2012/05/31 by Iwo Bialynicki-Birula, Iwo Białynicki‐Birula, Zofia Bialynicka-Birula +1 · 29 citations
Computer Science · Mathematics · Physics and Astronomy · #Entropic uncertainty #Heisenberg picture #Mathematical analysis #Mathematics #Momentum (technical analysis) #Noncommutative and Quantum Gravity Theories #Operator (biology) #Photon #Physics #Position operator #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Space (punctuation) #Uncertainty principle #quant-ph
paper · pdf · doi:10.1103/physreva.86.022118
published in Physical Review A 86(2) (American Physical Society) · Sequel to PRL, 108, 140401 (2012); Submitted to PRA
arxiv created 2012/08/26 · openalex publication_date 2012/08/30 · arxiv updated 2015/06/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound 3\ensuremathℏ/2 in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.