Quantum scattering

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

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.

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 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.

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 $S_l(k)=e^{2i{\delta }_l(k)}$.

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 $S_l=e^{2i{\delta }_l}$.

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 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 $4\pi a_s^2$.

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 $\hslash$.

At an interface where a finite potential changes discontinuously, continuity of a radial wavefunction and its derivative is equivalent to matching $u^{\prime }/u$ on the two sides. This removes the arbitrary normalization constants and gives an equation for the scattering phase shift.

The hard-sphere limit of a repulsive spherical potential makes penetration into a sphere of radius $a$ vanish. Its low-energy S-wave scattering length tends to $a$, and its quantum total cross-section tends to $4\pi a^2$.

The Lippmann-Schwinger equation rewrites the Schrodinger differential equation as an integral equation using a free Green function and a prescribed incoming state.

A scattering amplitude is the coefficient of an outgoing asymptotic wave relative to the specified incoming wave.

Near a one-particle pole, a scattering amplitude factors into amplitudes coupling each side to the intermediate state, summed over its physical polarizations. The pole position gives its squared mass, while its residue encodes its spin and couplings.

Crossing symmetry relates scattering amplitudes by analytic continuation exchanging incoming and outgoing particles. A symmetric four-point scalar amplitude has equivalent descriptions under permutations of the Mandelstam variables.

With relativistically normalized external states, a two-particle initial state has differential cross-section $d\sigma =|\mathcal{M}{|}^{2}d{\Phi }_n/\mathcal{F}$, where $d{\Phi }_n$ is the Lorentz-invariant phase-space measure and $\mathcal{F}$ is the invariant flux factor. For identical unobserved final particles, divide the full labeled phase-space integral by their permutation multiplicity.

At leading electromagnetic order and with negligible electron mass, the photon-channel relativistic cross-section is ${\sigma }_h=4{\pi }^2\alpha {\rho }_h=16{\pi }^3{\alpha }^2{\rho }_J/s$, in the stated hadronic photon spectral density and hadronic electromagnetic-current spectral density conventions. Initial electron and positron spins are averaged; final hadronic states are summed inclusively. Electroweak channels beyond photon exchange are separate contributions.

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is $R=N_c{\sum }_f{Q}_f^2$. It depends on flavor charges as well as the number of active flavors; it is not simply $N_c{N}_f$.

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude $|\lambda |$. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is ${\lambda }^2/(128{\pi }^{2}s)$. On a hemisphere representing each event once, the density is ${\lambda }^2/(64{\pi }^{2}s)$. Both conventions give total event cross-section ${\lambda }^2/(32\pi s)$.

For two incident particles in relativistic normalization, this is $2\sqrt{\kappa (s,m_1^2,m_2^2)}$, where $\kappa$ is the Källén function. For equal masses it is $2\sqrt{s(s-4m^2)}$.

The on-shell measure $d^{3}p/(2E)$ is Lorentz invariant. The product measure together with the four-dimensional Dirac delta distribution enforces four-momentum conservation. For equal masses in two-body scattering, $d{\Phi }_2=\sqrt{1-4m^2/s}d\Omega /(32{\pi }^2)$.

For two outgoing particles with centre-of-mass momentum magnitude $k$ and total energy $\sqrt{s}$, integrating the four-momentum delta function gives $d{\Phi }_2/d\Omega =k/(16{\pi }^2\sqrt{s})$. A factor $1/2!$ for identical outgoing particles is inserted when counting events over the full phase space, not into this labelled kinematic measure itself.

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

When the full Lorentz-invariant phase-space measure treats $n$ identical final particles as labeled, each physical final configuration is counted $n!$ times. The unlabelled relativistic scattering cross-section therefore includes $1/n!$. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.

The momentum transfer in a scattering process is the difference between the incoming and outgoing wavevectors, commonly $Q=k-k^{\prime }$ up to a sign convention and a factor of $\hslash$.

Elastic scattering preserves the total kinetic energy. For one particle scattered by a fixed target, the incoming and outgoing wavevectors therefore have equal magnitudes.

Diffraction is interference produced when waves scatter from spatial structure. A periodic structure concentrates the scattered intensity at reciprocal-lattice conditions.

In the Born approximation, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

For elastic scattering with $k-k^{\prime }=q\in {\Lambda }^{\ast }$ and $|k|=|k^{\prime }|=k$,The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

Bragg scattering is coherent elastic scattering from a periodic lattice, enhanced when the wavevector transfer is a reciprocal lattice vector. Finite crystals have narrow rather than infinitely sharp reciprocal-space peaks.

After analytic continuation in momentum, a pole at $k=i\kappa$ with $\kappa >0$ gives negative energy and spatial decay, and therefore represents a bound state.