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 . corresponding to this field can't be generally specified in terms of quantities and , as the unobserved negative energy 2) Corresponding to states of a photon with negative make a contribution in decompositions (32) – (35). energy E ()(k) kc pc (which are theoretically possible) the orthonormal bivectors, answering to the helicity So, at the level of the postulate we can say that the single- 1, are photon state can be described bivector (36) – (37) with a plus sign for the real photons and with the minus sign for the
() hypothetical photons with negative energy. () (r,t) (Oe) e 1 (k) i (kr k ct) 1 (r,t) k, 1 e bv; k , 1 3/ 2 0 The bivector (36) – (37) also plays an important role in the 0 (2) description of the single-photon state.
() 0 (Oe) e 1(k) i (kr kct) 0 We can denote the intensities, giving the contributions to (r,t) () e bv; k , 1 (r,t) 3/2 the integrals (32) - (35) according to the following relations: k ,1 (2) 1 () () () () E(r,t) E, 1 (r,t) E,1(r, t) E,1(r,t) E, 1(r, t) Taking into account (9), the basis vectors k, (r,t) and (r,t) , also being in turn eigenfunctions of operators of () () () () k, H(r,t) H, 1(r,t) H, 1(r, t) H, 1(r,t) H, 1(r, t) the energy, momentum and helicity ˆ (sˆpˆ ) /p (in vector representation), satisfy to the orthonormality relations Then from (36), (38) important connections follow:
() () () () 3 2 d r (r,t) (r,t) (kk) (Oe) Ε, 1(r, t) Ε, 1(r, t) H, 1(r, t) H, 1(r, t) k, k, 3 2 () () () () d r (r,t) (r,t) (kk) (Oe) 1 (r,t) 1 (r,t) 1 (r,t) 1 (r, t) k, k , Therefore bivectors (24) – (27) satisfy to the orthonormality Together with the formulas (43) – (44) associating with the relations decompositions (33), (35), the relations (40), (41) give [5, 27, 32] the superposition principle for the intensities E and H: d 3r () (r,t) () (r, t) (Oe) 2 (kk) bv; k, bv; k , () () Ε (r, t) Ε, 1(r,t) Ε, 1(r,t)
210 H (r, t) H() (r, t) H() (r, t) energy mean value in the state (1) and the Hamilton operator , 1 , 1 (17) (and also the equation (16), and relations (30)):
Using specified bivectors, it is possible to write the spatial () () () () () density distribution of photon energy in the state (37): E Hˆ i bv t 2 3 () () 3 () () () 1 kc b(k,) d k E (k) P (k) d k E (r,t) bv (r,t) bv (r,t) 8
1 () () () () what coincides with (52) for positive energy. 1 (r,t) 1 (r,t) 1 (r,t) 1 (r,t) 8 To conclude this paragraph we mention that from the equation (16) the continuity equation follows [5, 27, 32] for 1 () 2 () 2 () 2 () 2 E (r,t) H (r,t) E (r,t) H (r,t) () 8 ,1 , 1 ,1 ,1 density of probability P (r,t) and of stream density () j P (r,t) of probability to find the photon in the vicinity of the This energy density is to be distinguished from the point r in a time point t: “classical” energy density, which, obviously, has the form () P (r,t) () div j P (r,t) 0 (cl) (r, t) 1 E(r, t) 2 H(r, t) 2 t E 8 where () () () 1 () (r,t) () (r,t) ()(r,t) () (r,t) (r,t) (r,t) (r, t) 16 bv bv bv bv P j ()(r,t) c () (r,t) ˆ () (r,t) From the practical point of view, the construction of the P bv bivector (36) – (37) isn't so much important in itself, how the finding of the coefficients B (k,) is important in the case, Although for arbitrary coefficients b (k, ) the equation (55) when the single-photon state is defined to be corresponding to doesn't possess of relativistic invariance property, nevertheless the state of the electromagnetic field, initially given by means it can be shown that this equation is relativistic invariant [27, of the classical intensities E and H. Further, composing, for 32] for the case of plane monochromatic waves. example, the vector Ε i Η and using the orthonormality relations (28), (30), it is possible to calculate the coefficients III. MODELING OF SHORT-PULS LASER RADIATION
1 3 () On the basis of the above stated general method of B (k ,1) d r k , 1 (r,t) (r, t) construction of wave function of a free photon in coordinate (Oe) 2 representation in [1–4] the most important wave packet in 1 d 3 r () (r, t) () (r,t) scientific and methodical aspects with Gaussian momentum (Oe) 2 bv; k , 1 distribution is considered, namely with the coefficients
Then with these coefficients we can calculate the coefficients b (k, 1) b (k,1) b (k, ) appearing in (1) by the formula 3 2 2 2 2 exp k x k y (k z k 0 ) ikr0 (Oe) 2 b (k , ) B (k , ) 2 8 kc where parameters k 0 (0, 0, k0 ) , r0 (x0 , y0 , z0 ) , The introduction of these coefficients most fully characterize the average values and the dispersions of the implements the principle of correspondence. Indeed, according corresponding physical quantities in the state of a photon (1) to (7), (24) – (27), (34), (37), (47), based on the classical and satisfy the normalization condition (2). electrodynamics, the total photon energy can be written as Parameterization (58) answers to the state of a photon with 2 zero average helicity as the 1 are presented in (58) with () 3 (cl) (Oe) 3 2 E E d r (r,t) d k B (k, ) equal probability. All characteristics can be broken into two E 8 categories: momentum-energy and space-time. According to 2 () quantum mechanics, values of these categories characteristics d 3k b (k,) kc d 3k (k)kc P answer to the corresponding uncertainty relations. New here, compared to the particles with mass, is the fact that the values () The integration of density E (r,t) yields just the same result. of characteristics of the second category essentially can On the other hand, the same result is obtained if we use the depend on "choice" of vectors e (k) . purely quantum-mechanical formula of calculating of the
211 A. Momentum-energy characteristics 2 E () c 2 2k 2 1 3 Applying (1), (58) and quantum-mechanical formula of 0 2 2 2 k0 calculation of average value of physical quantity F, the operator of which is equal Fˆ , namely Using (67) and (68), it is possible to calculate dispersion and uncertainty of energy of a photon in state (1):
() () () () () F Fˆ Fˆ 2 2 D E () E () E D E E () where at the values 1 are defined by B. Space-time characteristics corresponding terms of the formula (1), we find at once average values of projections of momentum and their squares: Requirements (8) – (13) are satisfied, e.g., for the following polarization vectors [1, 3–5]: () () () px py 0 pz k0 1 (1 cos ) cos 2 e (k) (1 cos )sin cos 2 2 I () 2 () 2 () 2 2 2 px p y p k sin cos 2 2 z 2 2 0 (1 cos )sin cos from where it follows that the average vector of a momentum e (k) cos (1 cos )cos 2 at 0 II 2 of the photon in a state of positive energy is directed along the sin sin axis z, and with negative is opposite to it:
() 1 (1 cos )cos 2 p k 0 k 0 e z . e (k) (1 cos )sin cos I The dispersions of the projections of a momentum vector on sin cos the xyz axes are defined by the parameter : (1 cos )sin cos 2 eII (k) cos (1 cos ) cos at 2 2 2 2 2 () () sin sin D p px px Dp Dp . x 2 2 y z 2 2 where the Cartesian components of the corresponding vectors Then uncertainty of the photon momentum projections in state in the configuration space are specified, expressed in terms of (1) are reduced to formulas the spherical coordinates of vector k in momentum space. Taking into account formulas (70) – (73) it is conveniently to carry out the calculation of space-time characteristics in p x D p p y pz momentum representation, using the formula (6). x 2 2 In particular for the average values of coordinates and their squares of a point of detection of the photon in state (1) we where the presence 2 is connected with such choice which obtain the following expressions: gives, according to (6) the simplest form of the momentum distribution in the state (1), namely Gaussian form () () () () x x0 y y0 z z0 ct nz () 2 2 2 2 P (k) b (k,1) b(k, 1) () 2 2 (2) 2 2 () x x0 A 1 с t nx 3 2 2 2 2 2 exp k x k y (k z k 0 ) 2 2 2 y () y 2 A (2) с 2 t 2 n () 0 2 2 y Applying the formulas (54), (65), (66) we find the average 2 2 () 2 2 (2) 2 2 () () energy of the photon in state (1), respectively with positive z z0 A 3 с t nz 2сt n z z0 and negative spectrum of its energy, 2 where 2 2 2 2 () exp( k0 ) 1 () 1 exp ( k0 ) E k0c 1 erf ( 1 k 0 ) , n 1 erf ( k ) 2 2 z 2 2 0 2 k0 k0 2 k0 k0 and also, similarly, the average square of energy of the photon () 2 () 2 1 () 2 in the state (1): n n 1 n x y 2 z
212 Thus, there is a transformation of the original shape of the 1 1 erfi ( k ) exp ( 2 k 2 ) initial wave packet (Fig. 1) to a certain "conical" shape (Fig. 2 2 2 2 k 0 0 2 k0 0 and 3). A (2) A (2) 1 A (2) A (2) 1 2 2 3 13 1 2 2k 2 A (2 ) 1 0 erfi ( k ) exp 2 k 2 1 3 2 2 k 0 0 2k 0 0 1 exp 2k 2 u 2 A (2) 2 2 exp 2 k 2 0 du 13 0 0 1 u
From (74) – (83) it follows that the dispersions Dx , Dy , Dz of coordinates of detection point of the photon which is in the state (1) parameterized by means of (58) are equal
2 2 2 D D A(2) с 2 t 2 n () A(2) с 2t 2 D x y 2 1 x 2 1 nx 2 D A (2) с 2t 2 D 4 z 2 3 n z
According to (84) the dispersions Dx and Dy are the same for the considered wave packet which is symmetric relatively of the z axis.
C. Analysis of modeling results As seen from (84), the rate of expansion of the wave Fig. 1. Distribution of electric field intensity E x at time moment t 0 : packet is the same in each plane xy, in accordance with the zc 0 , x y z 0, 0120 cm . fact that the wave packet (1) with the parameterization (58) remains symmetric relatively of the z axis. As characteristics of speed of this expansion it is possible to use periods x , y ,
z during which initial dispersions (at t = 0 ) are doubled along the directions x, y, z. From (84) we find: x(t 0) y(t 0) z(t 0) x y z 5 c n x c n y c n z Since even in a simple form of distribution (58) it is not possible analytically to obtain an expression for the probability density in configuration space, we carry out the analysis of the evolution of the considered wave packet by means of calculation of the intensity of electric field, using the Fig. 2. Distribution of electric field intensity E at time moment t : initial formula (1), (42), (51). Not equal to zero in this case the x z zc 0, 268 cm , x y 0, 0179 cm , z 0,00169 cm . projection of intensity Ex is only; it characterizes a certain way, the spatial probability density. The spatial "form" of a wave packet in the initial time is “spherical”. We will give results of numerical calculation for the packet corresponding to the duration of radiation rad 80 fs with the central wavelength of 10 microns. Parameter 0.001692 was calculated using the uncertainty relation for energy and time E t / 2 , where t was assumed equal to rad /2 , and E was determined by the formulas (67) – (69).
On an axis of symmetry of a packet, the density of probability of photon detection in the vicinity of the center of the packet (with coordinate zc ) moves practically with velocity of light in vacuum. The farther from the axis, the lower the velocity of probability density is in the direction of Fig. 3. Distribution of electric field intensity E x at time moment t 2 z : the average velocity (along z axis) of wave packet. zc 0, 536 cm , x y 0, 0358 cm , z 0,00268 cm .
213 Speed of this transformation is the more, the less initial vector dominance also exhibited by the photons, their "radius" of a wave packet (1), according to the general propagation in vacuum can be associated also with other, more representations of quantum mechanics. Fig. 1–3 show the complex virtual processes. distributions of the most significant projection of the intensity of electric field E x computed respectively in three different V. THE YOUNG’S INTERFERENCE EXPERIMENT 11 moments of time: t 0 , t z 14,9 y 0,89810 c and It is known that when problem of wave-particle duality is t 2 , where is the time of expansion of the packet (85) discussed, then for example in experiment such as the Young's z z experiment, to explain the wave properties of particles having along an z axis. mass, the wave function in the coordinate representation is attracted. The similar picture of diffraction and interference observed with the light is explained, appealing to the classical IV. THE MAIN FORMULA OF WAVE-PARTICLE DUALITY electrodynamics based on Maxwell's equations. So, in the case AND NATURE OF PHOTON of Young's experiment the explanation of the interference In our opinion the constructed photon quantum mechanics result is reduced to establishment of the phases difference of substantially removes a problem of wave-particle duality of two monochromatic waves emitted by the slits 1 and 2 (see quantum "particles". Main "formula" of wave-particle duality Fig. 4b), entering the observation point P on the screen. of light and particles can be formulated as follows [5, 32, 35]: 1. Photons and microparticles at interaction behave as a corpuscles, transferring and transmitting (to other particles) in a certain quantity as dynamic characteristics (energy, momentum, angular momentum), and "internal" (mass, electric charge, spin, etc.). In particular, such transfer is carried out at hit of a photon or microparticle in quite dot detector (or a point on the screen) with coordinate r at time point t . The fact of hit of "all particle entirely" in the dot detector is characteristic for a corpuscle, but not for some real wave. 2. However photons and microparticles propagate in space by "wave rules", that is their distribution in space is described Fig. 4. Propagation of the photon in space: a) spins flip wave in physical by the wave function. In particular, density of probability of vacuum at Planck distances; b) penetration of the wave function of a detection in space of the nonrelativistic particle with a nonzero photon at a time through two slits in Young's experiment. mass is postulated by a formula (r,t) (r,t) 2 , and a photon by (3). This probability density also causes the hit of a A similar phase kr kct is available in each term of photon and microparticle in the dot detector. A characteristic the photon wave function (1). If the radiation is more or less interferential picture on the screen corresponds to distribution monochromatic, then in the expression for the probability of (r,t) along the screen. density (3), obviously, arises the member proportional to the Nevertheless, electromagnetic radiation even in the case of cosine of the phases difference 1 2 k (R1 R2 ) of waves small lengths and obviously expressed "corpuscular these two waves (emitted by the slits 1 and 2). This also properties", is impossible to consider as a stream of the certain provides an explanation for the occurrence of interference "created", "dot" particles, similar to the massive particles. In fringes, similarly to the explanation of classical our view, the photon is a quasi-particle, and light is a result of electrodynamics. the propagation of a spin wave in physical vacuum, the Thus, introducing in consideration the photon wave structure and nature of which have to be considered at the function in the coordinate representation, we have the Planck distance [5, 32, 36]. This question is closely related to opportunity to explain the wave phenomena on a uniform the structure of the leptons and other fundamental particles on basis for all quantum particles. This especially becomes the same distances. According to [37–39] the center of an relevant when an experiment involving photons emitted electron is extreme maximon, that is the quantum nonsingular obviously alone (for the first time single photons have been object creating round itself an extreme Kerr-Newman metric. reliably fixed in [40]). It has spin s = 1 / 2 and approximately Planck mass, charge and radius. For most observed phenomena involving photons it is VI. CONCLUSION possible to give the following interpretation of their propagation in vacuum. In the photon propagation the middle- The results of our modeling of photon wave packet ordered (in time and space) alternate spins flip occurs (during propagation allow to illustrate the possibility of a single- the Planck time TP for each flip, see Fig.4a) of virtual vacuum photon approach to the description of electromagnetic extreme maximons, which creates the effect of the spin wave, phenomena. In particular, it appears that those aspects of and in "macroscopic scale" produce manifestation the interference and diffraction such as the interference pattern of corpuscular-wave properties of photons. However due to the Young's double-slit experiment, which were described in the
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