Heisenberg's original derivation of the uncertainty principle and its universally valid reformulations

TL;DR

This paper revisits Heisenberg's original derivation of the uncertainty principle, clarifies historical assumptions, and proposes a universally valid reformulation based on modern quantum measurement theory.

Contribution

It clarifies the historical derivation of the uncertainty principle and introduces a new, universally valid reformulation applicable under general quantum measurement assumptions.

Findings

Heisenberg's original proof relied on an outdated postulate.

The repeatability hypothesis was crucial in Heisenberg's derivation but is now obsolete.

A new reformulation of the uncertainty principle is proposed, valid under general measurement conditions.

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 Schr\"{o}dinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.