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Results in Physics

Quantum flux associated with a photon on the basis of the Heisenberg uncertainty relation

Ziya Saglam, Bahadır Boyacıoğlu, Servet Bayram, Ashok Chatterjee

Published Jan 1, 2016• 2 citations
Publisher

Abstract

By using the Heisenberg uncertainty relation for a photon in a uniform magnetic field, we have shown that the quantum flux is associated with a photon ± hc e . Applying the Bohr–Sommerfeld quantization argument for a photon, we have also shown that the uncertainty relation for a photon is exactly the same as the uncertainty relation for an electron.

DOI
10.1016/j.rinp.2016.04.009

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By using the Heisenberg uncertainty relation for a photon in a uniform magnetic field, we have shown that the quantum flux is associated with a photon ± hc e . Applying the Bohr–Sommerfeld quantization argument for a photon, we have also shown that the uncertainty relation for a photon is exactly the same as the uncertainty relation for an electron.
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.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.
We have calculated the changes in the quantized magnetic fluxes corresponding to the electronic orbits of Dirac hydrogen atom within the framework of Dirac theory and found that the flux change between the final state and the initial state is equal to (±Φ 0 or zero).We have also calculated the non-zero matrix elements for dipole transitions (such as 1s↔2p and 2p↔3d) for the solutions of Dirac hydrogen atom. We find a complete agreement among all the above results. The combined result is that photon should carry an intrinsic flux quantum of (Φ 0 =hc/e).
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 $3\ensuremath{\hbar}/2$ 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.
The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.