14: Scattering Theory
- Page ID
- 15819
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Historically, data regarding quantum phenomena has been obtained from two main sources. Firstly, from the study of spectroscopic lines, and, secondly, from scattering experiments. We have already developed theories that account for some aspects of the spectra of hydrogen, and hydrogen-like, atoms. Let us now examine the quantum theory of scattering.
- 14.1: Fundamentals of Scattering Theory
- This page covers time-independent, energy-conserving scattering by defining a Hamiltonian with a localized potential. It presents the incident wavefunction, Schrödinger’s equation, and the Helmholtz equation, leading to solutions for both incident and scattered waves. The differential scattering cross-section is defined in relation to particle flux, highlighting key relationships such as energy conservation in scattering processes.
- 14.2: Born Approximation
- This page explores the Born approximation in quantum scattering theory, simplifying the scattering function \(f({\bf k},{\bf k}')\) for weak scattering by using the incident wavefunction. It derives expressions connecting \(f({\bf k},{\bf k}')\) to the Fourier transform of the Yukawa potential \(V({\bf r})\) and formulates the scattering cross-section while outlining the validity conditions for the approximation.
- 14.3: Partial Waves
- This page covers the scattering of wavefunctions in a spherically symmetric potential, detailing how the incident wavefunction leads to a scattered wavefunction that depends on the polar angle \(\theta\). It discusses the scattering amplitude \(f(\theta)\), linking it to phase shifts \(\delta_l\) and spherical harmonics.
- 14.4: Optical Theorem
- This page discusses the optical theorem, which links the total scattering cross-section to the imaginary part of the scattering amplitude at zero momentum transfer. It emphasizes the importance of forward scattering for interference with the incident wave and its impact on the probability current in that direction.
- 14.5: Determination of Phase-Shifts
- This page covers the evaluation of phase shifts, \(\delta_l\), for a spherically symmetric potential that is effective only within a specific range. It details solutions to the Schrödinger equation both inside and outside the potential, ensuring continuity of wavefunctions and their derivatives. The page derives logarithmic derivatives, \(\beta_{l+}\) and \(\beta_{l-}\), demonstrating their equivalence and presenting a method for computing \(\delta_l\) via equation inversion.
- 14.6: Hard-Sphere Scattering
- This page examines wave scattering by a hard sphere, detailing phase shifts and the S-wave radial wavefunction. It notes that low-energy S-wave scattering results in a total cross-section four times the geometric cross-section, diverging from classical expectations. At high energies, multiple partial wave contributions lead to an effective total cross-section of \(2\pi a^2\), highlighting the intricate nature of wave interactions around the sphere.
- 14.7: Low-Energy Scattering
- This page discusses low-energy \(S\)-wave scattering in the context of a finite potential well, emphasizing that the total scattering cross-section can be derived from the phase shift. When the potential depth is much greater than the incident particle energy, the cross-section remains energy-independent. The Ramsauer-Townsend effect is highlighted, where specific potential values result in minimal scattering, a phenomenon that has been experimentally observed.
- 14.8: Resonances
- This page discusses resonance behavior in scattering related to the \(S\)-wave or \(l\)th partial wave achieving a phase shift of \(\pi/2\). This leads to increased cross-section dependence on energy, compared to non-resonant scattering, and relates to a bound state in a potential well, creating a metastable state.
Thumbnail: Collimated homogeneous beam of monoenergetic particles, long wavepacket which is approximately a planewave, but strictly does not extend to infinity in all directions, is incident on a target and subsequently scattered into the detector subtending a solid angle. The detector is assumed to be far away from the scattering center. (Department of Physics Wiki @ Florida State University).