Radiocarbon dating uses the loss of carbon-14 relative to stable carbon after carbon exchange stops. It can measure either an isotope ratio by accelerator mass spectrometry or radioactive activity per fixed carbon content. Interpreting an age requires radiocarbon calibration and control of contamination and reservoir differences.

With no initial radiogenic argon-40 and complete daughter retention, the surviving potassium-40 population is $N_K=N_{K,0}{e}^{-\lambda t}$ and the accumulated daughter population is $N_{\mathrm{Ar}}=b(N_{K,0}-N_K)$. Here $\lambda$ is the total nuclear decay constant and $b$ is the argon-producing branching fraction. Their ratio gives the displayed age. The result measures closure of the mineral's isotope system under those assumptions.

The probability that an event of a specified kind produces a registered count in a specified arrangement. For a radiation detector, geometry, intervening material and the detector response can all affect detection efficiency.

The becquerel, symbol $\mathrm{Bq}$, is the SI unit of radioactive activity, with $1\mathrm{Bq}=1{\mathrm{s}}^{-1}$. An radioactive activity of one becquerel means one expected nuclear transformation per second under the stated conditions, not a guarantee of one event in every individual second.

Measured count rate need not equal radioactive activity. A particle detector registers only some decays because of geometry and detection efficiency, and one transformation can emit several particles or none detectable by that instrument. Activity also differs from deposited energy or absorbed dose: those depend on the emitted particles, their energies and where their interactions occur.

In his 1964 paper, Murray Gell-Mann proposed fractional-charge quarks, constructing baryons and mesons from them. He also distinguished a mathematical constituent model from the hypothesis of finite-mass physical particles. The successful bookkeeping did not itself settle that distinction.^[ref]

George Zweig's 1964 CERN model used three constituents called aces, with fractional electric charge and baryon number $1/3$. Its abstract also proposed an experimental search.^[ref] The theoretical construction and the experiment that might discriminate between interpretations are separate parts of the argument. The Omega baryon discovery experiment illustrates the power of testing an incomplete particle pattern against new observations.

For a three-light-flavor example, take up quarks, down quarks and strange quarks with the following additive quantum numbers. Here $Q$ denotes electric charge in units of the positive elementary charge $e$, $B$ is baryon number, $S$ is strangeness, $I_3$ is the third component of isospin, and $Y=B+S$ is flavor hypercharge. An antiquark reverses all five displayed values. These are flavor coordinates, not spatial coordinates or electroweak hypercharge.

Adding these columns immediately explains familiar hadrons. A proton with valence quarks $uud$ has $Q=2/3+2/3-1/3=1$ and $B=1$; a neutron with $udd$ has $Q=0$ and $B=1$. A positive kaon with $u\bar{s}$ has $Q=2/3+1/3=1$, $B=0$ and $S=+1$. Thus the same constituent rules describe both baryons and mesons.

Two hadrons can have identical additive labels while differing in their internal state. The Lambda baryon and neutral Sigma baryon both have valence content $uds$, yet their total isospin differs. Likewise, the neutral pion flavor direction $(u\bar{u}-d\bar{d})/\sqrt{2}$ and the Eta octet state $(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ both have $Q=Y=0$, but their inner product vanishes. Expanding that inner product gives $(1-1+0)/\sqrt{12}=0$. Constituent counting must therefore be supplemented by quantum superposition and the symmetry of the state.

The constituent-quark model describes a low-energy hadron using effective massive quarks and phenomenological interactions. Its constituent quark masses include strong-interaction dressing and differ from short-distance running quark masses.

For three light flavours and two spin states, an approximate $SU(6)$ spin-flavour symmetry acts on the six-dimensional single-quark spin-flavour space. A spatial ground-state three-quark colour singlet requires the symmetric representation of dimension $56$. Under flavour symmetry and ordinary spin rotations, it decomposes as $(10,4)\oplus (8,2)$: the baryon decuplet has spin $3/2$ and the baryon octet spin $1/2$. The product of spin and flavour factors can require a correlated sum, rather than a single unentangled product.

Consider an original one-dimensional effective mass model with $V(x)=0$ for $|x|<a$ and $V(x)=V_0>0$ outside. Take the same positive effective mass $m^{\ast }$ everywhere, an abrupt interface, and no interactions. The wavefunction represents a transverse envelope; free motion parallel to the layer is outside this calculation. A double heterostructure laser need not be thin enough for this quantization to dominate its operation.

Set $s=x/a$, $ϵ={\hslash }^2/(2m^{\ast }{a}^2)$, $z_0^2=V_0/ϵ$, $z^2=E/ϵ$ and $w=\sqrt{z_0^2-z^2}$. The time-independent Schrödinger equation becomesHere $u(s)$ denotes the dimensionless potential; a bound state has $0<z<z_0$. Matching the wavefunction and its derivative at $s=1$ givesFor the even bound state, the interior solution is $A\cos (zs)$ and the exterior solution is $A\cos z{e}^{-w(|s|-1)}$. For the odd bound state, replace $\cos (zs)$ by $\sin (zs)$ and $\cos z$ by $\mathrm{sgn}(s)\sin z$. The physical wavefunction is $\psi (x)=\varphi (x/a)/\sqrt{a}$.

Wavefunction normalization includes both exterior tails, not just the active layer. Direct integration givesThe first two terms are the interior integral; the last is the total exterior integral. Thus the interior probability is $P_{\mathrm{in}}=A^2[1\pm \sin (2z)/(2z)]$, with the upper sign for even parity.

Choose $z_0=2$. There is one even root in $(0,\pi /2)$ and one odd root in $(\pi /2,2)$; higher branches start beyond $z_0$. Each left-hand side increases on its stated interval while $\sqrt{4-z^2}$ decreases, proving uniqueness there. Numerically,These are calculated values of the chosen finite square well. The excited bound state lies closer to the barrier and has a longer exterior tail: being bound does not mean having unit probability inside the well.

This model demonstrates electronic confinement, not optical refractive index confinement or a complete laser. Unequal effective masses, band bending, interface disorder and interactions require a more detailed model. In particular, continuity of the ordinary derivative above uses the chosen constant effective mass assumption.

A particle resonance is an unstable excitation inferred from an enhanced scattering or decay response near a characteristic energy. It is associated with a pole of the analytically continued scattering amplitude, rather than a permanently normalizable state that survives indefinitely.

For an isolated narrow resonance, a Breit-Wigner form relates the peak scale to a mass and a width, with lifetime approximately $\hslash /\Gamma$. Different final states probe different partial widths and branching fractions. Interference with nonresonant amplitudes, nearby thresholds and detector resolution can shift or distort the observed peak, so a bump's width need not equal the intrinsic decay width without a model. Conservation laws and angular distributions help identify which quantum numbers are compatible with the signal.

In inelastic scattering, the participants exchange energy with internal degrees of freedom, so their total translational kinetic energy need not be conserved.

X-ray scattering techniques use the change in a scattered photon's direction and, in inelastic measurements, its energy to probe matter. Inelastic X-ray scattering transfers energy to excitations such as phonons; elastic methods probe spatial structure without that energy transfer.

The angle between specified incoming and outgoing directions in a collision; the reference frame and outgoing particle must be stated.

A specified asymptotic configuration of the particles and their internal quantum states in quantum scattering. An incoming channel can scatter into several outgoing channels; S-matrix elements give their probability amplitudes with a stated probability current normalization. A channel can be energetically inaccessible even when other channels are open.

The change in incident and outgoing electron wavefunctions caused by the target Coulomb potential. It can invalidate a plane wave treatment in the Born approximation, especially for a large target atomic number.

For constant mass $m$, a finite jump from potential energy zero at $x<0$ to $V_0>0$ at $x>0$ requires continuity of the wavefunction and its first derivative. With incidence only from the left and $E>V_0$, put $k=\sqrt{2mE}/\hslash$ and $q=\sqrt{2m(E-V_0)}/\hslash$. Matching $A+B=C$ and $k(A-B)=qC$ gives reflected probability amplitude $B/A=(k-q)/(k+q)$ and transmission amplitude $C/A=2k/(k+q)$. The transmission probability is $(q/k)|C/A{|}^2$, conserving probability current.

For $0<E<V_0$, replace $q$ by $i\kappa$, where $\kappa =\sqrt{2m(V_0-E)}/\hslash$ is the tunnelling decay constant. The right-hand wavefunction is $C{e}^{-\kappa x}$, with zero probability current. Its probability density decays on length $1/(2\kappa )$, while its amplitude decays on length $1/\kappa$. The semi-infinite raised region gives total reflection, unlike a finite potential barrier allowing quantum tunnelling into a second allowed region.

A scattering cross-section divides the scattered rate per target by the incident particle flux. Its differential form resolves that rate by outgoing direction. It has units of area for three-dimensional scattering and length for two-dimensional scattering. In two dimensions, the convention $\psi \sim e^{ikr\cos \theta }+e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$ gives $d\sigma /d\theta =|f(\theta ){|}^2/k$ by comparing outgoing and incident probability currents.

For a quantum rotor with angle period $2\pi$, let $u_m=e^{im\varphi }/\sqrt{2\pi }$ and prepare $\psi (\varphi ,0)=\sqrt{4/(3\pi )}{\cos }^2\varphi$. Its only nonzero angular-momentum amplitudes are $c_0=\sqrt{2/3}$ and $c_{\pm 2}=1/\sqrt{6}$. Their energies are $0$ and $2{\hslash }^2/I$, respectively, soThe state revives after $T=\pi I/\hslash$ and its density at $T/2$ is the initial density rotated by $\pi /2$. At $T/4$ there are four equal angular maxima. This is interference between a stationary component and a degenerate pair of counter-rotating components; it is not a uniformly rotating rigid profile. Angular-momentum and energy probabilities stay constant.

In 1977, H. Jeff Kimble, Mario Dagenais, and Leonard Mandel excited a dilute sodium atomic beam with a resonant dye laser. Apertures restricted the observed atoms; a microscope objective collected fluorescence perpendicular to both beams. A beam splitter sent the collected light to two photomultipliers. Shaped pulses drove the start and stop inputs of a time-to-digital converter; its delay histogram rose away from zero delay, indicating photon antibunching. Turning the atomic beam off measured scattered-laser backgrounds. More than one contributing atom and scattered light limited the comparison with an ideal isolated two-level system; the paper explicitly treated corrections.^[ref]

A simple reason for suppressed immediate re-emission is visible in an ideal two-level system. After spontaneous emission, the atom is in $|g⟩$. Its emission operator ${\sigma }_{-}=|g⟩⟨e|$ obeys ${\sigma }_{-}^2=0$, so two instantaneous emissions are impossible. Excitation must rebuild the excited-state population before another emission. This argument explains the zero-delay idealization; the actual photon counting record additionally depends on collection, background, timing, and the number of contributing atoms.

Robert Hanbury Brown and Richard Q. Twiss sought angular sizes through correlations of detected light intensities. Their method multiplies detector outputs after detection, whereas a Michelson interferometer combines fields before detection.^[ref]

A monochromatic discharge source illuminated a half-silvered beam splitter and two photodetectors. The averaged current product showed an excess near coincidence. Moving one detector introduced a delay that removed it, supplying a control. Slow response weakened the peak. This photon bunching is compatible with fluctuating classical light intensity.^[ref]