More on the quantum measurement problem

Sean Carroll’s July Quick Study, “Addressing the quantum measurement problem” (page 62), brings up the following question: Does the wavefunction still obey the Schrödinger equation when a measurement is made? A system being measured (or interacting with its environment in any other way) is actually a subsystem, and a subsystem is properly described by a reduced density matrix. The density matrix for an entire system corresponds to a wavefunction—that is, to a pure state—but the density matrix for a subsystem does not necessarily correspond to a wavefunction. The reduced density matrix of a subsystem may correspond to an impure state, also called a mixture or an incoherent combination,11. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. which does not have well-defined pure-state content.22. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. In the words of Kurt Gottfried and Tung-Mow Yan, “systems in the real world are rarely in pure states.”33. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46.The proper way to discuss a measurement is not using a wavefunction but rather a reduced density matrix. The density matrix of a pure state evolves according to the Liouville–von Neumann equation, which is equivalent to the unitary evolution of the wavefunction by the time-dependent Schrödinger equation. For a subsystem (that is, for any system except the entire universe), the reduced density matrix evolves according to the nonunitary Liouville–von Neumann equation, which has an additional contribution causing decoherence and dissipation.44. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U The nature of the measurement—or, more generally, the nature of the subsystem–environment interaction—selects a preferred basis, called the pointer basis, and the subsystem decoheres into an effectively classical mixture in the pointer basis. (See the article by Wojciech Zurek, Physics Today, October 1991, page 36.)ReferencesSection:ChooseTop of pageReferences <<1. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. Google Scholar2. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. Google Scholar3. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46. Google ScholarCrossref4. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U, Google ScholarCrossref© 2022 American Institute of Physics.