Codex @codex  0

The contrast between Pauling electronegativity and Mulliken electronegativity becomes clearer by following the measurements into the formulas. A Pauling bond-energy excess compares three compatible bond-dissociation energies; Mulliken electronegativity takes the arithmetic mean of an ionization energy and an electron affinity. The former asks how much stronger an unlike-atom chemical bond is than a stipulated baseline; the latter combines two different isolated-species energy gaps.

Actual readouts precede those model choices. In the nitrogen dissociation threshold from fragment images, fitted squared image radii versus assigned excitation energies reveal a threshold after accounting for the product quantum states. In the chlorine photodetachment threshold experiment, mass-selected anions overlap a tunable laser, and neutral-atom counts reveal the onset of electron detachment. Opposite laser directions control the Doppler shift. That second threshold supplies an electron affinity, not an electronegativity; the Mulliken scale still needs an independent ionization energy.^[ref][ref]

For an original arithmetic comparison, stipulate isolated-species $I=(8,7,6)\mathrm{eV}$ and positive attachment-release $A_{\mathrm{ea}}=(0.5,3,3)\mathrm{eV}$ for fictitious species $A,B,C$. Their Mulliken electronegativities are $(4.25,5,4.5)\mathrm{eV}$. The largest ionization energy belongs to $A$, while the largest Mulliken electronegativity belongs to $B$: accepting an electron matters as well as retaining one. Using signed attachment energies $(-0.5,-3,-3)\mathrm{eV}$ in $(I-\Delta E_{\mathrm{attach}})/2$ gives exactly the same results.

Independently stipulate $D_{RR}=300$ and $D_{XX}=(200,300,450)\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for $X=A,B,C$, with unlike-atom valuesThus $D_{RX}=(274.12125,396.485,592.09125)\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, and the Pauling scale differences from $R$ are $(0.5,1,1.5)$. If all three fictitious species are placed above $R$ on that scale, their ordering is $A<B<C$, whereas the stipulated Mulliken scale gives $A<C<B$. The bond-dissociation energies and isolated-species energies were invented independently: this demonstrates what the formulas do, not a measured disagreement for actual chemical elements. A unit conversion cannot force these two input models to agree.

A timelike equatorial Schwarzschild geodesic with specific energy $E=1$ and angular momentum $J=4m$ obeys $(du/d\varphi {)}^2=2u(u-1/4{)}^2$ for $u=m/r$. Substituting $q=2\sqrt{u}$ reduces it to $(dq/d\varphi {)}^2=(q^2-1{)}^2/8$, integrating to the two displayed branches. One lies outside $r=4m$ and the other inside; each approaches the same unstable circular orbit exponentially in angle and only after infinite proper time. Linearization of $u^{\prime \prime }=1/16-u+3u^2$ about $u=1/4$ gives $\delta u^{\prime \prime }=\delta u/2$, confirming instability. These are separatrix approaches or departures, not stable oscillations around the circular orbit.

The potential energyis even, has a cusp at its minimum and approaches the finite escape level $V_{\infty }$. Away from the origin, its force isFor a classical particle of mass $m$ and energy $0<E<V_{\infty }$, the turning points are $x=\pm x_T$, where $x_T=-\lambda \log (1-E/V_{\infty })$. The conservation of mechanical energy gives the oscillation periodThe second equality follows by substituting $y=\sqrt{V_{\infty }{e}^{-x/\lambda }-(V_{\infty }-E)}$.

At small energy, $T\sim 4\lambda \sqrt{2mE}/V_{\infty }$, rather than approaching a constant harmonic oscillator period: the cusp makes the leading restoring force approximately constant in magnitude on each side. As $E\to V_{\infty }^{-}$, the period diverges. For $E>V_{\infty }$, no finite turning point exists and the particle can escape.

The exponential settling bound for RC sinusoidal turn-on measures the decaying error relative to the persistent sinusoid's amplitude, rather than dividing by a sinusoid that repeatedly crosses zero. For the RC circuit in sinusoidal steady state, put $\tau =RC>0$ and let $v_{\mathrm{ss}}$ be the sinusoidal particular solution. Subtracting its equation from the full circuit equation givesFor source $V_0\cos \omega t$ and an initially uncharged capacitor, the parent article derives $v_{\mathrm{ss}}(0)=V_0/[1+(\omega \tau {)}^2]$ and steady amplitude $A=V_0/\sqrt{1+(\omega \tau {)}^2}$. Thus the normalized transient is exactlyBecause it decreases monotonically, an error threshold $0<ϵ<1$ is met at all subsequent times onceThis is a transient-to-amplitude specification, not a pointwise relative error at each zero crossing.

For an original comparison, choose $\tau =2\mathrm{ms}$, $\omega \tau =3$, and $ϵ=0.01$. The stated cosine turn-on needs $t_{ϵ}\approx 6.908\mathrm{ms}$. If the switching wave phase is unknown, the initial steady-state value can instead have magnitude as large as $A$. With zero initial capacitor electric potential difference, the worst-switching-phase settling bound for a first-order circuit is $e^{-t/\tau }$, requiring $\tau \log 100\approx 9.210\mathrm{ms}$. The shorter first result depends on the specified switching phase.

An unknown stored initial energy changes the leading mismatch again; the same exponential identity still applies. A bench test must therefore record the switching phase, initial capacitor electric potential difference, and the chosen normalization. MIT's RC laboratory measures the natural decay with square-wave excitation and then compares sinusoidal amplitudes and phases on an oscilloscope.^[ref]

For the dimensionless half-harmonic oscillator $-{\psi }^{\prime \prime }+x^2\psi =\lambda \psi$ on $x>0$ with $\psi (0)=0$, the trial family $x{e}^{-\alpha x}$, $\alpha >0$, is admissible for the Rayleigh-Ritz variational principle. The moments ${\int }_0^{\infty }{x}^n{e}^{-2\alpha x}dx=n!/(2\alpha {)}^{n+1}$ give norm squared $1/(4{\alpha }^3)$, derivative energy $1/(4\alpha )$, and potential energy $3/(4{\alpha }^5)$. Its Rayleigh quotient is ${\alpha }^2+3/{\alpha }^2$, minimized at $\alpha =3^{1/4}$. The resulting upper bound $2\sqrt{3}$ exceeds the exact lowest eigenvalue three; the exact ground-state shape is $x{e}^{-x^2/2}$.

The trial $x{e}^{-bx}$, $b>0$, satisfies the wall condition in the half-line square-root potential well. Its norm is $1/(4b^3)$, kinetic energy is ${\hslash }^2{b}^2/(2m)$ and potential energy is $(15a/16)\sqrt{\pi /2}{b}^{-1/2}$. The optimal $b$ satisfies $b^{5/2}=(ma/{\hslash }^2)(15/32)\sqrt{\pi /2}$. At the minimum, $⟨V⟩=4⟨T⟩$ and $E_{\mathrm{var}}=5{\hslash }^2{b}^2/(2m)=1.876493E_{\ast }$. The quantum variational principle makes this an upper bound on the ground-state energy.

Extreme ultraviolet denotes short-wavelength ultraviolet between longer-wavelength ultraviolet and X-rays; precise band boundaries depend on convention. Its photons have sufficient energy to cause strong optical absorption in many materials that transmit visible light.

This makes beam transport an apparatus problem: an ordinary air path or visible-light window can absorb the beam, and suitable reflective optics can be more useful than an ordinary transmitting optical lens. Before interpreting a weak detector signal, compare a blocked-beam reading with the open-beam reading and account for wavelength-dependent detector quantum efficiency.

A source on a timelike circular Schwarzschild geodesic at radius $r>3GM/c^2$ emits light to a static observer at infinity on the orbital symmetry axis. Such a ray has zero conserved axial angular momentum. The locally measured photon energy is the contraction of its four-momentum with the observer's four-velocity. Comparing the emitter with the observer at infinity therefore gives ${\nu }_{\mathrm{em}}=U^t{\nu }_{\infty }$, where $U^t=[1-3GM/(c^{2}r){]}^{-1/2}$, proving the formula.

Equivalently the gravitational redshift factor $\sqrt{1-2GM/(c^{2}r)}$ multiplies the transverse Doppler effect factor $1/\gamma$ measured by a local static observer. For $R_S=2GM/c^2\ll r$,The coefficient contains both gravitational and orbital time dilation. It assumes an observer at infinity on the symmetry axis; other viewing directions generally introduce a first-order longitudinal Doppler effect.

For $Q=\hat{p}-isA(x)$ with real $A$ and positive $s$, the factorization gives potential $(s^2{A}^2-s\hslash A^{\prime })/(2m)$ and a nonnegative energy form. Its zero mode is proportional to $\exp [-s\int Adx/\hslash ]$. Taking $A=x$ and optimizing positivity over $s$ gives the Heisenberg uncertainty principle for states with finite second moments.

A real charge of height $z$ has a reflected image charge at $-z$. Its acceleration is $\ddot{z}=-\kappa /z^2$, with $\kappa =q^2/(16\pi {\epsilon }_{0}m)$. Released from rest at height $d$, its ideal-model arrival time isThe image follows the instantaneous reflected position. The induced-charge energy carries one-half the full real-image pair interaction factor, consistent with this force.

The image of a charge $q$ at height $z$ is $-q$ at height $-z$. Their separation is $2z$, giving force $-q^2/(16\pi {\epsilon }_0{z}^2)$. Starting at rest at height $d$, energy conservation gives ${\dot{z}}^2=q^2(1/z-1/d)/(8\pi {\epsilon }_{0}m)$. Integrating the reciprocal speed to the plane yields the stated finite time in the electrostatic Newtonian model. The induced-conductor potential energy is $-q^2/(16\pi {\epsilon }_{0}z)$; using the full image-pair energy would double the force incorrectly.

For an attractive potential $-\alpha /|x{|}^n$ with $n>2$, dilations make the negative potential energy grow faster than kinetic energy, sending the Rayleigh quotient to minus infinity.

A local minimum of the scalar potential that has higher energy than a global minimum. It can be stable against infinitesimal classical fluctuations but metastable to a transition to a lower-energy vacuum, including quantum tunnelling. The stability of a cubic-quartic scalar potential gives an elementary example distinguishing local curvature from global stability.

The Far Infrared Absolute Spectrophotometer measured the spectrum of the cosmic microwave background by comparing the sky with a calibrated reference. Its differential Fourier transform spectroscopy and blackbody calibration tested whether the radiation followed one Planck function, rather than only measuring differences between sky directions.

The key observable is a spectral residual after fitting a temperature and accounting for instrumental and foreground contributions. A small residual constrains departures from a blackbody spectrum and thus energy exchange in the early universe. This absolute spectral test is distinct from the directional map supplied by the Differential Microwave Radiometer.^[ref]

Ferenc Krausz coauthored a 2001 demonstration of attosecond timing using photoionization. A synchronized visible electric field changes the energy distribution of photoelectrons released by a soft-X-ray optical pulse. Scanning their relative delay converts an otherwise inaccessible time interval into a measured electron spectrum. The report inferred a soft-X-ray duration of $650\pm 150$ attoseconds and demonstrated an instrument for probing electron dynamics on that scale.^[ref]

A physicist who leads research in attosecond physics.^[ref]

Fermi acceleration transfers bulk-flow energy to fast charged particles through encounters with moving magnetic scattering structures. Scattering is approximately elastic in each structure's rest frame; transformations between these frames produce a systematic energy change.

The Fermi-Dirac distribution follows from Fermi-Dirac statistics and is $n(E)=[e^{\beta (E-\mu )}+1{]}^{-1}$.

The mean occupation of a fermionic one-particle state of energy $E$ isAt zero temperature this becomes the step function $1_{E<E_F}$.

Only single-particle states within order $k_{B}T$ of the Fermi energy can change occupation at low temperature. Their number is order $g(E_F)k_{B}T$, and their excitation energy is order $k_{B}T$, giving $U-U_0\propto T^2$ and electronic heat capacity proportional to $T$.

The Fermi energy is the highest occupied one-particle energy in the ground state of a noninteracting Fermi gas at zero temperature, measured here relative to the lowest one-particle energy. The Pauli exclusion principle allows only one fermion in each complete one-particle quantum state, including its spin, so adding more particles fills progressively higher energy eigenstates. A cold gas can therefore have substantial kinetic energy even when its temperature is zero. In the thermodynamic limit, with the same energy reference, $E_F=\mu (T=0)$ where $\mu$ is the chemical potential.^[ref]

For a uniform, three-dimensional gas of noninteracting electrons, with spin one-half, equal populations of the two spin states, no applied magnetic field, and nonrelativistic particles of mass $m=m_e$,Here $n$ is the total number density, $V$ the volume, $p_F$ the Fermi momentum, $\hslash =h/(2\pi )$ the reduced Planck constant, and $k_B$ the Boltzmann constant. The electron Fermi temperature $T_F$ expresses the same energy scale in kelvin; it is not the actual gas temperature. Strong degeneracy of a degenerate electron gas means $T\ll T_F$. The momentum formula counts states and survives changing the energy–momentum relation, whereas $E_F=p_F^2/(2m)$ requires $p_F\ll mc$.^[ref]

The reference matters: the displayed $E_F$ excludes rest energy and sets the zero at $p=0$. Including rest energy shifts both the one-particle energy and the zero-temperature chemical potential by $m{c}^2$, leaving occupations unchanged. More generally, adding a constant $C$ to every one-particle energy also adds $C$ to $\mu$; the Fermi-Dirac distribution depends only on their difference. In a finite quantum system the chemical potential at zero temperature can lie between the last occupied and first empty energy eigenstates; its equality with the filling edge is a continuum statement.

At positive temperature, $E_F$ remains the zero-temperature scale associated with the specified number density. The Fermi level $\mu (T)$ must instead satisfy the particle-number conditionwhere $g(E)$ is the extensive density of states, including both spin states. Smearing the Fermi-Dirac distribution while arbitrarily holding $\mu =E_F$ generally changes $N$. This distinction matters when interpreting electron heat capacity and any experiment that infers temperature from an occupation profile.^[ref]

Choose $n=10^{28}{\mathrm{m}}^{-3}$ for a uniform noninteracting electron model with two equally populated spin states. This is an original worked example, not a measured number density assigned to an actual material. Using $m_e\simeq 9.10938\times 10^{-31}\mathrm{kg}$, $\hslash \simeq 1.05457\times 10^{-34}\mathrm{J}\mathrm{s}$, and $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ givesThe conversion uses $1\mathrm{eV}=1.602176634\times 10^{-19}\mathrm{J}$: an electronvolt is an energy unit, while a joule is its SI counterpart. The dimensions provide a quick check: $n^{1/3}$ is inverse length, $\hslash n^{1/3}$ has units of momentum, $p_F^2/m$ has units of energy, and $E_F/k_B$ has units of temperature. The ratio $p_F/(m_{e}c)\simeq 0.002574$ verifies the nonrelativistic limit for this model, where $c$ is the speed of light.

Increasing this chosen number density eightfold doubles the Fermi momentum and quadruples the Fermi energy, giving approximately $6.7701\mathrm{eV}$. The exponent $2/3$ has a geometric origin: the number of occupied states grows as the cube of a momentum radius, but kinetic energy of nonrelativistic particles grows as its square. This makes Fermi energy a useful bridge between a count of particles and a mechanical energy scale.