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In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.

The energy component of the difference between the known initial four-momentum and the measured visible final four-momenta. In equal-beam electron-Positron collisions it is $\sqrt{s}-{\sum }_i{E}_i$ in natural units. It can arise from escaping neutrinos, but also from detector losses; it is distinct from the missing invariant mass and missing transverse momentum.

Use natural units $c=1$ and the Minkowski metric with signs $(+,-,-,-)$. In the ideal centre-of-momentum frame, assume a known initial four-momentum $P=(\sqrt{s},0)$, no unaccounted radiation and exact visible reconstruction. DefineThe missing four-momentum in electron-positron collisions is a subtraction of measured quantities. It is not itself a particle detector hit. With $M_{e\mu }^2=(p_e+p_{\mu }{)}^2$, expansion gives the useful identityFor any collection of future-directed massless invisible four-momenta $q_i=E_i(1,{\hat{\mathbf{n}}}_i)$,Thus a positive missing invariant mass excludes exactly one massless invisible particle under the stated assumptions. It does not exclude one massive invisible particle, identify a neutrino flavor, or determine the number of massless invisible particles beyond that lower bound. Several collinear massless particles can even have zero combined invariant mass. Detector resolution and unaccounted radiation invalidate a literal event-by-event interpretation of these ideal inequalities.

For an original exact example in natural units, choose total energy $6\mathrm{GeV}$ and visible massless four-momenta, in $\mathrm{GeV}$,Two possible invisible massless four-momenta are $(2;-1/2,-1/2,\sqrt{14}/2)$ and $(2;-1/2,-1/2,-\sqrt{14}/2)$. Splitting either into two positive collinear parts gives a three-particle interpretation with identical recoil. Alternatively, one invisible particle of mass $\sqrt{14}\mathrm{GeV}$ has that recoil. This example tests the ambiguity of the subtraction, without asserting any of these alternatives is a tau lepton decay.

With $(+---)$ signature and $c=1$, $T^0{}_0=(E^2+B^2)/(2{\mu }_0)$ while $T^0{}_i=-(\mathbf{E}\times \mathbf{B}{)}_i/{\mu }_0$. Raising the second index gives $T^{0i}=S_i$, where $\mathbf{S}$ is the Poynting vector, and symmetry gives the physical energy-flux components $T^{i0}$. The negative mixed component is a metric-sign effect, not reversed energy transport. In SI coordinates $x^0=ct$, $T^{i0}=S_i/c$.

A mobility edge separates localized from extended energy eigenstates on the energy axis. Three-dimensional disordered models can have both. In the infinite one-dimensional tight-binding model with independent continuous random site energies, all states are localized for any nonzero disorder, with probability one. This statement excludes specially correlated disorder.^[ref]

A mobility edge is not a band gap: states can exist on both sides of $E_c$, but differ in spatial extent. For electrons, the location of the Fermi energy relative to a mobility edge matters for low-temperature electrical conductivity. A three-dimensional sample need not have an extended interval at every disorder strength. In particular, the finite one-dimensional calculations below neither locate a three-dimensional mobility edge nor establish a nonzero universal critical disorder.

An energy interval with no extended bulk energy eigenstates available for transport. It can contain localized eigenstates and have a nonzero density of states; a mobility gap need not be a spectral gap.

For a uniform fixed-end mechanical string of length $L$, linear mass density $\mu$ and wave speed $c$, let its $n$th normal mode be $y_n=[A_n\cos {\omega }_{n}t+B_n\sin {\omega }_{n}t]\sin (n\pi x/L)$, where ${\omega }_n=cn\pi /L$. The conserved energy is the sum of kinetic energy and elastic potential energy,The sine and cosine squared integrals are $L/2$, so substitution proves the title formula. Orthogonality makes energies of distinct normal modes add. For zero initial displacement and initial modal velocity $V_n$, $A_n=0$ and $B_n=V_n/{\omega }_n$, giving $E_n=\mu L{V}_n^2/4$.

Radiation from rotational and vibrational transitions of molecules removes gas thermal energy. In primordial gas below the atomic threshold, molecular hydrogen is a principal coolant; hydrogen deuteride can matter in suitable chemical conditions. Molecular formation, photodissociation, optical depth and density-dependent level populations control the rate. This allows some small systems below the atomic virial temperature threshold to form stars.

A molecular motor couples an energy supply to directed mechanical motion. ATP synthase provides a rotary example.

Molecular orbital theory describes the electronic arrangement of a molecule using molecular orbitals: one-electron spatial wavefunctions extending over the combined atomic nuclei. Filling bonding molecular orbitals can stabilize a chemical bond; filling antibonding molecular orbitals opposes that stabilization. A molecular orbital is a building block of a many-electron description, rather than the entire many-electron wavefunction. See Robert W. Field, MIT 5.61, lecture 24, pp. 1–4: molecular orbitals and the LCAO approximation.

First make the interference visible with an exactly normalized one-dimensional toy. Replace the three-dimensional $1s$ atomic orbitals of the hydrogen atom by two Gaussian functions centred at $x=\pm a$, with dimensionless $x$ and $a>0$:The normalized sum and difference are ${\psi }_{\pm }=({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm S)}$, following the same linear combination of atomic orbitals algebra. At $a=1$, the orbital overlap is $S=e^{-1}\approx 0.367879$. At the midpoint, ${\varphi }_A(0)={\varphi }_B(0)\approx 0.455581$, givingBoth probability densities have total integral one: interference redistributes probability, rather than creating an extra electron. These are toy wavefunctions, not actual hydrogen atom orbitals or predictions of molecular hydrogen energies.

Now fill the two $1s$-derived molecular orbitals in the minimal model. The lower bonding molecular orbital is labelled ${\sigma }_g$; the upper antibonding molecular orbital is ${\sigma }_u^{\ast }$. The Pauli exclusion principle permits at most two electrons in each spatial molecular orbital, with opposite spin projections along a chosen axis. For the lowest electron configuration in this model, fill ${\sigma }_g$ first. The subscripts $g$ and $u$ denote even and odd behaviour under inversion through the midpoint; the star marks an antibonding molecular orbital. See Flowers et al., Chemistry 2e, 8.4: occupancy and bond order and Field, MIT 5.61, lecture 25, pp. 1–3: bonding, antibonding, and symmetry labels.

With $N_b$ electrons in bonding molecular orbitals and $N_a$ in antibonding molecular orbitals, the model's bond order is $b=(N_b-N_a)/2$. The three elementary counts are:

The zero bond order says this minimal model gives no net covalent bond for ${\mathrm{He}}_2$. It does not rule out every bound pair of helium chemical atoms: a very weak Van der Waals force can bind a neutral helium dimer, which has been observed experimentally. See Zeller et al., Imaging the He2 quantum halo state using a free electron laser, PNAS 113, 14651–14655 (2016), p. 14651: van der Waals binding and Luo et al., The weakest bond: Experimental observation of helium dimer, Journal of Chemical Physics 98, 3564–3567 (1993). The bond order count does not include that weak binding mechanism.

The useful surprise is that sharing alone does not determine a chemical bond: the relative quantum phase and the occupancy matter together. The same two shapes explain why adding an electron to a bonding molecular orbital increases the model's bond order, while adding one to an antibonding molecular orbital decreases it. Molecular orbital theory also accounts for the unpaired electrons and paramagnetism of molecular oxygen, which its all-paired Lewis structure misses. See Chemistry 2e, 8.4: the molecular orbital description of oxygen.

The single-molecule state sum $q={\sum }_s{e}^{-\beta E_s}$ for specified energy references and external constraints. In a dilute classical ideal gas, indistinguishability gives $Q_N=q^N/N!$; translation supplies the factor proportional to volume.

For a translation-invariant quadratic bosonic quantum field, anomalous terms couple $\mathbf{k}$ to $-\mathbf{k}$. A full-momentum sum uses half the pair-creation coefficient in each term, because both labels describe the same anomalous pair. After a canonical Bogoliubov transformation, however, the two members carry distinct quasiparticle occupations; each contributes its own oscillator energy. The vacuum correction is $\frac{1}{2}{\sum }_{k\ne 0}(E_k-A_k)$ in a full-zone sum. Switching to one representative from each pair requires changing both the occupation sum and constant consistently.

For a periodic box of volume $V$, expanding $\psi (\mathbf{r})=V^{-1/2}{\sum }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{r}}{a}_{\mathbf{k}}$ in $H_{\mathrm{int}}=\frac{u}{2}\int d^{3}r{\psi }^{†}{\psi }^{†}\psi \psi$ gives the displayed normal ordering. The spatial integral is $V{\delta }_{{\mathbf{k}}_1+{\mathbf{k}}_2,{\mathbf{k}}_3+{\mathbf{k}}_4}$ by Fourier orthogonality, enforcing momentum conservation and leaving the factor $V^{-1}$. The coupling $u$ in three dimensions has units of energy times volume.

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.

When a simply connected $SU(5)$ gauge group undergoes spontaneous symmetry breaking to the connected physical Standard Model subgroupthe vacuum manifold is $G/H$. The long exact sequence of homotopy groups of a fibration gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Thus the theory admits sectors with nonzero integer magnetic monopole topological charge; a charged configuration cannot relax continuously to the vacuum while its asymptotic field remains on the vacuum manifold. This classifies permitted stable defects. Their actual formation and abundance also depend on the transition and subsequent dynamics, as in the Kibble mechanism.^[ref]

In natural units, let $\eta$ be the magnitude setting the broken-symmetry Higgs field, $g$ its gauge coupling and $\lambda$ a characteristic dimensionless coefficient of the Higgs field potential. Up to normalization-dependent constants, the massive gauge bosons and radial Higgs field excitation have masses $g\eta$ and $\sqrt{\lambda }\eta$. Their inverse masses set the gauge and scalar core radii, respectively:Rescaling a static classical magnetic monopole profile by $g\eta$ gives its energy in the form $M_M=(4\pi \eta /g)f$ for a dimensionless profile factor $f$; its value depends on dimensionless ratios of scalar couplings to $g^2$, the Higgs field potential and the charge sector. Weak gauge coupling therefore makes these defects much heavier than the associated massive gauge bosons, rather than assigning them the elementary-particle scale $g\eta$.^[ref]

// Target: cosmology.bigb

The temperature derivative of the Fermi occupation has opposite signs below and above $\mu$. Pairing energies $\mu \pm x$ leaves a positive contribution because the density of states increases with energy. The unpaired higher-energy tail is also positive. Hence the density increases at fixed $\mu$ for an unbounded three-dimensional parabolic band.

The Quantum Hamiltonian with kinetic energy $p^2/(2\mu )$ and Morse potential has bound energies of the displayed form, where $A=\hslash {\omega }_e$, $B=A^2/(4D_e)>0$ and ${\omega }_e=a\sqrt{2D_e/\mu }$. Adjacent energy differences are $A-2B(n+1)$, distinct and positive over the physical bound-level range. A finite truncation with nonzero adjacent dipole couplings therefore satisfies connected nondegenerate transition chain generates special unitary control. Controllability of such a truncation does not by itself prove controllability on the full space including dissociation states.

The Morse potential models an anharmonic molecular bond with dissociation energy $D_e>0$, equilibrium separation $x_e$, and inverse length $a>0$. Unlike a quantum harmonic oscillator, it has finitely many bound levels and a dissociation continuum.

Rudolf Ludwig Mössbauer found the effect during his Heidelberg work in 1957.^[ref] His 1961 lecture Recoilless nuclear resonance absorption of gamma radiation describes crystalline preparations using the $129\mathrm{keV}$ transition of the iridium isotope ${}^{191}\mathrm{Ir}$. Separate cryostats, lead collimation and a particle scintillation detector measured transmission. Liquid-air cooling of both preparations enhanced absorption; cooling only the source against a room-temperature absorber slightly reduced it. A rotating source cryostat supplied nearly constant projected velocities over selected arcs. Centimetres per second suppressed absorption relative to zero velocity; a nonresonant absorber supplied the comparison. This tested a narrow common emission/absorption energy beyond the detector's resolution. Temperature data alone had not established its width. The lecture cites the initial and direct-verification publications from 1958.^[ref]

A nucleus bound in a solid can emit or absorb a photon without exciting lattice vibrations, transferring momentum to the solid as a whole. The resulting recoil energy is negligible compared with the isolated-nucleus recoil, making very narrow resonant emission and absorption lines observable. The Pound-Rebka experiment uses this effect to resolve the small frequency shift from gravitational redshift.

The striking feature is the combination of a large photon energy and a tiny natural linewidth: an atomic nucleus can act as an extraordinarily selective energy filter. The Mössbauer effect supplies a narrow shared emission line and absorption line even when nuclear recoil would separate the corresponding free-nucleus lines. A slow mechanical motion can then tune their overlap through the Doppler effect. The Pound-Rebka experiment is an application to general relativity; the discovery experiment below concerns the recoil-free process itself.