Photon counting registers discrete detector events and their times. A beam splitter sends the selected light to two photodetectors; a coincidence circuit or time-recording system compares the outputs. With a vacuum input at the unused optical port, splitter fractions $R,T$ and independent quantum efficiencies ${\eta }_1,{\eta }_2$, a selected input photon reaches the two recorded channels with probabilities $p_1=R{\eta }_1$ and $p_2=T{\eta }_2$. Conditional on input occupation $N$, each pair of distinct photons contributes $p_1{p}_2$ to the cross-channel pair moment. ThereforeTheir normalized ratio recovers $g^{(2)}(0)$. This assumes number-resolving counts, or sufficiently dilute gates for threshold detectors to approximate them. One photon cannot create both channel counts in this ideal model.

Photon statistics is the probability distribution of the number operator in specified field modes, or the distribution of registered counts in a specified gate. Those distributions agree only for an appropriate ideal measurement. Mode selection, losses, and detector dead time matter. For a one-mode density matrix, $p_n=⟨n|\rho |n⟩$ andThese two sums already distinguish preparations with the same expected value.

Fix a detected position and polarization of waves. Write the electric field operator as $\hat{E}(t)={\hat{E}}^{(+)}(t)+{\hat{E}}^{(-)}(t)$, where the positive-frequency part contains annihilation operators and ${\hat{E}}^{(-)}=({\hat{E}}^{(+)}{)}^{†}$. An ideal absorbing photodetector responds towith density matrix $\rho$. The diagonal gives the single-count rate up to the detector coupling. Multiple absorptions require higher, normal-ordered field products. Roy J. Glauber used these correlations to develop the quantum account of optical coherence.^[ref]

The ordering follows from a detection amplitude. If absorption takes an initial pure state $|i⟩$ to an unobserved final state $|f⟩$, its field factor is $⟨f|{\hat{E}}^{(+)}|i⟩$. Squaring its complex modulus and summing over a complete orthonormal basis of final states gives $⟨i|{\hat{E}}^{(-)}{\hat{E}}^{(+)}|i⟩$. Averaging over initial preparations replaces this expression by a trace. Thus a detector that absorbs cannot count the Fock vacuum, although the vacuum has quantum fluctuations.

For one selected normal mode, ${\hat{E}}^{(+)}(t)=\mathcal{E}u(t)a$. All Fock states with nonzero occupation, coherent states, and thermal light in this same mode havewhere both denominators are nonzero. Its magnitude is one even when the expected value of the field vanishes. Perfect first-order optical coherence therefore does not identify a coherent state. Pair detection supplies additional information.

Quantum simulation uses a controllable quantum system to reproduce selected properties or dynamics of another quantum system. The motivation is that the Hilbert space of many interacting particles grows rapidly, making a full classical representation expensive.

In an analog simulation, an engineered Hamiltonian approximates the target interactions directly. In a digital simulation, a sequence of quantum gates approximates the target unitary time evolution. For example, interacting spins can model a XXZ spin chain, with measured correlations compared to the target theory. A useful result requires identifying what is reproduced, controlling preparation and measurement errors, and estimating the effect of imperfect interactions. Reproducing a few observables can be valuable even when the simulator does not reproduce every microscopic degree of freedom.

For $g$ copies of a positive-energy branch $E=\hslash v|\mathbf{k}|$ in area $A$, the density of states is $D(E)=gAE/(2\pi {\hslash }^2{v}^2)$. At zero chemical potential, the thermal energy is $3g\zeta (3)A(k_{B}T{)}^3/(4\pi {\hslash }^2{v}^2)$, giving the displayed fixed-zero-chemical-potential heat capacity. If a symmetric filled negative-energy band also supplies holes, its excitation energy contributes an equal amount. Counting cones, spin species and holes explicitly is necessary before assigning an absolute numerical coefficient.

The oscillator length fixes the spatial scale of a quantum harmonic oscillator of mass $m$ and angular frequency $\omega$. Its normalized ground wavefunction is ${\phi }_0(x)={\pi }^{-1/4}{\ell }^{-1/2}\exp [-x^2/(2{\ell }^2)]$, so $|{\phi }_0(0){|}^2=1/(\sqrt{\pi }\ell )$ and $⟨x^2⟩={\ell }^2/2$. The definition of $\ell$ is distinct from the root-mean-square displacement $\ell /\sqrt{2}$.

For a number operator eigenstate $|n⟩$, the annihilation operator and creation operator satisfy $⟨a⟩=⟨a^{†}⟩=⟨a^2⟩=⟨(a^{†}{)}^2⟩=0$, $⟨a^{†}a⟩=n$ and $⟨a{a}^{†}⟩=n+1$. Inserting $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ and $p=i\sqrt{m\hslash \omega /2}(a^{†}-a)$ gives the displayed second moments and zero means.

Thus the kinetic energy and potential energy each have mean $\hslash \omega (n+1/2)/2$, and the uncertainty product is $\Delta x\Delta p=\hslash (n+1/2)$. Among number states only the ground state saturates the Heisenberg uncertainty relation. A quantum superposition can have nonzero position and momentum means despite each constituent number state's zero means.

In the isotropic three-dimensional quantum harmonic oscillator, the first excited Cartesian energy eigenstates are proportional to $x,y,z$ times the common Gaussian $e^{-r^2/(2a_0^2)}$. If ${\varphi }_x,{\varphi }_y,{\varphi }_z$ are normalized with the same real phase convention, their combinations in the title are normalized angular momentum eigenstates with $l=1$ and $m=1,0,-1$.

Indeed, $z=r\cos \theta$ and $x\pm iy=r\sin \theta e^{\pm i\phi }$ identify a common radial factor times the spherical harmonics $Y_{10},Y_{1,\pm 1}$. The transformation is unitary, so it changes the basis rather than the threefold orbital degeneracy.

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 $3\hslash \omega /2$.

In the flat-surface, constant-gravitational acceleration model, a coasting projectile conserves horizontal velocity. If it arrives at the top of its trajectory with horizontal speed $v_o$, its earlier velocity at angle $\theta$ above the horizontal has vertical component $v_o\tan \theta$. Its remaining height gain is therefore $v_o^2{\tan }^2\theta /(2g)$. The displayed burnout height follows. This is a local projectile motion approximation, not an exact Kepler orbit. Computing $v_o=\sqrt{g(R_E+h_o)}$ assumes constant gravitational magnitude also at the orbit; computing $v_o=\sqrt{g_0{R}_E^2/(R_E+h_o)}$ instead assumes an inverse-square law with surface value $g_0$. Those assumptions must be distinguished.

An amplitude spectral density is the square root of a power spectral density. For an electric potential difference expressed in volts and frequency in hertz, its units are $\mathrm{V}/\sqrt{\mathrm{Hz}}$.

For an incompressible irrotational two-dimensional potential flow, an analytic complex potential of a two-dimensional potential flow combines the velocity potential $\Phi$ and streamfunction $\Psi$. The Cauchy-Riemann equations give $u={\Phi }_x={\Psi }_y$ and $v={\Phi }_y=-{\Psi }_x$, hence $W^{\prime }(z)=u-iv$. Curves of constant $\Psi$ are streamlines. An analytic coordinate change $z=f(\zeta )$ transforms the complex velocity by $dW/dz=(dW/d\zeta )/f^{\prime }(\zeta )$ wherever $f^{\prime }\ne 0$.

Potassium-40 is a long-lived radioactive isotope with 19 protons and 21 neutrons. It can decay to calcium-40 through beta-minus decay, or to argon-40 through electron capture and a rare beta-plus decay. The argon-producing branch supports potassium-argon dating.

A position vector is the vector from a chosen origin to a point. Its time derivative is the velocity; differences of position vectors give displacement.

A polysaccharide that forms a porous gel on cooling from solution. In agarose gel electrophoresis, the agarose network provides a size-dependent obstacle to DNA migration in an electric field.

Agar is a polysaccharide mixture used to solidify many cell culture medium formulations.

Repeat the random evaluation test for a polynomial identity with fresh independent random variables, keeping the same fixed polynomial. Reject as soon as any polynomial evaluation is nonzero; accept only if all $k$ values vanish. An identity is always accepted, preserving one-sided error. If a nonzero polynomial has single-trial false acceptance probability $p_1$, independence gives $p_k=p_1^k$. The Schwartz-Zippel lemma bounds this by $(d/|S|{)}^k$ when $d<|S|$.

For $G(x)=(x-1)(x-2)$ over the finite field ${\mathbb{F}}_5$, two uniform independent choices $(a,b)$ have $25$ equally likely ordered outcomes, each with probability $1/25$. Both polynomial evaluations are zero only at $(1,1),(1,2),(2,1),(2,2)$. Hence false acceptance falls from $2/5$ in one trial to $4/25$ in two trials, and correct rejection rises from $3/5$ to $21/25$.

If the second choice reuses the first, the pair is $(a,a)$ and only five diagonal outcomes are possible. Two still falsely accept, so the error remains $2/5$. Choosing fresh independent points is precisely what makes the full $25$-cell grid and the product of probabilities valid. For a target error $0<\epsilon <1$, this example needsThe ceiling function makes the trial count an integer. Four trials give $(2/5{)}^4=16/625>1/100$, while six give $64/15625<1/100$; five still give $32/3125>1/100$, so six is the first count below one percent. This is an exact error guarantee for each run on $G$, rather than an observed success rate.

Let $P$ be a nonzero multivariate polynomial over a field $K$, with total degree at most $d\ge 0$. Choose a nonempty finite set $S\subseteq K$. For independent random variables $a_1,\dots ,a_n$, each with a uniform distribution on a finite set on $S$, the Schwartz-Zippel lemma givesNonzeroness is essential; the zero polynomial always evaluates to zero. The total degree sums exponents within each monomial, rather than taking their maximum: $x^2{y}^3+y$ has total degree five. The field need not be finite, but $S$ must be finite. The estimate reduces error only when $d/|S|<1$.^[ref]

The proof begins with the root bound for a polynomial. For several variables, write $P={\sum }_{j=0}^m{Q}_j(x_1,\dots ,x_{n-1})x_n^j$, with $Q_m\ne 0$. The total degree of $Q_m$ is at most $d-m$. By induction, $Q_m$ vanishes with probability at most $(d-m)/|S|$. Otherwise the remaining univariate polynomial has at most $m$ polynomial roots. Adding the two contributions gives $d/|S|$.^[ref]

The algorithm samples a point, performs exact polynomial evaluation, rejects the identity if the value is nonzero, and otherwise accepts. The lemma bounds false acceptance for every fixed nonzero input. For the polynomial identity testing over a finite field example, $G$ has $d=2$ and $|S|=5$, so its exact $2/5$ error attains the bound. Numerical near-zero values from floating-point arithmetic do not satisfy this exact algebraic contract. A polynomial-time algorithm claim additionally needs efficient sampling and evaluation, including the bit cost of representing and operating on field elements. Counting field operations alone does not establish that cost.

Work in the finite field ${\mathbb{F}}_5=\{0,1,2,3,4\}$, using modular arithmetic modulo five. Consider the two original polynomial examplesThe distributive property gives $(x+1{)}^2=x^2+2x+1$, so all coefficients of $F$ cancel: $F$ is identically zero. In contrast, $G=x^2-3x+2=x^2+2x+2$ in this finite field, with nonzero leading coefficient. Its roots of a polynomial are exactly one and two: a product in a field is zero only if a factor is zero.

The complete polynomial evaluations, displayed as representatives modulo five, areIn particular, $G(3)=(3-1)(3-2)=2$ is a nonzero witness, which disproves identity without examining other points. At one or two, a zero value fails to reveal that $G$ is nonzero. A random variable $a$ with a uniform distribution on a finite set ${\mathbb{F}}_5$ hits those two polynomial roots with probability $2/5$.

A crucial limit appears with $H(x)=x^5-x$. Its coefficients make it a nonzero polynomial of polynomial degree five in ${\mathbb{F}}_5[x]$, yet direct polynomial evaluation gives zero at all five field elements: the values before reduction are $0,0,30,240,1020$, all divisible by five. Thus $H$ induces the same polynomial function as the zero polynomial. Any number of trials restricted to ${\mathbb{F}}_5$ will falsely accept this identity. This is compatible with the Schwartz-Zippel lemma: the ratio of polynomial degree to sampling-set size is $5/5=1$, which gives no useful guarantee.

When a polynomial's representation permits polynomial evaluation over a larger field extension, a larger sampling set can restore the useful degree ratio. Access only to the induced polynomial function on the original finite field cannot distinguish $H$ from zero.^[ref]