Chapter 10 Further Reading

Primary References

J.J. Sakurai and Jim Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge, 2021), Chapter 4

Sakurai's Chapter 4 ("Symmetry in Quantum Mechanics") is the definitive textbook treatment. It covers symmetry operations and the group structure of quantum mechanics, translation symmetry and momentum, rotation symmetry and angular momentum, discrete symmetries (parity, lattice translation, time reversal), and symmetry considerations in bound states. Sakurai's approach is elegant and assumes Dirac notation throughout, making it a natural extension of our Chapter 10. His treatment of time reversal is particularly thorough — he derives the antiunitary property from first principles and works through the spin-1/2 case in detail.

Recommended sections: 4.1 (Symmetries, Conservation Laws, and Degeneracies), 4.2 (Discrete Symmetries: Parity), 4.3 (Discrete Symmetries: Lattice Translation), 4.4 (The Time-Reversal Discrete Symmetry).

Ramamurti Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 1994), Chapters 11 and 12

Shankar develops symmetry in two chapters: Chapter 11 covers the general theory of symmetry (conservation laws, generators, translation, rotation, parity) and Chapter 12 develops the rotation group and angular momentum in detail. Shankar is unusually explicit about the group-theoretic underpinnings — he introduces Lie groups and Lie algebras at a level accessible to physics students. His derivation of the translation operator from the generator is pedagogically excellent.

Recommended sections: 11.1--11.4 (general theory), 11.5 (parity), 12.1--12.3 (rotation group and angular momentum algebra). Chapter 12 connects directly to our Chapter 12.

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloe, Quantum Mechanics, Vol. 1 (Wiley, 1977), Complements to Chapter VI

Cohen-Tannoudji treats symmetry through a series of "complements" (extended worked examples) attached to his chapter on angular momentum. The complements on rotation invariance, parity, and selection rules are exceptionally thorough, with detailed derivations that leave no steps implicit. This is the reference to consult when you want mathematical rigor at the undergraduate level.

Recommended complements: D-VI (rotation invariance), E-VI (parity), F-VI (selection rules).

Secondary References

David J. Griffiths and Darrell F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge, 2018), Sections 6.1--6.2 and Problem 6.33

Griffiths treats symmetry more briefly than Sakurai or Shankar, but his discussion of translation and rotation symmetry (Section 6.1, "Symmetries") and discrete symmetries (Section 6.2, "Degeneracy") is characteristically clear and well-motivated. Problem 6.33, on the relationship between symmetry and degeneracy, is an excellent exercise that reinforces the ideas of our Chapter 10.

Leslie E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed. (World Scientific, 2014), Chapters 3 and 4

Ballentine's treatment is the most mathematically sophisticated at the textbook level. Chapter 3 covers the general principles of symmetry in quantum mechanics, including a careful discussion of Wigner's theorem and the distinction between unitary and antiunitary operators. Chapter 4 develops the rotation group and angular momentum. Ballentine is especially good on the Galilean group and its projective representations — a topic that most textbooks avoid.

Kurt Gottfried and Tung-Mow Yan, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003), Chapter 7

Gottfried and Yan provide an advanced treatment that bridges the gap between textbook quantum mechanics and the group-theoretic methods used in particle physics. Their discussion of symmetry breaking (both explicit and spontaneous) is unusually thorough for a quantum mechanics textbook.

Group Theory for Physicists

Howard Georgi, Lie Algebras in Particle Physics, 2nd ed. (Westview, 1999)

The standard reference for the group theory used in particle physics. Georgi develops the theory of Lie groups and their representations with a focus on physical applications: $SU(2)$, $SU(3)$, the Lorentz group, and gauge groups. The first three chapters cover the material needed to deepen the understanding of our Chapter 10. Later chapters are essential reading for anyone interested in the Standard Model.

Recommended for: Students planning to continue to quantum field theory or particle physics.

Zhong-Qi Ma, Group Theory for Physicists, 2nd ed. (World Scientific, 2019)

A comprehensive and modern treatment of group theory with extensive applications to quantum mechanics. Ma covers finite groups, Lie groups, representation theory, and their applications to atomic, molecular, and solid-state physics. Particularly useful for the crystal symmetry material previewed in Section 10.8.

Michael Tinkham, Group Theory and Quantum Mechanics (Dover, 2003)

A classic text, originally published in 1964, that applies group theory specifically to quantum mechanical problems: atomic spectra, molecular vibrations, and crystal field theory. Tinkham's treatment of point groups and their representations is directly relevant to the crystal symmetry discussion in Section 10.8 and the full development in Chapter 26.

Historical and Biographical

Dwight E. Neuenschwander, Emmy Noether's Wonderful Theorem, 2nd ed. (Johns Hopkins, 2017)

An accessible and thorough account of Noether's theorem, written for advanced undergraduates. Neuenschwander derives both Noether theorems (the first for global symmetries, the second for local symmetries) and discusses applications ranging from classical mechanics to quantum field theory. The historical context is excellent.

Yvette Kosmann-Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (Springer, 2011)

A scholarly history of Noether's theorems and their impact on twentieth-century physics. Kosmann-Schwarzbach traces the development from Noether's 1918 paper through the applications in general relativity, quantum mechanics, and gauge theory. This is the definitive historical account.

Christine C. Sutton, Spaceship Neutrino (Cambridge, 1992), Chapter 6

A well-written popular account of the parity violation discovery, including the contributions of Lee, Yang, and Wu. Sutton provides the human story behind the physics, including the controversies about credit and the Nobel Prize.

Chien-Shiung Wu and S.A. Moszkowski, Beta Decay (Interscience, 1966)

Wu's own technical account of beta decay physics, including the experiment that overthrew parity. Essential primary source material for anyone interested in the detailed experimental and theoretical aspects of parity violation.

Condensed Matter Applications

Neil W. Ashcroft and N. David Mermin, Solid State Physics (Brooks/Cole, 1976), Chapters 8--9

The standard reference for Bloch's theorem and energy bands. Ashcroft and Mermin develop the consequences of crystal symmetry for electronic structure in exhaustive detail. Their Chapters 8 (Bloch's theorem) and 9 (electrons in a weak periodic potential) are the direct continuation of our Section 10.8.

Charles Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2004), Chapter 7

Kittel's treatment is more concise than Ashcroft and Mermin but covers the essentials of Bloch's theorem, energy bands, and the Brillouin zone. A good first reading before tackling the more detailed treatments.

  • Before Chapter 11: Review Sakurai 4.1 for the symmetry–degeneracy connection, which will be essential when we discuss entanglement in composite systems.
  • Before Chapter 12: Read Shankar 12.1--12.3 or Sakurai 3.1--3.5 for the full development of the angular momentum algebra from the rotation group generators. Everything in Chapter 12 follows from the commutation relations $[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$ derived in this chapter.
  • For crystal symmetries: Ashcroft and Mermin Chapter 8 provides the background for Chapter 26.
  • For parity violation: Wu's original paper (Phys. Rev. 105, 1413, 1957) is short, clear, and historically important.