The low-temperature heat capacity of a three-dimensional free electron gas is $({\pi }^2/2)N_{e}kT/T_F$. Expressed per mole of metal atoms with $z$ model conduction electrons each, this becomes the displayed result. Expressed per mole of conduction electrons, set $z=1$. A thermal-window counting argument predicts the $T/T_F$ scaling but gives only an order-one coefficient; it must still retain the number of electrons when changing the molar basis.

Reversing the phase on one half of an otherwise uniform slit makes its aperture transmission odd. Its Fourier amplitude is $4{\sin }^2(qb/4)/(iq)$, where $q=k\sin \theta$, so the axis is dark despite the removable denominator singularity. The largest symmetric lobes satisfy $\tan s=2s$, with first positive root approximately $1.16556$. Doubling an ideal half-wave layer restores equal phases and the ordinary bright-centred single-slit pattern. Total transmitted power is conserved if the layer has no absorption or reflection.

The maxima of the full two-slit intensity need not lie at maxima of its cosine-squared interference factor, because the single-slit envelope has a nonzero slope. Differentiating its logarithm gives $[\pi w/(\lambda L)](\cot u-1/u)=[\pi d/(\lambda L)]\tan v$, where $u=\pi wX/(\lambda L)$ and $v=\pi dX/(\lambda L)$. Solving this equation in the appropriate fringe gives the true maximum. The usual fringe-center estimate is useful when the envelope varies slowly across a fringe.

Let a coherent array of identical narrow slits have total far-field complex amplitude zero at a chosen direction. Reversing one slit's phase by $\pi$ changes its amplitude from $a_j$ to $-a_j$, so the new total is $-2a_j$. Its intensity is four times the single-slit intensity, independent of which slit was changed. For $N$ equally spaced slits, adjacent phase step $2\pi m/N$ gives such a zero when $m$ is not a multiple of $N$. A finite slit envelope multiplies the common intensity factor.

The relevant aperture size $b$ spans the entire coherent array, including the separated elements, rather than one element or a single baseline increment. A quadratic propagation-phase error is small only when $b^2/(\lambda R)\ll 1$ under the paraxial approximation. The often-used boundary $2b^2/\lambda$ is a convention for an allowed phase error, not a sharp physical transition. Individual-element patterns still supply an angular envelope in the far field.

For equal-frequency harmonic ground and excited molecular potentials displaced by $\Delta Q$, the low-temperature vibrational overlap has the Poisson weights in the title, with $S=M{\omega }_v(\Delta Q{)}^2/(2\hslash )$. Optical lines occur at $E_{00}+\nu \hslash {\omega }_v$, apart from broadening and slowly varying electronic or optical prefactors. The normalized weights sum to one and have mean vibrational excitation $S$. Unequal frequencies, several vibrations, hot initial states and intermolecular coupling modify this simple Franck-Condon principle model.

For nonzero denominators, $a/b=c/d$ exactly when $ad=bc$. For positive denominators, $a/b<c/d$ exactly when $ad<bc$. This compares fractions with exact integer arithmetic, avoiding rounding from floating-point arithmetic.

Exact arithmetic on rational numbers represented by integer numerators and nonzero integer denominators. Common factors can be removed with the greatest common divisor. Exact rational arithmetic avoids rounding error when deciding equality or integrality.

A unit fraction is $1/n$ for a positive integer $n$. Its numerator is one, so it is already a reduced fraction; the denominator determines its value uniquely. For example, $1/2$ and $1/7$ are unit fractions, while $2/7$ is not.

Every unit fraction lies in $(0,1]$, including $1/1=1$. A rational number can have several expressions as a sum of unit fractions: $1/2=1/3+1/6$, for example. An Egyptian fraction requires such a finite representation, usually with distinct denominators.

A plane curve generated by recursively folding or by a suitable L-system. One drawing convention uses the productions $a\mapsto aRbFR$ and $b\mapsto LFaLb$, where $F$ moves forward, $L,R$ turn by $90^{\circ }$, and $a,b$ expand but do not draw. Summaries of whole expanded blocks permit a position after a large number of forward steps to be found by descending only into the block containing that step.

A Pythagoras tree is a recursively constructed geometric fractal. Begin with a square; erect a right triangle on one side and construct outward squares on its two legs, then repeat on those squares. For the isosceles construction, both child squares have side length $1/\sqrt{2}$ times the parent side. The Pythagorean theorem makes the sum of their areas equal the parent area. The sum of areas of all squares at one generation is therefore constant, but overlapping descendants mean that summing generation areas does not give the area of the visible union.

On $0\le x\le L$, with $L>0$, the tent $f(x)=\min (x,L-x)$ has Fourier sine series coefficients $b_n=4L\sin (n\pi /2)/({\pi }^2{n}^2)$. Split the coefficient integral at $L/2$ and integrate its two linear pieces by parts; their cosine endpoint terms cancel and their sine terms add. Reflection about the midpoint makes every even-index coefficient zero, and the remaining odd-index coefficients alternate in sign. The $n^{-2}$ bound gives absolute convergence and uniform convergence; the series equals the continuous tent, including its zero endpoints.

On a rectangle $(0,L_x)\times (0,L_y)$, products of the one-dimensional Fourier sine basis give an orthogonal basis of L2 space. Fourier orthogonality in each variable givesFor a constant function one, these coefficients are $4[1-(-1{)}^m][1-(-1{)}^n]/({\pi }^{2}mn)$: only odd indices occur. The product modes vanish on the rectangle's boundary, making them natural Dirichlet Laplacian eigenfunctions. In the heat equation, their amplitudes decay as $e^{-\kappa {\pi }^2(m^2/L_x^2+n^2/L_y^2)t}$. Initial data not vanishing at the boundary are still represented in the interior and in the square-integrable sense; the positive-time boundary values need not agree continuously with the initially assigned boundary values.

For an integrable function on $[-L,L]$, with $L>0$ and integer $n\ge 1$, the displayed Fourier coefficient multiplies the mode $\sin (n\pi x/L)$ in its real Fourier series. Fourier orthogonality fixes the normalization because the squared mode has integral $L$. If $f$ is odd, the product with the sine is even, so $b_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$. These are also the coefficients of the Fourier sine series of the restriction to $(0,L)$.

The wave phase labels a physical stage of an oscillation at a spacetime event. Treating it as a Lorentz scalar makes its gradient $K_{\mu }={\partial }_{\mu }\Phi$ a covector; raising the index gives the four-wavevector $K^{\mu }=(\omega /c,\mathbf{k})$. A boost along $x$ gives ${\omega }^{\prime }=\gamma (\omega -v{k}_x)$ and $k_x^{\prime }=\gamma (k_x-v\omega /c^2)$. Thus angular frequency and wavevector transform together rather than independently.

All four interacting waves have the same frequency. Counterpropagating equal-frequency pumps can give a source proportional to the conjugate signal field, providing phase conjugation. The pump product must have the appropriate uniform spatial phase, and finite apertures and phase mismatch limit reconstruction.

For a photon and a massive particle with zero total momentum, their momenta are opposite and equal. Conservation of total energy fixes the same momentum magnitude before and after elastic scattering, independently of direction. A right-angle photon turn therefore makes the recoiling particle move perpendicular to the original photon direction in that frame. Transform both velocity components, or the complete four-momentum, to obtain the laboratory recoil; a transverse component is reduced by the boost Lorentz factor when the particle has zero longitudinal velocity in the zero-momentum frame.

A calibration constraint assigning zero probability to ages beyond a specified limit. Sequence data cannot overcome an excluded age under that prior.