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2 results

  • 15 - Identical-Particle Systems: Quantum Statistics. The Density Matrix

  • Ana María Cetto, Universidad Nacional Autónoma de México, Mexico, Luis de la Peña, Universidad Nacional Autónoma de México, Mexico
  • Book:
    Quantum Mechanics
    Print publication:
    18 December 2025, pp 392-436
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  • Summary

    Chapter 15 is devoted to systems of identical particles and the associated quantum statistics. It is shown how the spin (whether integer or half-integer) determines fundamental features of the system’s behavior that cannot be obtained by approximate methods, in particular the exchange effects characteristic of states of two or more identical particles. The spin-symmetry relation is discussed in detail for both the Bose–Einstein and the Fermi–Dirac statistics. Quantum collective phenomena that are a defining feature of condensed matter, including phonons, superconductivity, Bose–Einstein condensates and the quantum Hall effect, are briefly introduced. Since most real quantum systems consist of subsystems in different quantum states that are mixtures of pure states, the chapter includes an introduction to the density matrix, its basic properties and its application to quantum statistics. It concludes with a critical discussion of decoherence and the significance of the collapse of the wave function.

Visible and Dark Matter in the Universe
  • A Short Primer on Astrophysical Dynamics
  • Giuseppe Bertin
  • Published online:
    08 December 2022
    Print publication:
    22 December 2022
  • This is a concise introduction to modern astrophysics for physicists, with a focus on galaxy dynamics and the discovery of dark matter halos in galaxies. Part I summarizes important discoveries in observational astronomy and astrophysics, in a manner accessible to those who are new to the topic. Building on this foundation, Part II describes the study of dark matter and provides more detail on galactic dynamics. Important physical concepts that form the basis of key astrophysical phenomena are explained, avoiding unnecessary technicalities and complex derivations. The approach is semi-empirical and emphasizes the importance of key measurements and observations in formulating fundamental theoretical questions and developing their solutions. Students are encouraged to develop a deep understanding of major discoveries and contemporary research topics, beyond the simple application of practical models and formulae, as a bridge to more advanced study in astrophysics.