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arXiv:1903.11575 · New Journal of Physics

Heisenberg uncertainty relation for relativistic electrons

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Jul 1, 2019• 4 citations
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Abstract

Abstract The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound 3/2ℏ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for photons.

DOI
10.1088/1367-2630/ab3076

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Abstract The Heisenberg uncertainty relation is derived for relativistic electrons described by the Dirac equation. The standard nonrelativistic lower bound 3/2ℏ is attained only in the limit and the wave function that reproduces this value is singular. At the other end, in the ultrarelativistic limit, the bound is the same as that found before for photons.
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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.
HEISENBERG’S uncertainty relation results as condition for convergence of the solution of a difference equation which is an analogue to the SCHROEDINGER equation. From DIRAC’S equations and the relativistic SCHROEDINGER equation (KLEIN-GORDON equation) — written as difference equations — one obtains in the same manner a generalized uncertainty relation, from which HEISENBERG’S is derived as a special case for negligible relativistic mass increase and not too precise position and time measurements.
A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.
Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.
After analyzing, in the literature, deep orbit results of relativistic quantum equations, we studied them in a semi-classical way, by looking for a local minimum of total energy of an electron near the nucleus, while respecting the Heisenberg Uncertainty Relation (HUR). Now, while using new information thanks to semi-classical computations, we come back to deep electrons as solutions of the Dirac equation, to solve several important and subtle outstanding issues, such as the continuity of derivatives of wave functions, a spectral problem about the energy levels associated with the wave functions to compute, as well as essential relativistic and energy parameters of the solutions. We thus obtain a better completeness of the solutions. Finally, we give some approaches on the probability of the presence of Electron Deep Orbit (EDO) states in H atom.