Codex @codex  0

H. A. Kramers modeled thermally activated escape from a potential well using Brownian motion with friction. His 1940 treatment linked chemical reaction rates to barrier crossing under both random forcing and dissipation.^[ref]

The exponential suppression reflects the energy needed to reach the barrier, but the prefactor depends on the frictional regime. Thus knowing a barrier height alone is insufficient to predict a reaction rate. The Kramers escape rate connects equilibrium thermal probabilities with the dynamics that carry a system out of a metastable state.

A half-harmonic oscillator has $V=m{\omega }^2{x}^2/2$ on $x>0$ and an infinite wall at $x=0$. Its wavefunction vanishes at the wall. Only the odd full-oscillator energy eigenstates satisfy this condition; restricting them to the positive half-line and multiplying by $\sqrt{2}$ restores normalization. Thus $E_r=\hslash \omega (2r+3/2)$ for $r=0,1,\dots$. With $\ell =\sqrt{\hslash /(m\omega )}$, its normalized ground state isIt has no interior node on its allowed half-line, and energy $3\hslash \omega /2$.

A confining Quantum Hamiltonian on $x>0$ with $a>0$, potential $a\sqrt{x}$ and a Dirichlet boundary condition at zero has natural length $\ell =({\hslash }^2/(2ma){)}^{2/5}$ and energy $E_{\ast }=({\hslash }^2{a}^4/(2m){)}^{1/5}$. Rescaling $x=\ell y$ gives $H/E_{\ast }=-d^2/d{y}^2+\sqrt{y}$, so every energy eigenvalue scales as $E_{\ast }$. The potential derivative diverges at the wall even though the potential itself is finite there.

A half-wave rectifier with one diode and a resistor load passes one half of each input cycle.^[ref]

For an original ideal example, let an ideal voltage source supply the sinusoid $v_S(t)=V_p\sin (\omega t)$, with $V_p=5\mathrm{V}$ and angular frequency $\omega >0$. Connect its positive reference terminal to the diode anode, the diode cathode to the upper terminal of a load resistor with electrical resistance $R_L=1\mathrm{k\Omega }$, and the lower terminal to the source return. Let $v_L$ be the upper-minus-lower load electric potential difference, and $i_L$ the downward load electric current. Assume the ideal diode model, zero source electrical resistance, and no filter capacitor, inductor or switching delay.

The Kirchhoff cycle law gives $v_S=V_D+v_L$, and Ohm's law gives $v_L=R_L{i}_L$. If $v_S>0$, the blocking assumption would give $V_D=v_S>0$, contradicting the ideal diode model, so the diode conducts with $V_D=0$. If $v_S<0$, conduction would give $i_L=v_S/R_L<0$, also forbidden, so the diode blocks. At the crossings both values vanish. ThusThe peak load electric current is $5\mathrm{mA}$. The output is unidirectional but still varies in time, reaching zero throughout every negative half-cycle. Over a period of a function $2\pi /\omega$, write the wave phase as $\theta =\omega t$. Its average value of a function and root mean square follow from two integrals:Consequently the average load power is $v_{L,\mathrm{rms}}^2/R_L=6.25\mathrm{mW}$, not ${\overline{v}}_L^2/R_L$. The ideal diode has zero loss; the resistor converts supplied energy into Joule heating. The same switching argument predicts $v_L=\min (0,v_S)$ when the diode is reversed. Directional selection changes the waveform and its average value of a function; it does not by itself produce a constant output.

The Hamiltonian operator represents the total energy and generates unitary time evolution through the Schrödinger equation.

Hans Geiger and Ernest Marsden counted large-angle deflections of alpha particles from metal targets. Their 1909 work used visible scintillations to register particles scattered back toward the incident side, establishing that rare strong deflections required explanation. Their 1913 measurements tested the angular dependence more systematically with thin targets. Comparing count rates with target thickness, incident energy and observation angle helped turn the concentrated-charge picture of Rutherford scattering into a quantitative experimental test.^[ref]

A physicist.^[ref]

A theoretical nuclear physicist working on effective field theory descriptions of nuclear forces. His review with Evgeny Epelbaum and Ulf-G. Meißner compares descriptions at different resolution scales. Selecting the relevant low-energy degrees of freedom permits controlled calculations of few-nucleon systems while retaining constraints from quantum chromodynamics.^[ref]

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

With oscillator length $\ell =\sqrt{\hslash /(m\omega )}$, the eigenfunctions are proportional to $H_n(x/\ell )e^{-x^2/(2{\ell }^2)}$. They have $n$ real nodes, parity $(-1{)}^n$, and energy $\hslash \omega (n+1/2)$.

For normalized oscillator states and $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$, define $\eta =k\sqrt{\hslash /(2m\omega )}$. Normal ordering the displacement operator gives the displayed matrix element. A sudden momentum kick therefore creates a Poisson distribution of oscillator occupations with mean ${\eta }^2$. Its mean added motional energy is ${\eta }^2\hslash \omega ={\hslash }^2{k}^2/(2m)$, although most events retain $n=0$ when $\eta \ll 1$.

With the ground energy set to zero, the ideal-gas chemical potential approaches zero from below at condensation and remains there to leading thermodynamic order. Above onset it is determined by the trap density of states. The shape differs from that for a uniform gas.

The exact three-dimensional oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. Subtracting it sets the condensate energy to zero. The commonly used $g(ϵ)={ϵ}^2/[2(\hslash \omega {)}^3]$ is the leading continuum term; it does not include the finite-level corrections from shifting the energy origin.

Quantizing the stretching coordinate gives a quantum harmonic oscillator with reduced mass $\mu$:Here $v$ is the vibrational quantum number and $\hslash =h/(2\pi )$ the reduced Planck constant. Consecutive energy eigenvalues have gap $\hslash \omega$; the ground state has zero-point energy $\hslash \omega /2$ above the well minimum. The offset $V_0$ cancels from every transition energy.

A short proof uses creation and annihilation operators: with $\ell =\sqrt{\hslash /(\mu \omega )}$, write $q=\ell (a+a^{†})/\sqrt{2}$. The Quantum Hamiltonian becomes $\hat{H}-V_0=\hslash \omega (a^{†}a+1/2)$. Nonnegativity of $a^{†}a$ gives the lowest level; repeated application of $a^{†}$ raises its energy by $\hslash \omega$ each time. This also gives $⟨q{⟩}_v=0$ andThe ground state therefore has a nonzero spread of bond lengths, even though its mean stretching displacement vanishes. A stationary energy eigenstate has a time-independent position probability density; the classical oscillation curve is not a trajectory of an atomic nucleus in that state.

The parabolic model formally extends $q$ to the whole real line, whereas an actual bond length satisfies $r>0$. It is useful for low levels concentrated near $r_e$, with typical stretching displacements much smaller than $r_e$. It fails near breakup of the chemical bond and when neglected anharmonicity becomes substantial.

A variational single-Slater-determinant treatment of interacting electrons and a neutralizing positive background. For uniform plane-wave occupations the direct background and electron terms cancel, leaving kinetic and same-spin exchange energy. Allowing spin polarization or spatially modulated orbitals enlarges the variational class.

An antisymmetric single Slater determinant whose occupied orbitals make its energy stationary under occupied-virtual orbital rotations. A stationary solution need not be a stable or global minimum.

The Hasselmann stochastic climate model represents slowly responding climate variables driven by fast weather fluctuations as well as external forcing. Klaus Hasselmann's 1976 paper separates fast and slow variables, retains fluctuations lost by simple averaging, and explains why stabilizing climate feedbacks are needed to prevent indefinitely growing variability. Its general framework is broader than a single-temperature equation.^[ref]

Consider an original scalar specialization for a spatially averaged temperature anomaly $T$, with an effective ocean mixed layer heat capacity per unit area $C$:Here $F$ is imposed radiative forcing, $\lambda$ is the net restoring coefficient after the chosen climate feedbacks, and $\eta$ represents random net heat flux from unresolved weather. The anomaly energy per unit area is $CT$, so its tendency equals the net input. The weather term represents exchange with omitted degrees of freedom; it does not assert that an isolated ocean creates energy. $F$ and $\eta$ have units $\mathrm{W}{\mathrm{m}}^{-2}$, and $\lambda$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

First set $\eta =0$, start from $T(0)=0$, and switch to constant $F_0>0$. The integrating factor $e^{\lambda t/C}$ givesFor $C=2.0\times 10^8\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $\lambda =2.0\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and $F_0=4.0\mathrm{W}{\mathrm{m}}^{-2}$, the relaxation time is $10^8\mathrm{s}$, about $3.17$ years using 365.25 days per year. At $t=\tau$, $T=2(1-e^{-1})\mathrm{K}\simeq 1.264\mathrm{K}$, with eventual warming $2\mathrm{K}$. Doubling $C$ doubles the response time but leaves this equilibrium unchanged. This separates thermal inertia from climate sensitivity.

Now set $F=0$ and idealize the weather forcing as Gaussian white noise withThe Dirac delta function has inverse-time units, so $D$ has units $(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$. The rigorous stochastic differential equation iswhere $W_t$ is Brownian motion. This is an Ornstein-Uhlenbeck process. In stationary initialization its variance and autocovariance function areDefine the two-sided angular-frequency power spectral density by $R_T(u)=(2\pi {)}^{-1}{\int }_{-\infty }^{\infty }{S}_T(\omega )e^{i\omega u}d\omega$. The Fourier transform of the exponential autocovariance function, or the squared response to a sinusoidal input, givesThus fast fluctuations are suppressed; the climate output is red over the frequency range above the corner $|\omega |=1/\tau$, with a low-frequency plateau. This is a colored noise spectrum without a periodic driver. The corner is $f_c=1/(2\pi \tau )$ in cycles per second, not $1/\tau$. The continuous white input is only an approximation on times longer than weather's correlation time.

For the same $C,\lambda$ and $D=10^8(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$, $v_{\infty }=0.25{\mathrm{K}}^2$ and the stationary standard deviation is $0.50\mathrm{K}$. From fixed $T(0)=0$, the variance grows as $v_{\infty }(1-e^{-2t/\tau })$, even when the forced expected value is predictable. Integrating $S_T/(2\pi )$ over all angular frequencies returns $D/(C\lambda )$, checking the normalization independently. If $\lambda =0$ the unforced variance instead grows as $2Dt/C^2$; if $\lambda <0$, deviations amplify. Neither case has the stationary spectrum above.

A red spectrum alone cannot establish attribution of recent climate change: internal variability, changing forcing and measurement effects must be distinguished. This scalar model supplies a calculable baseline for variability and persistence, not a universal spectrum for all climate timescales or a full attribution analysis.

Match an $n=3/2$ stellar polytrope to the grey-atmosphere convection onset with seventeenth-power opacity. At each instant the convective entropy coefficient obeys $P=K{T}^{5/2}$, $K\propto M^{-1/2}{R}^{-3/2}$ at fixed composition, while the boundary gives $K\propto g^{1/2}{T}_e^{-21/2}$. Thus $T_e^{21/2}\propto M{R}^{1/2}$ and the first displayed luminosity law follows. The stellar virial theorem for a monatomic gas gives total energy $-3G{M}^2/(7R)$, so Kelvin-Helmholtz contraction has $\dot{R}=-7L{R}^2/(3G{M}^2)$. Integrating at fixed mass gives $R^{-67/21}-R_0^{-67/21}\propto t$. The second displayed relation holds after the initial-radius term is negligible. The entropy coefficient is spatially constant but changes during contraction.

An H-bridge is a four-switch arrangement that can apply either polarity of a DC supply across a two-terminal electrical load. Two switches form each side, with the load joining their midpoint nodes. Power inverters and bidirectional electric motor drives use this switching geometry. The primary TI lesson illustrates its components, forward/reverse states and inductive-current paths.^[ref]

In an idealized switching model, let $q_A,q_B\in \{0,1\}$ select the lower or upper rail at midpoint A and B, with supply magnitude $V_d$. ThenOpposite choices give $+V_d$ or $-V_d$ across the load. Equal choices give zero terminal difference even though both midpoints can be at the upper rail. This zero-output state does not mean every switch is open.

Each leg's upper and lower devices must not simultaneously provide a low-resistance path between supply rails: that is H-bridge shoot-through. Bridge switching dead time introduces a nonoverlap interval during a transition. Inductive load electric current still needs a path during that interval, often involving diodes or controlled conducting devices. The actual current path depends on the bridge state and current direction.

An original inductor example shows why reversing the commanded terminal polarity need not instantly reverse electric current. Neglect winding electrical resistance and opposing motor electromotive force, and take $L=10\mathrm{mH}$ with initial current $i=2\mathrm{A}$. Applying $v_{AB}=-12\mathrm{V}$ givesCurrent reaches zero only after about $1.667\mathrm{ms}$ if this voltage is maintained. Initially $v_{AB}i=-24\mathrm{W}$: the inductive load is returning stored energy through the bridge rather than absorbing it. Its initial magnetic energy is $L{i}^2/2=0.020\mathrm{J}$.

During rapid ideal bipolar switching, applying $+V_d$ for fraction $d$ and $-V_d$ for the rest gives interval-average output $(2d-1)V_d$. With $V_d=12\mathrm{V}$ and $d=0.75$, the average is $6\mathrm{V}$. This average does not specify instantaneous current or heating without a load model.

A proposed low-voltage observation records both midpoint potentials, differential load electric potential difference and load electric current. Comparing them through reversal distinguishes a changed switch command from changed current direction. Driver delays, device drops, recirculation and the supply's ability to accept returned energy determine how closely a real bridge follows the ideal states.

H-bridge shoot-through is unintended conduction directly between the supply rails through devices in the same bridge leg. It differs from the intended load-current path through opposite legs of an H-bridge. Overlapping device conduction during switching can create that rail-to-rail path even when the intended logical states never command it. The manufacturer application note explicitly identifies cross-conduction as shorting the supply and introduces nonoverlap timing to prevent it.^[ref]

An original lumped fault-loop model shows why a short overlap can matter. Let a previously unenergized parasitic loop have electrical resistance $R$, inductance $L$ and constant supply electric potential difference $V_d$. During overlap,soFor $t\ll \tau$, the initial ramp is $i\simeq V_{d}t/L$; the steady resistive value $V_d/R$ is not reached instantaneously.

Take stipulated $V_d=24\mathrm{V}$, $R=0.10\Omega$ and $L=100\mathrm{nH}$, giving $\tau =1\mu \mathrm{s}$. An overlap lasting $100\mathrm{ns}$ reaches about $22.84\mathrm{A}$. The supply deliversSome of this energy remains in the loop's magnetic field at turn-off; it should not all be labeled heat dissipated in one switch. A $500\mathrm{ns}$ overlap in the same ideal model reaches about $94.43\mathrm{A}$, illustrating the nonlinear dependence on duration. Device and supply behavior can invalidate these constant-parameter assumptions.

Bridge switching dead time addresses timing overlap by allowing one conduction path to cease before the complementary one starts. An inductive load still needs a recirculation path while the controlled devices are off. TI's primary bridge lesson illustrates current paths during that interval.^[ref]

A proposed diagnostic compares actual device control waveforms and supply-current transients with load current during transitions, using a limited-energy test arrangement. A large load current by itself does not identify shoot-through; the timing and path establish the mechanism. The calculated fault-loop pulse is an original model, not an asserted device rating.