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Heisenberg's Original Derivation of the Uncertainty Principle and its Universally Valid Reformulations


Affiliations
1 Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan
 

Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement, but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the present author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.

Keywords

Error-Disturbance Relations, Quantum Root Mean Square, Quantum Measurement, Repeatability Hypothesis, Uncertainty Principle.
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  • Heisenberg's Original Derivation of the Uncertainty Principle and its Universally Valid Reformulations

Abstract Views: 430  |  PDF Views: 273

Authors

Masanao Ozawa
Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan

Abstract


Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement, but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the present author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.

Keywords


Error-Disturbance Relations, Quantum Root Mean Square, Quantum Measurement, Repeatability Hypothesis, Uncertainty Principle.

References





DOI: https://doi.org/10.18520/cs%2Fv109%2Fi11%2F2006-2016