-electron energy band inside the band gap. The Wannier-Mott exciton supplies a useful large-radius model; the Frenkel exciton describes the opposite, tightly localized limit.^[ref]

In Giant Rydberg Excitons in Cuprous Oxide by Tomasz Kazimierczuk and others, a polished $34\mu \mathrm{m}$ copper(I) oxide crystal was mounted without strain at $1.2\mathrm{K}$. A tunable single-frequency dye laser scanned photon energy; balanced detection compared transmitted light with a reference beam. Transmission dips resolved spectral lines assigned to bound excitons through principal quantum number $n=25$. Their convergence towards an unbound threshold supports a hydrogen-like internal series, with small systematic deviations.^[ref]

Controls mattered: restricting broadband illumination improved resolution of high-$n$ excitons; lowering laser power or expanding its spot reduced interaction-induced suppression. Balanced detection suppressed source fluctuations. The measurement probes optical resonances, not individual electron-electron hole trajectories or directly imaged radii. Electric-dipole selection rules favor P-envelope states in this material, unlike the hypothetical permitted lowest-state transition above. Finite resolution and crystal inhomogeneity affect spectral lines; the inferred binding series requires a continuum threshold and a model.^[ref]

The exciton binding energy is the positive energy needed to separate a bound electron and electron hole into the relevant unbound scattering states. Compare the exciton energy $E_X$ with the continuum threshold $E_{\mathrm{cont}}$ at the same total crystal momentum and for the same energy bands. The negative internal energy eigenvalue $-E_B$ and the positive separation cost $E_B$ are different sign conventions for related quantities. An exciton may move freely through a perfect crystal while remaining internally bound: “free exciton” does not mean an unbound pair.

The first two optically active hydrogenic levels have separation $32R^{\ast }/9$ in the planar model, versus $3R^{\ast }/4$ in the bulk model. The separation cancels an uncertain common continuum threshold, making it useful when rounded band parameters do not reproduce every absolute peak energy.

For the ideal Wannier-Mott exciton in energy bands whose edges occur at the same crystal momentum, the pair excitation energy, measured from the unexcited crystal, isHere $\hslash \mathbf{K}$ is the pair's total crystal momentum; the last term is center of mass kinetic energy. For a permitted photon absorption transition with negligible recoil energy, $\hslash \omega \simeq E_g-E_{B,n}$. An electric-dipole selection rule and a nonzero transition matrix element are still needed: a bound state need not produce an observable spectral line. If the relevant energy band edges have different values of crystal momentum, a phonon can also be needed to conserve crystal momentum; subtracting a binding scale from a band gap alone does not specify the optical threshold.

For example, the lowest internal bound state in this hypothetical model costs $1.489115\mathrm{eV}$ to create optically, yet needs only $0.010885\mathrm{eV}$ more to dissociate. The large creation energy and the small exciton binding energy answer different physical questions. An electron-hole recombination event can instead release excitation energy by photon emission or to other degrees of freedom; an emitted photon need not reproduce the original absorption energy if relaxation intervened.

Consider an original noninteracting model with three orthonormal spatial orbitals $A,B,C$ at energies $0,\Delta ,3\Delta$, where $\Delta >0$. Both spin projections are available at each energy, so there are six spin orbitals. The Quantum Hamiltonian is $H=h\otimes I+I\otimes h$, with no interaction between particles and no dependence on spin. Comparing its allowed ground states isolates the effect of fermionic exchange symmetry.

For two distinguishable spin-one-half particles, the spatial ground state is $A\otimes A$ and any of four independent two-spin states is possible: $E_0=0$ and ground-state degeneracy four. For two identical electrons, both can still occupy $A$, using $A\uparrow$ and $A\downarrow$, but there is just one allowed ground state: the symmetric spatial factor times the spin-one-half singlet state,Exclusion has reduced the ground-state degeneracy without raising the ground state energy.

Now restrict both particles to the same spin projection $\uparrow$. Distinguishable particles can still both occupy $A\uparrow$, so $E_0=0$. Identical electrons must occupy different spatial orbitals; the lowest choice is $A,B$, withThe spin factor is a spin-one-half triplet state, and the spatial factor is antisymmetric. These are ground states within the restricted parallel-spin sector, not the unrestricted two-electron ground state. Both restricted models have ground-state degeneracy one.

The first excited two-electron energy in the unrestricted model is $\Delta$, with four independent quantum states: one occupant in $A$, one in $B$, and two spin choices for each. A basis adapted to particle exchange operators separates these into one spin-one-half singlet state with symmetric spatial factor and three spin-one-half triplet states with antisymmetric spatial factor. All four have equal energy in this model. Singlet versus triplet does not by itself determine an energy gap; an interaction or a spin-dependent term is needed to split them.

As a further count, the full two-electron space has dimension $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$. Its energies in units of $\Delta$ and corresponding multiplicities are $(0,1,2,3,4,6)$ and $(1,4,1,4,4,1)$. For distinguishable spin-one-half particles the ordered tensor product instead has dimension $6^2=36$, with multiplicities $(4,8,4,8,8,4)$ at those same energies. These are exact model counts, not atomic measurements. The higher restricted ground state energy follows from removing forbidden quantum states; the Quantum Hamiltonian contains no additional “exclusion force”.

The expected energy of a random physical state is the expectation of its energy observable. For discrete states with probabilities $p_i$ and energies $E_i$, it is $U={\sum }_i{p}_i{E}_i$, provided the sum is defined. In the canonical ensemble, $p_i=e^{-\beta E_i}/Z$ gives $U=-{\partial }_{\beta }\log Z$ for energy levels independent of $\beta$, where $Z$ is the partition function. This distinguishes an average over possible states from the energy measured in one realization; the variance quantifies the remaining fluctuations.