The Uncertainty Principle Revisited

TL;DR

This paper revisits the quantum uncertainty principle by analyzing successive measurements of observables using an extended von Neumann model, revealing how measurement disturbance affects the statistical distribution of outcomes and establishing a connection with the observables' commutator.

Contribution

It introduces a generalized uncertainty relation based on successive measurements and their disturbance effects, extending the understanding of quantum measurement limitations.

Findings

Derived a new uncertainty relation linked to the commutator of observables.

Illustrated the relation for position and momentum measurements.

Showed the impact of measurement disturbance on outcome distributions.

Abstract

We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {\em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $\delta a$ around $a^{(n)}$. The width of this statistical distribution as function of $\delta a$ constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.