A many-particle trial state built from one-particle factors, without the additional correlations of a general many-body wavefunction. For identical bosons occupying one normalized orbital $\phi$, the symmetric fixed-number form is $(a_{\phi }^{†}{)}^N|0⟩/\sqrt{N!}$. Its field expectation is zero because annihilation changes particle number, while its one-body density matrix has the macroscopic eigenvalue $N$. An antisymmetrized multi-orbital fermionic trial state instead belongs to the Hartree-Fock construction.

A variational product of one-electron orbitals produces coupled one-electron equations, each containing the electrostatic density of the other occupied orbitals. Iterating orbitals and potentials gives a self-consistent field. The product omits fermionic exchange; an antisymmetric Slater determinant instead leads to Hartree-Fock theory.

Crystallization favored by greater total accessible configurational entropy, despite increased spatial order. Ordered cages at sufficiently high packing fraction can allow more positional/free-volume configurations than a disordered crowded fluid.

The two-dimensional counterpart of a hard sphere: two equal discs of radius $r$ cannot have centre separation less than $2r$. Their pair-excluded area is $4\pi r^2$, but the second-virial excluded-area parameter per particle is half that value because a pair would otherwise be counted twice.

A change in the number of adjacent ring atoms coordinating a metal, such as ${\eta }^5\to {\eta }^3$. It changes the coordination ligand donation and can open a site without loss of the entire ring coordination ligand.

The fast Hadamard transform computes multiplication by a tensor power of the two-by-two Hadamard matrix using repeated pairwise sums and differences. For $N=2^n$ input amplitudes, one stage replaces each selected pair $(a,b)$ by $(a+b,a-b)$, with factors of $1/\sqrt{2}$ when using the unitary convention.

There are $n={\log }_{2}N$ stages, each requiring order $N$ arithmetic operations, giving order $N\log N$ classical work instead of a generic dense matrix multiplication. A quantum circuit implements the corresponding unitary $H^{\otimes n}$ with one Hadamard gate on each of $n$ qubits. This compact circuit description does not imply that all $N$ resulting amplitudes can be read out in one measurement; the output statistics still follow the Born rule.

For normalized marked and uniform states $|w⟩,|s⟩$ with real overlap $a\in (0,1)$, the two reflections $(2|s⟩⟨s|-I)(I-2|w⟩⟨w|)$ have the displayed matrix in that ordered nonorthogonal basis. The coordinate Gram matrix is $S=\left(\begin{array}{cc}1& a\\ a& 1\end{array}\right)$ and $G^{†}SG=S$. With $a=\sin t$, its eigenvalues are $e^{\pm 2it}$. Applied $r$ times to $|s⟩$, it gives marked-state success ${\sin }^2((2r+1)t)$. In these coordinates the overlap with the marked state is the first coefficient plus $a$ times the second; it is not generally the first coefficient alone.

A periodic rectangular grid whose opposite boundaries are identified. On an $m$ by $n$ grid, coordinates are taken modulo $m$ and $n$, so each horizontal or vertical step wraps around at a boundary.

For $0<x<1$, the homogeneous Cauchy-Euler equation has solutions $x,x^{-1}$. The left-vanishing solution $x$ and right-vanishing solution $x-x^{-1}$ give the unweighted Green function $G(x,s)=x(1-s^{-2})/2$ for $x\le s$ and $(x-x^{-1})/2$ for $x\ge s$. It is continuous at $x=s$ and its first derivative has jump $1/s^2$, so $L_{x}G=\delta (x-s)$.

For continuous forcing $f$ on $[0,1]$, its integral isThis is bounded at zero and vanishes at one, but $u(0^{+})=-f(0)$. The first singular term uses the average $x^{-1}{\int }_0^{x}f\to f(0)$. In the other, the substitution $t=x/s$ gives $x{\int }_x^{1}f(s)s^{-2}ds={\int }_x^{1}f(x/t)dt\to f(0)$ by dominated convergence. Thus the two zero endpoint values require $f(0)=0$. When compatible, the solution is unique among bounded solutions because no nonzero combination of $x,x^{-1}$ meets both homogeneous endpoint conditions. This example shows why pointwise endpoint behavior of a Green function does not itself justify passing a limit through its integral.

The homogeneous Cauchy-Euler equation $Ly=0$ has fundamental solutions $x,x^{-1}$ on $x>0$, with Wronskian $-2/x$. Requiring decay at both endpoints gives the unweighted Green's functionRequiring zero initial value and derivative at zero instead gives the forward kernelBoth kernels are continuous and have first-derivative jump one at $x=\xi$, so $L_{x}G=\delta (x-\xi )$. Away from that point they solve the homogeneous equation. Integrating against the forcing with ordinary measure $d\xi$ yields the respective solutions whenever the integral and endpoint limits converge. Their difference is $G_0-G_D=x/2$, independent of $\xi$, so the solutions differ by the homogeneous term $(x/2)\int f(\xi )d\xi$ for integrable forcing. For example, the two choices for an indicator forcing on $(0,1)$ differ by $x/2$. Specifying a differential operator without its endpoint conditions does not specify its Green kernel.

For zero initial value and slope, the causal Green function is $G_0(x,s)=H(x-s)\sin [k(x-s)]/k$, with $k\ne 0$ and $H$ the Heaviside step function. For zero values at endpoints $a<b$, put $\kappa =|k|$. The Dirichlet Green function isBoth are continuous at the source and have a first-derivative jump of one, giving $L_{x}G=\delta (x-s)$. The Dirichlet inverse exists when $\sin [\kappa (b-a)]\ne 0$; a zero denominator signals a nontrivial homogeneous mode and a Fredholm solvability condition. The endpoint kernel is the zero-conjugation case of the Dirichlet Green function for a conjugated harmonic oscillator. Boundary conditions determine which kernel must be integrated against a forcing term.