Part of the Undergraduate Lecture Notes in Physics book series (ULNP)


In this chapter, we ask what happens when a quantum system is in contact with its environment. To describe this, we need a more general concept of the quantum state, namely the density matrix. We study the density matrix, and how it describes both classical and quantum uncertainty. This also naturally leads to a description of quantum systems at a non-zero temperature. We conclude this chapter with a short discussion on how entropy arises in quantum mechanics.

Supplementary material (193 kb)
Supplementary material 1: A Mach–Zehnder interferometer with a removable beam splitter (zip 193 KB) (261 kb)
Supplementary material 2: A Mach–Zehnder interferometer where the photon interacts with the environment (zip 261 KB) (499 kb)
Supplementary material 3: Locating a particle in a box (zip 498 KB) (530 kb)
Supplementary material 4: Erasing information will increase the entropy (zip 530 KB)


  1. R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183 (1961)MathSciNetCrossRefGoogle Scholar
  2. L. Szilard, Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeitschrift für Physik 53, 840 (1929)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of SheffieldSheffieldUK

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