Further Reading — Chapter 5: Quantum Mechanics Review

The quantum mechanics reviewed in this chapter is covered in varying depth by the standard quantum mechanics textbooks. The references below are organized by topic, with emphasis on the treatments most relevant to nuclear physics applications.

Quantum Mechanics Textbooks (Tier 1)

Griffiths, D. J. & Schroeter, D. F. — Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018)

The standard undergraduate QM textbook. Chapter 4 covers angular momentum and the addition of angular momenta, including a clear derivation of the Clebsch-Gordan coefficients for simple cases. Chapter 7 develops time-dependent perturbation theory and Fermi's golden rule. Chapter 5 treats identical particles and the Pauli exclusion principle. The WKB approximation is covered in Chapter 8 with application to tunneling. Griffiths is the ideal review text if your QM is rusty — his explanations are exceptionally clear and physically motivated. If you studied from this book as an undergraduate, rereading the relevant sections before tackling Chapters 6--9 of the present text will be time well spent.

Sakurai, J. J. & Napolitano, J. — Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, 2021)

The standard graduate QM textbook. Sakurai's treatment of angular momentum (Chapter 3) is the definitive pedagogical account: angular momentum commutation relations, eigenvalues, orbital angular momentum, addition of angular momenta, Clebsch-Gordan coefficients, Wigner-Eckart theorem, 3j and 6j symbols. Chapter 5 develops time-dependent perturbation theory through Fermi's golden rule with more mathematical rigor than Griffiths. The treatment of identical particles and permutation symmetry (Chapter 7) is thorough. The reader who masters Sakurai's Chapter 3 will have no difficulty with the angular momentum algebra in any nuclear physics text. The third edition includes modern topics and improved problem sets.

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

An alternative to Sakurai at the advanced undergraduate/beginning graduate level. Shankar's strength is in providing physical intuition alongside the formalism. The chapters on angular momentum (12, 15) and perturbation theory (17) are well written. The WKB approximation (Chapter 16) receives a more extended treatment than in most texts, including a careful discussion of connection formulas and validity conditions. Shankar is particularly good for readers who want to understand why the formalism works, not just how to use it.

Angular Momentum and Clebsch-Gordan Coefficients

Edmonds, A. R. — Angular Momentum in Quantum Mechanics, (Princeton University Press, 1957; reprinted 1996)

The classic monograph on angular momentum coupling theory. Compact, rigorous, and complete. Covers 3j, 6j, and 9j symbols, the Wigner-Eckart theorem, and rotation matrices in detail. This is the reference that nuclear physicists have used for decades. Not a teaching text — it is a reference for those who need to use the formalism professionally. The symmetry properties and special values tabulated here are invaluable.

Varshalovich, D. A., Moskalev, A. N. & Khersonskii, V. K. — Quantum Theory of Angular Momentum (World Scientific, 1988)

The most comprehensive collection of angular momentum formulas, tables, and special values in a single volume. Includes explicit formulas and tables for 3j, 6j, and 9j symbols, as well as rotation matrices and spherical harmonics. An essential desk reference for anyone doing serious nuclear structure calculations. Not for learning — for looking things up.

Fermi's Golden Rule and Perturbation Theory

Merzbacher, E. — Quantum Mechanics, 3rd ed. (Wiley, 1998)

Merzbacher's treatment of time-dependent perturbation theory (Chapters 19--20) is among the most careful available. The derivation of Fermi's golden rule is done with full attention to the mathematical subtleties of the long-time limit and the density of states. The discussion of the conditions under which the golden rule is valid is more thorough than in most texts.

Fermi, E. — Nuclear Physics, revised ed. (University of Chicago Press, 1950)

Fermi's own lecture notes, compiled by his students at the University of Chicago. The derivation of what became known as "Fermi's golden rule" appears in a characteristically clear and economical form. The notes also provide early applications to nuclear beta decay. A historically fascinating document that shows how Fermi himself thought about these problems. Available in many libraries and as a reprint.

WKB Approximation and Tunneling

Gamow, G. — "Zur Quantentheorie des Atomkernes," Zeitschrift fur Physik 51, 204 (1928)

The original paper in which Gamow applied quantum tunneling to alpha decay — one of the first successful applications of quantum mechanics to nuclear physics. Remarkably readable even today. Gamow's physical insight — that the enormous range of alpha decay lifetimes could be explained by the exponential sensitivity of barrier penetration — was a landmark in the development of nuclear physics.

Gurney, R. W. & Condon, E. U. — "Wave Mechanics and Radioactive Disintegration," Nature 122, 439 (1928)

Published independently of Gamow and reaching the same conclusion by the same method. The Gamow-Gurney-Condon theory of alpha decay remains the basis of all modern treatments.

Landau, L. D. & Lifshitz, E. M. — Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Pergamon Press, 1977)

Chapter 7 provides the most rigorous treatment of the quasi-classical (WKB) approximation available in a standard textbook, including the derivation of connection formulas, validity conditions, and applications to tunneling. The treatment of the Coulomb barrier and the Gamow factor is particularly complete. Demanding but definitive.

Nuclear Physics Context

Krane, K. S. — Introductory Nuclear Physics (Wiley, 1987)

Chapter 2 reviews the relevant quantum mechanics in a nuclear physics context, covering angular momentum coupling, parity, and identical particles. While the notation and conventions differ slightly from this book, Krane's review is concise and focused on nuclear applications. The reader who wants a second perspective on the same material will find Krane useful.

de Shalit, A. & Feshbach, H. — Theoretical Nuclear Physics, Vol. I: Nuclear Structure (Wiley, 1974)

Chapters 1--4 develop the quantum mechanical formalism specifically tailored for nuclear structure calculations. The treatment of angular momentum coupling, tensor operators, and the Wigner-Eckart theorem in a nuclear physics context is more detailed than any of the general QM texts. For the reader who intends to pursue nuclear structure theory, this is the essential reference — though it is not for the faint of heart.

Computational Tools

The sympy.physics.quantum.cg module in Python provides symbolic Clebsch-Gordan coefficients and Wigner symbols. The scipy.special module does not include CG coefficients directly, but the py3nj package (available via pip install py3nj) provides fast numerical 3j, 6j, and 9j symbols suitable for large-scale nuclear structure calculations. The code provided in this chapter's code/ directory implements CG coefficients from scratch as a learning exercise, and also demonstrates the use of sympy for verification.