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The Young's Interference Experiment in the Light of the Single-Photon

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The Young's Interference Experiment in the Light of the Single-Photon Information Technologies in Science, Management, Social Sphere and Medicine (ITSMSSM 2016) The Young’s Interference Experiment in the Light of the Single-photon Modeling of the Laser Radiation A.P. Davydov T.P. Zlydneva Dept. of Applied and Theoretical Physics of Physical- Dept. of Applied Mathematics and Informatics of Physical- Mathematical Faculty Mathematical Faculty Nosov Magnitogorsk State Technical University Magnitogorsk, Russia E-mail: [email protected] E-mail: [email protected] Abstract—The basic principles of quantum mechanics of coordinate representation. Nevertheless, we will specify further the photon, describing its single-particle states by means of that this concept characterizes the photon only conditionally, the wave function in the coordinate representation are owing to the fact that it still can't be considered as the certain given. This wave function is a wave packet constructed by created particle-corpuscle. In our opinion, the photon the superposition of the basic bivectors which are propagation is a consequence of excitation and propagation in eigenfunctions of the operators of energy, momentum and space of some excited quantum state of physical vacuum. First helicity. The results of modeling in the space-time of such of all, the excitation in the vacuum of certain spin wave like a wave packet with a Gaussian distribution on the magnon in a solid body is most probable. Although the nature of the corresponding excitation can be quite complex as it is momentum of the photon corresponding femtosecond laser evidenced e. g. by the manifestation of vector dominance in the radiation are discussed. On the basis of the general ideas interaction of photon with hadrons through strong interaction. about the evolution of this packet and within the framework of the constructed photon quantum mechanics the quantum-mechanical approach is proposed to explain II. PHOTON WAVE FUNCTION IN COORDINATE the wave phenomena exhibited by light such as the light REPRESENTATION interference in Young's experiment. It is emphasized that For a long time it was considered [6–13] that the photon actually the photon isn't some “created” quantum wave function can not be built in configuration space although corpuscle, but it’s a quasi-particle that is a result of the in momentum representation it is applied in many areas. The propagation of spin waves in physical vacuum, the nature reason for this is the zero (“rest”) mass of the photon. In of which should be considered at the Planck distances modern experiments (transfer of optical signals on quantum simultaneously with the structure of the leptons and other communication channels, “quantum teleportation”, fundamental particles. “paradoxes” with single photons, the problems of spatial entanglement and quantum computing, etc.) there is a need for Keywords—Schrödinger equation; Maxwell’s equations; wave association of photons with the localized carriers of elementary packet; probability density; bivector; detector; extreme maximon; units of information. Therefore the building of photon wave Planck’s parameters; wave-particle duality. function in coordinate representation becomes again actual “at the new level of knowledge”. Then, knowing the wave function, it is also possible with the quantum-mechanical point I. INTRODUCTION of view to explain the interference, diffraction and polarization It is known that the light and also the microparticles having of electromagnetic waves. Without the full review here, we mass exhibit similar corpuscular and wave properties. will refer on [7–19] where anyway this subject is touched, the However, if for example for electrons these properties can be term "wave function of a photon" is used, but nevertheless explained at least formally by means of wave function in wave function of a photon, normalized on unit probability, isn't coordinate representation, within usual quantum mechanics given in coordinate representation. Apparently, the first works (without secondary quantization), then for photons the in which the idea of a possible photon “localization” described diffraction and interference are still explained in the language by a probability density, defined by the normalized per unit of classical electrodynamics. The purpose of this paper is to wave function, has been proposed are [20–23]. Further make the first step in the direction to correct this situation. development of the theory and justification of building of Namely, based on illustrative results of the previous [1–5] coordinate single-particle wave function of the photon was modeling of the space-time propagation of the photon wave performed in [24–32] and others. packet corresponding to a short-pulse laser radiation, we will offer below an explanation of the occurrence of an interference According to [5, 27, 32] the photon wave function in pattern with single photons in the experiment of type of the coordinate representation has the following form: two-slit interference Young's experiment, having applied with respect to the photon the concept of wave function in © 2016. The authors - Published by Atlantis Press 208 e 1 (k) i (kr k ct ) 1 e (n) e (n) [ e (k)] e (k) e (k) () (r, t) b (k,1) e d 3k (2 ) 3 / 2 0 | e | | e | 1 (e n) (e n) (e e ) 0 e 1 (k) i (kr k ct) 0 3 b(k, 1) e d k e (n) e (n) e n e (2 ) 3 /2 1 n e e i e e e (k) sˆ e (k) where the top sign of all indexes corresponds to positive energy of photon, and the lower sign answers to negative energy, where ˆ is the operator of the photon spin in vector hypothetically possible; coefficients b (k, ) are basically s representation: arbitrary, but if the photon state initially set by means of the electric (E) and magnetic (H) fields intensities (in Gaussian ˆ ˆ ˆ ˆ s e xs x e y s y e z s z System), b (k, ) are expressed through them, as well as they 0 i 0 0 ez e y satisfy to the normalization condition for the wave packet (1): 0 0 0 0 0 i e 0 0 i e 0 0 0 e i 0 0 i e 0 e x y z z x 0 i 0 i 0 0 0 0 0 () e y ex 0 () () d 3r () (r,t) () (r,t) d 3r (r,t) P where e x e y e z are the basis vectors of xyz axes. d 3k ()(k,t) () (k,t) d 3k ()(k) 1 2 P The photon wave function ()(r,t) in coordinate representation satisfies to the equation of the Schrödinger Here photon detection probability density in coordinate and equation type momentum space (more precisely in the wave vector k p / () space, where p is photon momentum), respectively is equal to (r,t) i Hˆ () (r,t) t bv () () () where P (r,t) (r,t) (r,t) c (sˆpˆ ) 0 Hˆ c (ˆ pˆ ) () () () bv bv ˆ ˆ P (k,t) (k,t) (k,t) s 0 (sp) moreover, in accordance with (1) the photon wave function in is the Hamiltonian operator of free photon (having spin s 1) in bivector representation, in which its spin vector operator is momentum representation is equal to equal to ˆ ˆ ˆ ˆ sˆ 0 () 1 i kr () 3 S e S e S e S (k ,t) e (r,t)d r x x y y z z 0 sˆ (2) 3/2 ˆ pˆ i is particle momentum operator; the matrix ˆ bv in ik ct 1 0 e b(k , 1)e (k) b (k ,1) e (k) bivector representation has form 1 0 1 1 sˆ 0 ˆ bv Therefore photon detection probability density in the 0 sˆ momentum space actually doesn't depend on time and it is Equation (16) is similar to the equation, which is satisfied () 2 2 (k) b(k,1) b (k,1) b (k,) by bivector bv , with which it is also possible to P describe [12] the photon state in the coordinate representation. where 1 takes two possible values of the photon helicity. Here the physical quantities and (in matrix form) are Equation (7) follows from (4) – (6), taking into account the E i H E i H properties of orthonormality of the complex polarization x x x x vectors E y i H y η E y i H y Ez i Hz Ez i Hz e (k) e (k) i e (k) 2 namely but they are independent quantities from each other [12]. The bivector satisfies to the equation of type (16): (e e ) e e bv () (r,t) In addition, with n = k / k the following relations take place: i bv Hˆ () (r,t) t bv bv 209 which is the consequence [5, 27, 32] of the Maxwell equations In addition, bivectors corresponding to the states with the written in the Majorana form [33, 12] opposite energy sign are orthogonal to each other regardless of the values k and : i c (sˆpˆ) i c(sˆpˆ ) (pˆ) 0 (pˆ) 0 t t d 3r () (r, t) () (r, t) 0 bv; k, bv; k, Four independent solutions of the equation (21) which are The relations (28) - (30) make it possible to decompose any automatically satisfying to the equations (22), and also being vectors and bivectors in the corresponding bases: eigenfunctions of the helicity operator () () ˆ ˆ (r,t) Ε(r,t) i Η(r,t) 1 (r,t) 1 (r,t) (S pˆ ) (Spˆ ) 1 (sˆpˆ ) 0 ˆ ˆ ˆ () () sp p p 0 (sp) B (k ,1) (r,t) d 3 k B (k ,1) (r,t) k, 1 k, 1 (in bivector representation) and the generalized eigenfunctions of the momentum operator, are the following [5, 27, 32]: () () (r,t) Ε(r,t) i Η(r,t) 1 (r,t) 1 (r,t) 1) Corresponding to states of a photon with positive () 3 () 3 () B(k,1) (r,t) d k B(k,1) (r,t) d k energy E (k) kc pc (wich are consistent with the k, 1 k, 1 special theory of relativity [34]) the orthonormal bivectors, (35) answering to a helicity 1, are Ε() (r,t) i H() (r,t) ()(r,t) ()(r,t) , 1 , 1 1 () bv () () () (r,t) (Oe) e 1(k) i (kr kct) 1 Ε , 1(r,t) i H, 1(r,t) 1 (r,t) () (r,t) k, 1 e bv; k ,1 3/2 0 0 (2 ) () 3 () 3 B(k,1) (r,t)d k B(k,1) (r,t) d k bv;k ,1 bv; k,1 (Oe) e (k) () 0 1 i (kr kct) 0 (r,t) () e bv; k ,1 (r,t) 3/2 1 k,1 (2) From (32) – (35) it is visible that if the free electromagnetic field is initially set by means of the some classically respectively, where (Oe) is unit of measure (Oersted) of values interpreted field intensities E and H, the single-photon state and .
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