The Heisenberg Equation of Motion

  • Dirk Dubbers
  • Hans-Jürgen Stöckmann
Part of the Graduate Texts in Physics book series (GTP)


We turn to an alternative description of quantum theory, equivalent to Schrödinger’s. Rather simple reasoning led Heisenberg to an equation of motion based on the commutation relations of the operators involved. These commutation relations are uniquely linked to the corresponding uncertainty relations. From Heisenberg’s equation, we derive a compact and pictorial equation for magnetic spin precession, valid for any spin quantum number. When applied to the expectation value of the spin operator, this equation turns out to be identical to the classical precession equation of the spinning top. Additional terms that account for spin relaxation lead to the empirical Bloch equations widely used in spin resonance and in quantum optics.


Commutation Relation Uncertainty Relation Pauli Matrice Bloch Equation Angular Momentum Operator 
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  1. Bloch, F.: Nuclear induction. Phys. Rev. 70, 460–474 (1946) ADSCrossRefGoogle Scholar
  2. Schrödinger, E.: Zum Heisenbergschen Unschärfeprinzip. Sitzungsber. Preuss. Akad. Wiss. (Phys.-Math. Kl.) 19, 296–303 (1930), for an English translation see arXiv:quant-ph/9903100v3 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Dirk Dubbers
    • 1
  • Hans-Jürgen Stöckmann
    • 2
  1. 1.Fak. Physik und Astronomie, Physikalisches InstitutUniversität HeidelbergHeidelbergGermany
  2. 2.Fachbereich Physik Philipps-Universität MarburgMarburgGermany

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