Simultaneous Measurement of Noncommuting Observables
- 23 December 1966
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review B
- Vol. 152 (4), 1103-1110
- https://doi.org/10.1103/physrev.152.1103
Abstract
The state of a quantum-mechanical system after a simultaneous measurement of conjugate variables such as coordinate and momentum is derived. In the case of the minimum-uncertainty, or ideal, simultaneous measurement, the state of the system after the measurement is a coherent state corresponding to that representing a minimum-uncertainty wave packet. An operator formalism to describe simultaneous measurements of noncommuting observables is introduced which parallels that familiar for a single measurement, and an expression for the joint probability distribution predicting the results of an ideal simultaneous measurement is derived and applied to the example of a harmonic oscillator. It is found that the minimum uncertainty in a simultaneous measurement of noncommuting variables has two causes: (1) the unavoidable perturbations introduced by the measuring process, and (2) the unavoidable lack of precision in the state of the system itself. For position and momentum measurements, these two independent uncertainties contribute to give a net minimum uncertainty of . It is also shown that the formalism of simultaneous measurement leads to a well-defined quantum-mechanical phase space.
Keywords
This publication has 4 references indexed in Scilit:
- On the Simultaneous Measurement of a Pair of Conjugate ObservablesBell System Technical Journal, 1965
- Transformation Having Applications in Quantum MechanicsJournal of Mathematical Physics, 1965
- Coherent and Incoherent States of the Radiation FieldPhysical Review B, 1963
- Information Theory and Statistical MechanicsPhysical Review B, 1957