Heisenberg uncertainty relations

R. Shankar

Iwo Białynicki‐Birula, Adam Prystupiuk

This work completes the program started by I. Bialynicki-Birula and Z. Bialynicka-Birula [Uncertainty relation for photons, Phys. Rev. Lett. 108, 140401 (2012); Heisenberg uncertainty relation for photons, Phys. Rev. A 86, 022118 (2012); Heisenberg uncertainty relation for relativistic electrons, New J. Phys. 21, 073036 (2019)] to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for relativistic fermions. The uncertainty relations for bosons are based on the energy density. It is shown that the uncertainty relations based on the time component of the four-current, as we have done previously for electrons, are untenable because they lead to contradictions.

Kazuo Fujikawa, Koichiro Umetsu

A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart et al. [Nat. Phys. 8, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs–Kelly uncertainty relation, which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs–Kelly uncertainty relation and the Heisenberg error–disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation, as is widely accepted.

R. Shankar

Harris Varghese, M. Ravendranadhan

The Robertson's formulation of the uncertainty relation is the most widely accepted form of the Heisenberg uncertainty relation (HUR). It gets modified when we consider it for entangled particles. But this formulation does not consider the measurement process itself. There are reformulation of the uncertainty relations called the generalized uncertainty relations by including the measurement process into the uncertainty relation. HUR gets modified for the case of entangled particles. Here it is shown that the limit of the Generalized Uncertainty Relation (GUR) also reduces for entangled particles. So, GUR also shows similar trend as that of the HUR for entangled particles. Also as entanglement increases, the uncertainty reduces and measurement becomes more precise.