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.
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.
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.
This paper, deals with the uncertainty relation for photons. In [Phys.Rev.Let.108, 140401 (2012)], and [1] the uncertainty relation was obtained as a sharp inequality by using the energy distribution on space. The relation we obtain here is an alternative to the one given in [Phys.Rev.Let.108, 140401 (2012)] by the use of the position of the center of the energy operator. The fact that the components of the center is non commutative affected the right hand side of the Heisenberg inequality. But this resolved by the increase of the photon energy. Furthermore we study the uncertainty of Heisenberg with respect to angular momentum and Foureir. We end the paper by giving some examples.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.
Here the probability density of relativistic particles coordinates, satisfying the formal conditions of the quantum mechanics and the special relativity, is determined (under textbooks view, such density does not exist). It is specified for the boson, neutral and charged, scalar and vector, including the photon, the electron. Uniqueness of this design has been proved. This is a relativistic invariant, describing the probability of the particle observation in points of space-time. The time is deprived of his role of a dynamic parameter of distribution and formally equalized with other coordinates. Also the meaning of the energy-momentum distribution is clarified. Conditions of such distributions observation are described. As applied to the quantum field these structures are transformed into new characteristics of particles distribution in space-time in addition to those for momentum states. Their operators for free fields are obtained. Under such model the Heisenberg's uncertainty relations are grounded for the relativistic particle; both particles and antiparticles turn out to be different states of a single particle; vacuum state of boson quantum field has zero energy.
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 $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.