From relativity to the photon

I. K. Karpenko

Ziya Saglam, Bahadır Boyacıoğlu, Servet Bayram +1 more

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.

Theodore P. Jorgensen

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

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.

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

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.

Walter M. Fishl