Position Uncertainty in the Heisenberg Uncertainty Relation

TL;DR

This paper investigates the interpretation of position uncertainty in the Heisenberg relation, emphasizing the distinction between measurement error and result uncertainty, and examines the validity of linear assumptions in measurement models.

Contribution

It clarifies the difference between measurement error and result uncertainty in the Heisenberg relation and analyzes the linearity assumption in position measurement models.

Findings

The Heisenberg relation holds when using measurement error as position uncertainty.

Measurement result uncertainty is the standard deviation, not the measurement error.

Linearity assumption validity is critically examined.

Abstract

Position measurements are examined under the assumption that object position x_t and probe position X_t just after the measurement are expressed by a linear combination of positions x_0 and X_0 just before the measurement. The Heisenberg uncertainty relation between the position uncertainty and momentum disturbance holds when the measurement error \epsilon(x_t) for the object position x_t is adopted as the position uncertainty. However, the uncertainty in the measurement result obtained for x_0 is the standard deviation of the measurement result, and not the measurement error \epsilon(x_0). This difference is due to the reduction of a wave packet. The validity of the linearity assumption is examined in detail.