arXiv:1002.3005 · Progress of Theoretical Physics

Position Uncertainty in the Heisenberg Uncertainty Relation

Satoshi Kosugi

Published Mar 1, 2010• 1 citation

Abstract

Position measurements are examined under the assumption that object position and probe position just after the measurement are expressed as a linear combination of positions and just before the measurement. The Heisenberg uncertainty relation between the position uncertainty and momentum disturbance holds when the measurement error ε(xt) for the object position is adopted as the position uncertainty. However, the uncertainty in the measurement result obtained for is the standard deviation of the measurement result, and not the measurement error ε(x0). This difference is due to the reduction of a wave packet. The validity of the linearity assumption is examined in detail.

DOI: 10.1143/ptp.123.431

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arXiv:1002.3005 · Progress of Theoretical Physics

Position Uncertainty in the Heisenberg Uncertainty Relation

Satoshi Kosugi

Published Mar 1, 2010• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1143/ptp.123.431

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Heisenbergの不確定性原理において,位置の測定値の不確定性として何を用いるべきかについて考察した.測定対象の位置としては,測定前の位置x0と測定後の位置xtの2つが考えられる.位置xtの測定値(xt)expの不確定性はこの測定値の測定誤差ε(xt)である.しかし位置x0の測定値(x0)expの不確定性は測定誤差ε(x0)ではなく,測定値(x0)expの標準偏差σ((x0)exp)である.位置の測定値の不確定性としてε(xt)またはσ((x0)exp)を用いた場合には,Heisenbergの不確定性関係は成立している.2つの位置情報の不確定性ε(xt)とσ((x0)exp)の非対称性を引き起こしているのは波束の収縮である.Heisenbergの不確定性原理において,位置の測定値の不確定性として何を用いるべきかについて考察した.測定対象の位置としては,測定前の位置x0と測定後の位置xtの2つが考えられる.位置xtの測定値(xt)expの不確定性はこの測定値の測定誤差ε(xt)である.しかし位置x0の測定値(x0)expの不確定性は測定誤差ε(x0)ではなく,測定値(x0)expの標準偏差σ((x0)exp)である.位置の測定値の不確定性としてε(xt)またはσ((x0)exp)を用いた場合には,Heisenbergの不確定性関係は成立している.2つの位置情報の不確定性ε(xt)とσ((x0)exp)の非対称性を引き起こしているのは波束の収縮である.

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ε(x_t)

i.e., the measurement error of the post-measurement observable

\hat{x}_t

, not the precision

ε(x_0)

i.e., that of the pre-measurement observable

\hat{x}_0

that determines the uncertainty of the observable

\hat{x}_t

. Therefore, Heisenberg's relation must be interpreted as one between the resolution

ε(x_t)

and the disturbance

η(y_0)

. The magnitude of the disturbance

η(y_0)

is independent of the definition of the measurement value of the observable

\hat{x}_t

. Heisenberg's error-disturbance relation is proven to hold true in general, when the measurement value of

\hat{x}_t

is defined in order to minimize the resolution

ε(x_t)

A prevailing formulation of the Heisenberg uncertainty principle is re-examined that measuring position with noise

\\epsilon

brings the momentum disturbance no less than

\\hbar/2\\epsilon

. With rigorous formulations of noise and disturbance, a model of a measuring interaction demonstrates that this formulation of the Heisenberg uncertainty principle is by no means universally valid. An experimental realization of the model is also suggested.