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Unusual commutation relations in physics

André Nauts, Fabien Gatti

The angular momentum operator L̂ of a particle obeys the usual commutation rule L̂×L̂=iℏL̂. In contrast, the total angular momentum operator Ĵ of a set of particles relative to moving axes (as, for instance, when mounted on a tumbling molecule) obeys the anomalous commutation rule Ĵ×Ĵ=−iℏĴ. We give a pedagogical and mathematically nonsophisticated description of intermediate cases for which partial angular momentum operators, relative to various moving frames, obey unusual commutation relations that are neither usual nor anomalous. Insight into the origin of these unusual commutation relations provides a means of guessing the type of commutation rule that will be obeyed. In particular, it gives a way to avoid, as far as possible, the unusual commutation relations, which lead to complicated and nonsystematic expressions. Some examples from molecular physics are presented.The angular momentum operator L̂ of a particle obeys the usual commutation rule L̂×L̂=iℏL̂. In contrast, the total angular momentum operator Ĵ of a set of particles relative to moving axes (as, for instance, when mounted on a tumbling molecule) obeys the anomalous commutation rule Ĵ×Ĵ=−iℏĴ. We give a pedagogical and mathematically nonsophisticated description of intermediate cases for which partial angular momentum operators, relative to various moving frames, obey unusual commutation relations that are neither usual nor anomalous. Insight into the origin of these unusual commutation relations provides a means of guessing the type of commutation rule that will be obeyed. In particular, it gives a way to avoid, as far as possible, the unusual commutation relations, which lead to complicated and nonsystematic expressions. Some examples from molecular physics are presented.

American Journal of PhysicsNov 18, 20106 citations

The Position-Momentum Commutator

José Francisco García Juliá

Sep 1, 20130 citations

THE MOMENTUM REPRESENTATION

T.T. TAYLOR

Elsevier eBooksJan 1, 19761 citation

Quantum-enhanced multi-parameter sensing in a single mode: simultaneous measuring displacements in position and momentum

Ting Rei Tan

Quantum metrology is often constrained by measurement backaction when jointly estimating two non-commuting observables—a canonical example is position and momentum. Here, we experimentally demonstrate a single-mode displacement sensor that estimates small shifts in both position and momentum with precision below the standard quantum limit (SQL). Our approach measures commuting modular position and modular momentum observables, enabled by highly nonclassical grid states (Gottesman–Kitaev–Preskill–type) prepared in the mechanical vibrational mode of a single trapped ion. Using a phase-estimation–based measurement, we extract both parameters in one experiment while evading the backaction that would ordinarily couple the estimates. We further extend the protocol to another naturally non-commuting pair—number and phase—and again observe sub-SQL performance.Quantum metrology is often constrained by measurement backaction when jointly estimating two non-commuting observables—a canonical example is position and momentum. Here, we experimentally demonstrate a single-mode displacement sensor that estimates small shifts in both position and momentum with precision below the standard quantum limit (SQL). Our approach measures commuting modular position and modular momentum observables, enabled by highly nonclassical grid states (Gottesman–Kitaev–Preskill–type) prepared in the mechanical vibrational mode of a single trapped ion. Using a phase-estimation–based measurement, we extract both parameters in one experiment while evading the backaction that would ordinarily couple the estimates. We further extend the protocol to another naturally non-commuting pair—number and phase—and again observe sub-SQL performance.

Optical and Quantum Sensing and Precision Metrology IIMar 5, 20260 citations

Momentum Representation and Applications

WORLD SCIENTIFIC eBooksSep 8, 20140 citations

Displacement operators: the classical face of their quantum phase

Amar C. Vutha, Eliot A. Bohr, Anthony Ransford +2 more

Abstract In quantum mechanics, the operator representing the displacement of a system in position or momentum is always accompanied by a path-dependent phase factor. In particular, two non-parallel displacements in phase space do not compose together in a simple way, and the order of these displacements leads to different displacement composition phase factors. These phase factors are often attributed to the nonzero commutator between quantum position and momentum operators, but such a mathematical explanation might be unsatisfactory to students who are after more physical insight. We present a couple of simple demonstrations, using classical wave mechanics and classical particle mechanics, that provide some physical intuition for the phase associated with displacement operators.Abstract In quantum mechanics, the operator representing the displacement of a system in position or momentum is always accompanied by a path-dependent phase factor. In particular, two non-parallel displacements in phase space do not compose together in a simple way, and the order of these displacements leads to different displacement composition phase factors. These phase factors are often attributed to the nonzero commutator between quantum position and momentum operators, but such a mathematical explanation might be unsatisfactory to students who are after more physical insight. We present a couple of simple demonstrations, using classical wave mechanics and classical particle mechanics, that provide some physical intuition for the phase associated with displacement operators.

arXiv:1702.01833European Journal of PhysicsNov 27, 20173 citations

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