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A topological insulator has an insulating bulk electronic spectrum and boundary electronic behavior constrained by a nontrivial bulk band topology and the relevant protecting symmetries. In common three-dimensional time-reversal-invariant examples, the conducting surface can be approximated at low energy by a Dirac cone. The number of surface cones and the range of the linear approximation depend on the material; boundary conduction and thermal bulk phonons therefore need separate state counts.

For $\alpha >0$ and $t\ne 0$, the conformal factor $t^2/\alpha$ makes spatial translation a Killing vector field. Using proper time, its conserved momentum is $P=(t^2/\alpha )dx/d\tau$. With $v=dx/dt$ and $d{s}^2=-c^{2}d{\tau }^2$ this becomesThus $v/\sqrt{1-v^2/c^2}$ alone is not constant. Set $b=\sqrt{\alpha }P/c$. Every unparametrized timelike geodesic in a connected nonzero-$t$ patch has the formThe nonvertical paths have $v=cb/\sqrt{t^2+b^2}$ and proper-time parametrizationThe chart excludes $t=0$, where its metric is degenerate; a vertical family must not be lost by dividing by the conserved momentum.

For a finite jump in a one-dimensional potential energy, with constant mass and no delta potential, both the wavefunction and its first spatial derivative are continuous. Integrating the time-independent Schrödinger equation across a vanishing interval makes the jump in ${\psi }^{\prime }$ vanish, since the integral of $(V-E)\psi$ vanishes. A jump in $\psi$ itself would produce a derivative of a Dirac delta function in ${\psi }^{\prime \prime }$, with no matching term in the equation. The second derivative can jump. This matching allows transmission and quantum tunnelling; a finite step is not a hard wall.

For a particle of fixed energy $E$, a classically allowed region has positive classical kinetic energy $E-V(x)$. For a constant potential energy, the stationary wavefunction is a combination of $e^{ikx}$ and $e^{-ikx}$, where $k=\sqrt{2m(E-V)}/\hslash$ is the wavenumber. In a varying potential energy this local value underlies the WKB approximation away from turning points.

A classically forbidden region would require negative classical kinetic energy, but a quantum wavefunction need not vanish there. For constant potential energy, it is $A{e}^{qx}+B{e}^{-qx}$ with $q=\sqrt{2m(V-E)}/\hslash$. Boundary matching determines both coefficients. Beside a hard wall at $D$, a nonzero forbidden-region solution is proportional to $\sinh [q(D-x)]$, and therefore has no interior nodes. Across a finite barrier, both exponentials are generally needed; a pure decaying exponential is appropriate only when the other boundary condition permits it.

For a weak time-independent perturbation switched on at zero time, first-order time-dependent perturbation theory gives a transition probabilityThe frequency kernel has area $2\pi t$ and width of order $1/t$. Against a smoothly varying continuum of final states, it approaches $2\pi t\delta ({\omega }_f-{\omega }_i)$, giving the rate $2\pi |V_{fi}{|}^2\delta ({\omega }_f-{\omega }_i)/{\hslash }^2$. Using $E=\hslash \omega$ gives the energy form above. One integrates the Dirac delta function against the appropriate density of states; it is not an ordinary finite number for one isolated resonant level.

The time must be long enough to resolve the smooth continuum and short enough that the total transition probability remains small. For continuum variation scale $\Delta \omega$ and total rate $\Gamma$, the usual window is $(\Delta \omega {)}^{-1}\ll t\ll {\Gamma }^{-1}$. A discrete finite spectrum also limits this approximation through its recurrences.

A stack of thylakoids.

Stacking places many thylakoid membranes in a compact region of a chloroplast. The membranes remain part of a connected internal system; the stack is a structural arrangement, whereas the thylakoid lumen is the enclosed aqueous compartment. ^[ref]

The fluid interior enclosed by the thylakoid membrane.

Separates thylakoid lumen from chloroplast stroma in a chloroplast.

The number of target chemical atoms per incident area along the beam. For a uniform foil at normal incidence, number density $n$ and thickness $t$ give $n_s=nt$, in ${\mathrm{m}}^{-2}$. At oblique incidence the traversed path must replace $t$. The single-encounter count model requires the chance of two significant deflections to be negligible and energy loss through the foil to be small. A small probability in one detector patch alone does not establish either condition.

The Peltier effect is heat transfer associated with an electric current crossing a junction between materials with different Peltier coefficients. At fixed junction temperature, its contribution changes sign when the conventional current reverses. A junction can absorb heat from its surroundings for one direction and release it for the other. Joule heating, proportional to the square of the electric current, accompanies this transport but does not reverse sign. Consequently, observing a warm device alone does not identify the Peltier effect.

This makes the Peltier effect useful as an electrically controlled heat pump: reversing the supply can switch an object's thermal role without changing any plumbing. In thermoelectric cooling, however, the useful cooling is the junction's absorbed heat minus returning heat conduction and its share of Joule heating. Increasing electric current eventually makes cooling worse. The distinction between transported heat and net cooling is the central practical lesson of the thermoelectric cooler energy balance.

A thermoelectric module connects thermoelectric material pairs between thermal faces. Electrical series connections and thermal parallel connections combine their transport responses for thermoelectric cooling or electrical generation.