Quantum scattering

Quantum scattering compares a wavefunction's incoming and outgoing asymptotic components.

For a stationary one-dimensional scattering state with incident probability current $j_{\mathrm{in}}>0$, the dwell time in a region $\Omega$ is its integrated probability density divided by incident flux. It measures occupancy per incoming rate, not a classical trajectory or a uniquely defined tunnelling traversal time.

For $0<E<V_0$, set $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_0-E)}/\hslash$. Continuity at the potential step gives the evanescent amplitude $c=2k/(k+i\kappa )$ for unit incident amplitude. Integrating $|c{|}^2{e}^{-2\kappa x}$ over $x>0$ gives $|c{|}^2/(2\kappa )$. Dividing by the incident probability current $\hslash k/m$ proves the displayed dwell time.

For a real one-dimensional potential, a real-axis transmission probability is a modulus square. Its algebraic complex continuation need not have only physical bound-state poles. In a square well, $t=e^{-ika}/D(k)$ while the continued probability is $1/[D(k)D(-k)]$. Zeros of the second factor can encode a different boundary condition. In units ${\hslash }^2/(2m)=1$, a well of width two and depth $1+{\cot }^{2}1$ has a continued-probability pole at $k=i\cot 1$ which satisfies neither decaying even nor decaying odd matching. The physical bound-state criterion uses the outgoing amplitude on its decaying sheet.

The decay probability or scattering cross section per solid angle, relative to stated polar and azimuthal axes. Spin correlations and angular-momentum selection can make it anisotropic; changing the polar axis reverses terms odd in its direction cosine.

The normalized difference of forward and backward cross sections relative to a specified incoming-particle axis. For $d\sigma /dc=K[(A+B)(1+c^2)+2(A-B)c]$ with $c=\cos \theta$, integration gives ${\sigma }_F-{\sigma }_B=2K(A-B)$ and ${\sigma }_F+{\sigma }_B=8K(A+B)/3$, hence $A_{\mathrm{FB}}=3(A-B)/(4(A+B))$. The pole forward-backward asymmetry for charged leptons is its specialized Z boson result.

For a constant-mass one-dimensional particle propagating through three regions of potentials $0,V_1,V_0$, let $k_j$ be the real positive wavevectors. A layer with $k_{1}a=\pi /2$ makes the two interface reflection amplitudes cancel exactly when $k_1^2=k_0{k}_2$. Equivalently $E-V_1=\sqrt{E(E-V_0)}$. For an increasing intermediate step this requires $E>V_0$ and $V_0/2<V_1<V_0$. The result follows from $r=(r_{01}+r_{12}{e}^{2i{k}_{1}a})/(1+r_{01}{r}_{12}{e}^{2i{k}_{1}a})$ and $r_{ij}=(k_i-k_j)/(k_i+k_j)$.

For a real point coupling $g$ between a three-dimensional continuum and one discrete state, $A=m{g}^2/(2\pi {\hslash }^2)$. After the short-distance level shift is absorbed, the isotropic amplitude has the displayed form. The imaginary part is fixed by the outgoing boundary condition, and $\mathrm{Im}(1/f_k)=-k$ supplies elastic unitarity.

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.

The plane spanned by incoming and outgoing propagation directions when those directions are not parallel or antiparallel. Polarization perpendicular or parallel to it is useful when expressing a differential scattering cross-section. The plane is not unique at exact forward or backward scattering.

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.

For a real positive decay coefficient alpha, a stationary exponential tail has zero probability current because $\mathrm{Im}({\psi }^{\ast }{\partial }_x\psi )=0$. Matching a subthreshold potential step can nevertheless give a nonzero coefficient T with $|T{|}^2>1$ relative to a unit incident plane-wave amplitude. This measures the density just beyond the step, not a flux-transmission probability. It describes penetration without a propagating outgoing channel in the semi-infinite high-potential region.

For $V_0<0$, $k=\sqrt{2mE}/\hslash$ tends to zero as $E\to 0^{+}$ while $q=\sqrt{2m(E+|V_0|)}/\hslash$ stays finite. Matching gives probability amplitudes $r=(k-q)/(k+q)$ and $t=2k/(k+q)$, so quantum reflection probability $R=r^2\to 1$ and transmission probability $T=(q/k)|t{|}^2\sim 4k/q\to 0$. The final region is classically allowed; reflection results from the abrupt spatial change in the wavefunction's wavenumber. These are limits of positive-energy incoming states, not a flux ratio evaluated at exactly zero incident current.

For a positive step potential and left incidence, take $E\to V_0^{+}$ at fixed incident probability amplitude $A$. The transmitted wavenumber tends to zero while the incident wavenumber tends to $k=\sqrt{2m{V}_0}/\hslash$. Wavefunction matching at a finite potential step gives reflected amplitude $A$ and transmitted amplitude $2A$, producing the spatial wavefunction in the title. The left probability density is $4|A{|}^2{\cos }^{2}kx$ and the right density $4|A{|}^2$. The transmitted probability current still tends to zero because its velocity tends to zero. Over adjacent intervals of length $2\pi /k$, the right integrated density is twice the left. These ratios use finite local integrals of a non-normalizable stationary scattering state.

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.

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

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.

Inelastic X-ray scattering measures energy and momentum transferred from X-rays to matter. With sufficiently high energy resolution, it can map a phonon dispersion relation by measuring inelastic peaks at different momentum transfers. Crystal momentum is determined modulo a reciprocal lattice vector.

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

For one-dimensional scattering from a real potential, write the left exterior wavefunction as $e^{ikx}+r{e}^{-ikx}$ with $k>0$. The reflected-to-incident probability-current ratio is $R=|r{|}^2$, since both components have the same speed and their currents have opposite signs. This probability differs from the complex reflection amplitude $r$. With transmission into a right exterior wavenumber $q$, the transmission probability is $T=(q/k)|t{|}^2$ and conservation of current gives $R+T=1$ when there is no absorption.

A quantum reflection amplitude is the complex ratio of the outgoing reflected wave to the incident wave in a chosen scattering convention. In one reflected channel with the same exterior speed, the reflection probability is $|r{|}^2$. Its phase affects interference even when its modulus is one; shifting the reference plane changes that phase.

A particle incident below a semi-infinite step has a unit-modulus quantum reflection amplitude but a nontrivial phase. Continuity of the oscillating and evanescent solutions yields the displayed coefficient, for the incident phase origin at the interface. Its decaying exterior wave has zero net probability current. The phase depends on the chosen scattering convention; the unit reflection probability does not.

For one-dimensional quantum scattering, $t$ is the complex amplitude of the outgoing transmitted wave relative to the incoming wave. The transmission probability is $|t{|}^2$ times the ratio of transmitted and incident flux velocities; with equal asymptotic dispersion it is $|t{|}^2$. A reflectionless potential has $r=0$ and a unit-modulus transmission amplitude whose phase shifts continuum quantization. The supersymmetric factorization of the one-soliton potential gives $t(k)=(k+im)/(k-im)$ for its single-bound-state potential.

The S-matrix maps incoming asymptotic states to outgoing asymptotic states. Its connected transition matrix elements contain an overall energy-momentum Dirac delta function and an invariant scattering amplitude. In quantum field theory, the LSZ reduction formula obtains these elements from amputated field correlators.

A contribution to the unitarity relation for the S-matrix obtained by placing a set of intermediate lines on shell and summing their intermediate states. In a physical-state formulation only physical states enter the sum. When covariant gauge boson index sums include auxiliary polarizations, matching Faddeev-Popov ghost field contributions are needed to reproduce the physical result in a Non-Abelian gauge theory.

A pole of a scattering matrix arising from the numerator $F(-k)$ rather than a zero of the incoming Jost function $F(k)$ can fail to correspond to a bound state. For $F(k)=(k-i\kappa )/(k+i\alpha )$, the pole at $i\kappa$ is associated with the incoming zero; the pole at $i\alpha$ comes from the numerator. The regular Eckart realization has $\alpha >\kappa$. The distinction and this explicit example are discussed in dipot.ulb.ac.be/dspace/bitstream/2013/373712/4/2306.12216.pdf.

For radial quantum scattering, the incoming coefficient of the regular solution defines the Jost function, with convention $S(k)=F(-k)/F(k)$. A zero at $k=i\kappa$, $\kappa >0$, removes the exponentially growing component and produces a bound state, subject to regularity and normalizability. Poles of the outgoing numerator alone need not produce bound states.

For real $u$, this scalar S-matrix block satisfies $S_u(\theta )S_u(-\theta )=1$ and Hermitian analyticity of a two-particle S-matrix. It therefore has modulus one for real rapidity differences. For $0<u<\pi$ its pole at $\theta =iu$ has residue $2i\sin u$ and admits a direct-channel bound-state pole interpretation. Its crossed block is $\cosh [(\theta -iu)/2]/\cosh [(\theta +iu)/2]$. The endpoints $u=0$ and $u=\pi$ are removable degeneracies giving constant amplitudes, so a particle cannot be inferred there from the uncancelled denominator alone. This block specifies an amplitude; a complete theory must also obey the remaining crossing symmetry and bootstrap fusion constraints.

For relativistic two-body scattering in one spatial dimension, this strip is the standard domain containing bound-state and crossed-channel poles. A stable-particle interpretation requires appropriate pole kinematics and residues; not every singularity automatically represents a new particle. Crossing relates rapidity $\vartheta$ to $i\pi -\vartheta$.

A physical-strip pole of a diagonal S-matrix can come from exchange of an existing particle in the crossed channel. For external masses $m_a,m_b$, use the momentum difference, rather than the sum, to identify the exchanged mass through the displayed invariant. Crossing symmetry turns the pole at $i\alpha$ into a direct pole at $i(\pi -\alpha )$ of the particle-antiparticle channel. For $m_B=2m\cos (u/2)$, the pole of $S_{BA}$ at $iu/2$ has $t=m^2$ and exchanges the existing charge-one particle; it is not a new charge-three bound state.

At relative rapidity $iu$, the analytically continued sum of the constituent four-momentum vectors is on the bound-state mass shell with the displayed mass. For equal constituents, rapidities $\chi \pm iu/2$ sum to $2m_a\cos (u/2)(\cosh \chi ,\sinh \chi )$. A pole at $u=0$ is at threshold rather than a strictly bound state.

In a real species basis, the two-particle S-matrix satisfies $S(-{\theta }^{\ast })=S(\theta {)}^{†}$ under the usual scattering analyticity assumptions. On the real rapidity axis this relates inverse-rapidity amplitudes to complex conjugates. Combined with the algebraic inverse relation $S(\theta )S(-\theta )=I$, it gives physical unitarity.

For a localized target and a plane wave incident along the $z$-axis, the large-radius scattering wavefunction has the formwhere $f$ is the scattering amplitude.

A scattering state is a state used to describe quantum scattering. Its asymptotic incoming and outgoing components determine a scattering amplitude.

A Green function for stationary scattering is chosen to contain waves travelling away from its source. In one dimension at positive energy it is proportional to $e^{ik|x-x^{\prime }|}$; the condition selects the $E+i0$ resolvent boundary value.

A partial wave is a component of a scattering state with definite orbital angular momentum. A central potential preserves each angular-momentum channel, so the scattering problem decomposes into independent partial waves.

A partial wave with orbital angular momentum $\ell =2$. Its orbital parity is positive. Coupling it to a spin-one pair can include total $J=1$, allowing a D wave component in the deuteron.

Partial-wave scattering decomposes a scattering state into partial waves with definite orbital angular momentum. For a central potential, each channel has its own scattering phase shift and Partial-wave S-matrix.

The partial-wave scattering channel with orbital angular momentum quantum number one. Its angular dependence is built from degree-one spherical harmonics. For a finite-range potential away from a threshold pole, the finite-range partial-wave threshold law gives phase shift ${\delta }_1=O(k^3)$ and a partial elastic scattering cross-section proportional to $k^4$ as the relative wavenumber $k$ tends to zero.

For a real central potential and the outgoing convention $e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. The cosine form of the Jacobi-Anger expansion fixes $w_0=1$ and $w_m=2$ for $m\ge 1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident plane wave gives the displayed scattering amplitude. Integrating $|f{|}^2/k$ gives the displayed scattering cross-section, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free plane wave at zero scattering phase shift.

The differential scattering cross-section is the effective incident area scattered per unit solid angle. It is defined by the ratio of the outgoing particle rate through $d\Omega$ to the incident flux.

For an azimuthally symmetric classical scattering map from impact parameter $b$ to scattering angle $\theta$,When several impact parameters produce the same angle, their nonnegative contributions are summed.

Ifthen the ratio of scattered radial probability current through $r^{2}d\Omega$ to the incident probability flux is

A reflected wave is an outgoing component travelling back toward the side from which the incident wave arrived.

The transmission probability is the ratio of transmitted to incident probability current. When the asymptotic wavenumbers agree, it is the squared modulus of the transmission amplitude.

For a reflection-invariant localized potential, order incoming amplitudes from the left and right and outgoing amplitudes toward the left and right. Thenwhere $r$ and $t$ are the reflection and transmission amplitudes. Probability-current conservation makes $S$ unitary for a real potential.

For a parity-symmetric one-dimensional potential, the even and odd parity states diagonalize the One-dimensional S-matrix:For a real potential, unitarity further gives $|S_{++}|=|S_{--}|=1$ at real positive energy.

If the left and right travelling-wave coefficients are related bythen

For a central potential, each angular-momentum channel has a scalar scattering coefficient $S_l(k)=e^{2i{\delta }_l(k)}$.

Each channel coefficient is a component of the scattering matrix in a basis adapted to rotational symmetry.

For elastic scattering by a real central potential, the phase shift ${\delta }_l(k)$ measures the phase difference between the asymptotic radial wave in angular-momentum channel $l$ and the corresponding free wave. Its channel Partial-wave S-matrix is $S_l=e^{2i{\delta }_l}$.

The phase shift parametrizes elastic partial-wave scattering, rather than general scattering phenomena.

For regular low-energy elastic scattering by a finite-range central potential, higher partial waves are suppressed by their centrifugal barriers. Matching the regular interior radial solution to small-argument exterior spherical Bessel functions gives the displayed scaling of scattering phase shift, in the generic absence of a threshold singularity. Since the channel amplitude is proportional to $\sin {\delta }_{\ell }/k$, it scales as $k^{2\ell }$. The $\ell =0$ term is isotropic and normally leads. A tuned vanishing leading amplitude or a threshold resonance requires examining the next terms rather than applying the generic statement blindly.

For a reflectionless scattering wavefunction whose two asymptotic phases differ by $\delta (k)$, a large periodic box gives $k_{n}L+\delta (k_n)=2\pi n$. Compared with the free value $k_n^{(0)}=2\pi n/L$, $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+O(L^{-2})$ away from thresholds. Discrete bound states and phase-branch changes alter mode indexing and must be counted when comparing complete spectra.

An S wave is the rotationally invariant angular-momentum channel $l=0$. At sufficiently low energy it normally dominates short-range scattering because higher partial waves are suppressed by their centrifugal barriers.

For $V(r)=V_0\delta (r-R)$, put $g=2m{V}_{0}R/{\hslash }^2$. The regular reduced radial wave function is proportional to $\sin kr$ inside the shell and to $\sin (kr+{\delta }_0)$ outside. It is continuous at $R$, while its radial derivative jumps by $(g/R)u(R)$. Dividing the jump condition by the common value gives $k\cot (kR+{\delta }_0)-k\cot (kR)=g/R$, which implies the displayed phase shift modulo $\pi$. At nodes, the undivided matching equations give the continuous limiting interpretation. For $g\ne 0$, an equivalent formula is $k\cot {\delta }_0=-k\cot (kR)-k(kR)/[g{\sin }^2(kR)]$. The exact S-wave scattering amplitude is $[k\cot {\delta }_0-ik{]}^{-1}$.

For a central potential and the convention $S_l=1+2i{f}_l=e^{2i{\delta }_l}$, the scattering amplitude isThe identity $|S_l|=1$ is equivalent to $\mathrm{Im}{f}_l=|f_l{|}^2$.

For elastic scattering by a central potential,At low momentum, the S wave normally supplies the leading term.

Partial-wave unitarity bounds each angular-momentum component of a scattering amplitude because its elastic S-matrix eigenvalue has modulus one. Perturbative violation of this bound signals that interactions become strong or that additional degrees of freedom must enter.

For a real central potential and real $k$, radial-current conservation gives $S_l(k{)}^{\ast }{S}_l(k)=1$. Invariance of the radial equation under $k\mapsto -k$ gives $S_l(k)S_l(-k)=1$, so a continuous phase convention has ${\delta }_l(-k)=-{\delta }_l(k)$.

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

This zero-range boundary condition replaces a short-range interaction by its S-wave scattering length. If $\psi =A/r+B+O(r)$, it requires $B=-A/a$. An outgoing spherical wave also has a constant $ik$ term; retaining that term gives the unitary scattering-length amplitude $f_0=-a/(1+ika)$. The model neglects effective-range corrections and is applied outside the microscopic range, not literally inside a finite potential.

Two separated short-range scatterers are coupled because the outgoing wave from either centre supplies part of the regular incoming field at the other. Applying the Bethe-Peierls boundary condition at both centres gives the displayed two-by-two linear system, where $s_j$ is the incident plane-wave phase. matrix inversion sums repeated scattering between the centres. Near a small determinant, effective-range corrections may matter even if the nominal microscopic range is short.

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.

The zero-energy parameter of the angular-momentum-zero partial wave. For a short-range interaction in three dimensions, its leading elastic scattering amplitude is $f_0(k)\simeq -a_s/(1+ik{a}_s)$ when effective-range corrections are negligible. For identical dilute bosons this parameter fixes the leading contact coupling $g_B=4\pi {\hslash }^2{a}_s/m_B$.

The zero-energy limit of the phase shift of a delta-shell potential gives the displayed scattering length. It vanishes at zero coupling and has a pole at $g=-1$. For repulsion $g>0$, $0<a_0<R$; for weak attraction $-1<g<0$, $a_0<0$; for $g<-1$, $a_0>R$. Both infinite-coupling limits tend to $R$. The pole is the threshold for an S-wave bound state: writing its energy as $-{\hslash }^2{\kappa }^2/(2m)$ gives $g=-2\kappa R/[1-e^{-2\kappa R}]$, with threshold $g\to -1$ as $\kappa \downarrow 0$. The delta-shell Born amplitude agrees with the exact low-energy amplitude only in their common weak-coupling and low-energy regime, $|g|\ll 1$ and $kR\ll 1$, where both tend to $-gR$.

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh (\gamma a)/\gamma$. The low-energy $s$-wave scattering amplitude is $-a_s$.

A zero-energy scattering resonance occurs when an S-wave bound state reaches the continuum threshold. The zero-energy radial solution then approaches a nonzero constant rather than a generic linear function, and the scattering length diverges and changes sign.

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

For the exterior wavethe partial-wave scattering matrix isThus $\tan {\delta }_0=-k/\lambda$, the scattering length is $1/\lambda$, and the pole at $k=i\lambda$ represents a bound state of energy $-{\hslash }^2{\lambda }^2/(2m)$ whose exterior wavefunction is proportional to $\mathrm{sech}(\lambda r)/r$.

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hslash$.

For an isolated particle resonance of spin $J_R$, the Breit-Wigner approximation relates a production-and-decay cross section to its partial decay widths. In natural units, $p_i$ is either incident particle's momentum magnitude in the centre-of-momentum frame, $g=(2J_R+1)/[(2J_a+1)(2J_b+1)]$ for unpolarized incident particles, and $\Gamma$ is the total width. The formula assumes the stated spin counting and neglects nonresonant backgrounds and their interference. Multiplication by $(\hslash c{)}^2$ converts an inverse-squared-energy cross section to area when momenta are expressed in energy units.

For a narrow vector meson formed and observed in $e^{+}{e}^{-}$, neglecting external masses and slowly varying kinematic factors gives ${\sigma }_{ee}=3\pi {\Gamma }_{ee}^2/[M^2((E-M{)}^2+{\Gamma }^2/4)]$. Integrating the Lorentz line shape gives the displayed result. A normalized instrumental convolution preserves this full area, although it changes the observed peak and width. A constant uncorrected acceptance multiplies the area and must be divided out before extracting the decay width.

At fixed entrance and exit partial decay widths, a resonance's peak cross section is proportional to their product divided by the square of its total width. Integrating a narrow Lorentzian over energy supplies one factor of the total width, leaving an inverse first power. Extracting a width from a peak ratio therefore differs from extracting it from an integrated ratio.

Inclusive formation sums over all resonance decays and has exit branching fraction one. Selecting a particular final channel instead multiplies by that channel's branching fraction. Thus resonant elastic scattering contains the square of the entrance branching fraction, while inclusive formation contains only one power.

Compare two entrance channels feeding the same isolated resonance and the same selected exit channel, each at its own resonance energy. The exit partial decay width and the resonant denominator cancel. Different entrance momenta, spin weights and entrance widths remain. The formula assumes the same resonance, the stated unpolarized spin average and negligible nonresonant interference; a ratio at one shared laboratory energy would generally be different.

For unpolarized entrance particles, average over their $(2J_a+1)(2J_b+1)$ spin orientations and sum over the isolated resonance's $2J_R+1$ magnetic substates. The resulting weight multiplies the partial-wave cross section. It can exceed one because orbital angular momentum participates; it is not a standalone probability of forming a state. Polarization or several unresolved resonance spins requires more detailed channel counting.

For unpolarized incident particles, the resonant cross section sums over the resonance's magnetic substates and averages over the incident spin states. This gives the displayed statistical weight under the conventional Breit-Wigner resonance cross section normalization. A spin-zero resonance formed from two spin-half particles has $g=1/4$, whereas a spin-one resonance has $g=3/4$. This counting is distinct from final-state color charge multiplicities.

At $E=E_R$, the Breit-Wigner resonance cross section loses its explicit dependence on the total decay width after writing ${\Gamma }_{ab}=B_{ab}\Gamma$ and ${\Gamma }_X=B_X\Gamma$. For an unpolarized ultrarelativistic electron-Positron initial state and a spin-one resonance, $g=3/4$ and $p_i=E_R/2$, giving ${\sigma }_{\mathrm{peak}}=12\pi B_e{B}_X/E_R^2$ in natural units. An elastic resonant channel therefore measures the square $B_e^2$, rather than one power of the electronic branching fraction.

A pole of the continued scattering matrix associated with outgoing boundary conditions and an unstable trapped state. A pole below the real energy axis gives time dependence $e^{-i{E}_{R}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, so the probability lifetime is $\hslash /\Gamma$. Bound-state poles instead lie on the positive imaginary wavenumber axis and produce square-integrable decaying spatial waves.

A hard wall at the origin and an attractive delta potential at distance $a$ form a leaky quantum cavity. For unit incident amplitude from outside, the interior standing wave has coefficient $A=-i\lambda x/[1-e^{2ix}+i\lambda x]$, where $x=ka$ and $\lambda ={\hslash }^2/(mqa)>0$. Its modulus squared is the title expression. A resonance pole $p_n=x_n-i{\gamma }_n$ produces an enhancement with a locally Lorentzian line shape, proportional to $[(x-x_n{)}^2+{\gamma }_n^2{]}^{-1}$. At fixed low $n$ and small $\lambda$, $x_n\simeq n\pi (1+\lambda /2+{\lambda }^2/4)$ and ${\gamma }_n\simeq (\lambda n\pi /2{)}^2$.