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Updated: Jan 19, 2026

Quantum Numbers- Principal, Azimuthal, Magnetic and Spin
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Quantum contextuality in the Copenhagen approach.

Gregg Jaeger1

  • 1Quantum Communication and Measurement Laboratory, Department of Electrical and Computer Engineering and Division of Natural Science and Mathematics, Boston University, Boston, MA, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 17, 2019
PubMed
Summary

Quantum contextuality originates in the Copenhagen interpretation, where experimental setups influence measurement outcomes. This arises from the non-commutativity of quantum observables, impacting hidden-variable theories.

Keywords:
Copenhagen interpretationcomplementaritycontextualitypotentialityquantum theory

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Area of Science:

  • Quantum mechanics
  • Foundations of physics
  • Quantum information

Background:

  • Quantum contextuality is rooted in the Copenhagen interpretation of quantum mechanics.
  • Measurement outcomes are influenced by the experimental arrangement and other measurements.
  • Bohr's complementarity and Heisenberg's indeterminacy principle highlight the role of experimental context.

Purpose of the Study:

  • To identify the origin and basis of quantum contextuality.
  • To connect contextuality in quantum mechanics to the Copenhagen approach.
  • To explore the relationship between contextuality, non-commutativity, and hidden-variable theories.

Main Methods:

  • Analysis of the Copenhagen interpretation's core principles (Bohr, Heisenberg).
  • Definition of context as an equivalence class of compatible measurements.
  • Examination of von Neumann's proof on non-commutative observables.

Main Results:

  • Quantum contextuality arises from the non-commutativity of quantum observables.
  • The experimental arrangement is integral to the significance of quantum measurements.
  • Contextuality is linked to different contexts arising from non-commutative observables.

Conclusions:

  • The Copenhagen approach inherently incorporates quantum contextuality.
  • Non-commutativity of observables is the mathematical basis for quantum contextuality.
  • This framework relates to significant developments in hidden-variable theories and quantum foundations.