The uncertainty relation for joint measurement of position and momentum

2004/05/30 by R. F. Werner, Werner, R. F. · 2 citations
Decision Sciences · Physics and Astronomy · #Advanced Frequency and Time Standards #FOS: Physical sciences #Quantum Physics (quant-ph) #Radioactive Decay and Measurement Techniques #Scientific Measurement and Uncertainty Evaluation

paper · pdf · doi:10.48550/arxiv.quant-ph/0405184

openalex publication_date 2004/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (ΔP)(ΔQ)≥ Cℏ, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision ΔQ of the position measurement, and the perturbation ΔP of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

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