Chapter 22 Further Reading: Scattering Theory

Primary Textbook References

Griffiths, D. J. --- Introduction to Quantum Mechanics (3rd ed., 2018)

  • Chapter 10 covers scattering theory at the intermediate level. Sections 10.1--10.3 (Born approximation, partial wave analysis, phase shifts) correspond directly to our Sections 22.3--22.6.
  • Griffiths's treatment of the Born approximation is particularly clear and includes worked examples for the soft-sphere potential and the Coulomb potential.
  • The discussion of hard-sphere scattering (Section 10.3.2) is excellent for building intuition about phase shifts.
  • If you found our treatment of the optical theorem (Section 22.8) too terse, Griffiths devotes several pages to explaining its physical significance.

Sakurai, J. J. & Napolitano, J. --- Modern Quantum Mechanics (3rd ed., 2021)

  • Chapter 6 provides the most comprehensive intermediate-level treatment of scattering theory. Sakurai works entirely in Dirac notation and develops the Lippmann-Schwinger equation and T-matrix formalism rigorously.
  • Section 6.2 on the Born approximation includes the Yukawa-to-Coulomb limit and the screened Coulomb potential, presented more formally than in our text.
  • Section 6.4 on the optical theorem gives a derivation from unitarity that complements our partial-wave proof.
  • The treatment of the S-matrix (Section 6.6) goes significantly beyond our introductory discussion in Section 22.9 and is recommended for students proceeding to quantum field theory.

Shankar, R. --- Principles of Quantum Mechanics (2nd ed., 1994)

  • Chapter 19 treats scattering theory in detail, with an emphasis on the Green's function approach and the formal theory.
  • Shankar's discussion of the Born series and its convergence is more rigorous than most intermediate texts.
  • The section on Coulomb scattering includes the exact parabolic-coordinate solution and a careful analysis of why the classical and quantum results agree.
  • Recommended for readers who want deeper mathematical foundations.

Cohen-Tannoudji, C., Diu, B., & Laloe, F. --- Quantum Mechanics (2019 reprint)

  • Chapter VIII and its complements provide an exhaustive treatment of scattering. The main text covers the basics; the complements treat special topics including the optical theorem, Levinson's theorem, and resonances.
  • Complement $\text{B}_{\text{VIII}}$ on the free spherical wave and partial wave decomposition is particularly thorough.
  • The French school's emphasis on mathematical rigor makes this the go-to reference for proofs of completeness relations, analyticity properties, and convergence of the partial wave series.

Advanced and Specialized References

Taylor, J. R. --- Scattering Theory: The Quantum Theory of Nonrelativistic Collisions (1972, Dover reprint 2006)

  • The definitive monograph on non-relativistic scattering theory. Covers everything from basic formalism through multichannel scattering, analytic properties of the S-matrix, and dispersion relations.
  • Chapter 3 (Green's functions and the Lippmann-Schwinger equation) and Chapter 12 (partial wave analysis) go well beyond any textbook treatment.
  • Chapter 13 on resonances and the Breit-Wigner formula is the most careful treatment at the graduate level.
  • Strongly recommended for anyone doing research involving scattering calculations.

Newton, R. G. --- Scattering Theory of Waves and Particles (2nd ed., 1982, Dover reprint 2013)

  • An encyclopedic treatment of scattering theory covering both quantum mechanics and classical wave scattering.
  • The chapters on analytic properties of the S-matrix, the inverse scattering problem, and Regge poles are unique to this book.
  • More mathematical than Taylor; recommended for theoretically inclined readers.

Goldberger, M. L. & Watson, K. M. --- Collision Theory (1964, Dover reprint 2004)

  • A classic reference for formal scattering theory, written by two of the architects of the S-matrix program.
  • The treatment of dispersion relations, crossing symmetry, and the analytic structure of scattering amplitudes goes far beyond what is covered in standard quantum mechanics texts.
  • Historical interest as well: this book represents the state of the art of scattering theory in the pre-QCD era.

Nuclear and Particle Physics Applications

Krane, K. S. --- Introductory Nuclear Physics (1988)

  • Chapters 4 and 11 apply scattering theory to nuclear physics. The treatment of neutron resonances, compound nucleus formation, and optical model calculations provides concrete applications of the formalism developed in this chapter.
  • The experimental data on neutron-nucleus cross sections (showing spectacular Breit-Wigner resonances) are particularly instructive.

Perkins, D. H. --- Introduction to High Energy Physics (4th ed., 2000)

  • Chapter 2 covers the kinematics and cross sections of scattering experiments in the relativistic regime.
  • The discussion of how deep inelastic scattering revealed the quark structure of the proton is a direct application of the Born approximation idea: the scattering cross section as a Fourier transform of the target structure.

Rutherford, E. --- "The Scattering of $\alpha$ and $\beta$ Particles by Matter and the Structure of the Atom," Philosophical Magazine 21, 669 (1911)

  • The original paper. Remarkably readable after more than a century. Rutherford derives the classical cross section, compares with Geiger and Marsden's data, and concludes that the atom has a compact, massive nucleus.
  • Worth reading as an example of how a single scattering experiment can overturn a paradigm (the Thomson "plum pudding" model).

Mathematical Background

Abramowitz, M. & Stegun, I. A. --- Handbook of Mathematical Functions (1965, Dover)

  • Chapter 10 (Bessel functions) and specifically Sections 10.1--10.3 on spherical Bessel functions provide all formulas and asymptotic expansions used in this chapter.
  • Free online at the DLMF (Digital Library of Mathematical Functions): https://dlmf.nist.gov/

Arfken, G. B., Weber, H. J., & Harris, F. E. --- Mathematical Methods for Physicists (7th ed., 2013)

  • Chapter 15 (Legendre polynomials), Chapter 16 (spherical harmonics), and Chapter 17 (Bessel functions) cover the special functions essential for scattering theory.
  • The addition theorem for spherical harmonics and the Rayleigh expansion are derived carefully.

Computational Resources

Joachain, C. J. --- Quantum Collision Theory (3rd ed., 1983)

  • Includes detailed algorithms for numerical computation of phase shifts, Born amplitudes, and partial wave cross sections.
  • The discussion of numerical methods for solving the radial Schrodinger equation and extracting phase shifts at the matching radius is directly applicable to the code developed in this chapter's toolkit.

Thompson, I. J. & Nunes, F. M. --- Nuclear Reactions for Astrophysics (2009)

  • A modern treatment of scattering calculations with emphasis on computational techniques. Includes discussion of coupled-channels methods, R-matrix theory, and modern approaches to nuclear scattering.
  • Useful for students interested in applying scattering theory to real nuclear physics problems.

Historical and Conceptual Articles

Born, M. --- "Zur Quantenmechanik der Stossvorgange," Zeitschrift fur Physik 37, 863 (1926)

  • The paper that introduced the Born approximation. Also the paper in which Born proposed the probabilistic interpretation of the wavefunction (the Born rule) as a footnote. One of the most consequential papers in the history of physics.

Breit, G. & Wigner, E. --- "Capture of Slow Neutrons," Physical Review 49, 519 (1936)

  • The original derivation of the Breit-Wigner resonance formula, in the context of slow neutron capture by nuclei.

Heisenberg, W. --- "Die 'beobachtbaren Grossen' in der Theorie der Elementarteilchen," Zeitschrift fur Physik 120, 513 (1943)

  • Heisenberg's foundational paper on the S-matrix program, arguing that the S-matrix (not the Hamiltonian) should be the fundamental object in particle physics. A landmark paper that influenced decades of theoretical development.

Chew, G. F. --- The Analytic S Matrix: A Basis for Nuclear Democracy (1966)

  • A short, readable introduction to the S-matrix bootstrap program, which attempted to derive particle physics from analyticity, unitarity, and crossing symmetry alone, without a Lagrangian. Although ultimately superseded by QCD, the ideas influenced the development of string theory.