Quantum Mechanics and Signed Particles, A New Formulation Has Come into Existence

Transcription

1 Quantum Mechanics and Signed Paticles, A New Fomulation Has Come into Existence J. M. Sellie 1 *, I. Dimov 1 1 Institute of Infomation and Communication Technologies, Bulgaian Academy of Sciences, * coespondence Abstact Quantum mechanics epesents a puzzling poblem since almost a centuy now. Indeed, in the standad appoach utilized by mainsteam scientists, systems ae descibed by means of wavefunctions, a still enigmatic concept. Recently a new and pomising appoach has been intoduced which elies on the novel concept of signed paticles. This novel fomulation is built on an unusual intepetation of the Wigne equation, a elatively unfamilia model equivalent to the standad appoach. The new theoy povides a singula physical pictue in tems of paticles inteacting with extenal foces by means of ceation of pais of signed paticles. The appoach theefoe educes to statistical calculations ove an ensemble of signed paticles which allow intuitive, and computationally convenient, time-dependent simulations fo both single- and many-body quantum poblems. As a matte of fact, and despite its elatively ecent appeaance, this appoach has aleady poven to be a vey vesatile and valuable tool in many diffeent contexts touching the ealms of Physics, Chemisty and Nanotechnologies. This aticle wants to be a shot intoduction to this exciting topic which comes with incedible pomises fo theoetical, applied Science and technology. Intoduction Quantum mechanics was ceated to explain a seies of expeimental obsevations in the ealm of elementay paticles, atoms and molecules, which classical mechanics had no hope to explain. The physical evidences fo baffling phenomena such as paticle-wave duality and enegy quantization wee puzzling a whole community of scientists. In spite of these difficulties, eventually a successful set of ules was ceated, able to theoetically epoduce and pedict the obseved featues of quantum systems. Shotly, this is how quantum mechanics was bon. This emakable achievement was in geat pat made possible thanks to the application of an equation povided by E. Schödinge, summaizing the desciption of quantum systems in tems of pobability amplitudes o wave-functions, a evolutionay concept at that time and still an enigmatic concept today. A physical (and heuistic) intepetation to this equation was povided by M. Bon, nowadays known as the standad o Copenhagen intepetation. This theoy emains, to this day, the most utilized appoach to the study and compehension of quantum systems. But this is a athe incomplete stoy. As a matte of fact, ight afte the bith of the Schödinge equation, othe fomulations of quantum mechanics appeaed which geatly helped in shedding light on aspects that wee hadly undestandable othewise. Among these altenatives, the wok Page 1 of 7

2 of E. Wigne stands out, being an intuitive model which povides a diect connection between classical and quantum physics, due to its stong similaities with classical statistical mechanics, and descibing systems in tems of (quasi) distibution functions, a concept expeimentalists ae quite familia with [1], [2]. Based on a athe unconventional intepetation of the Wigne appoach to quantum mechanics, a peculia and pomising new fomulation involving signed paticles has ecently come into existence [2]. This novel theoy povides a set of few ules which, applied ecusively, allows time-dependent simulations of single- and many-body poblems, consideed to be one of the most computationally demanding poblem in Physics. The theoy offes significant advantages in tems of intuitiveness, computational implementation and paallelization. In fact, despite its elatively ecent appeaance, this appoach has aleady poven to be a vey vesatile and valuable tool fo scientists. It has aleady been utilized to explain, e.g., the appeaance of quantum decoheence in silicon mateial, and the appeaance of Femi (o exchange-coelation) holes fo indistinguishable electons, two daunting poblems in the standad appoach. This pape intoduces, fom a simplified (but still accuate) pespective the signed paticle fomulation of quantum mechanics. These ae exciting times fo quantum mechanics, and the signed paticle fomulation pomises to evolutionize, once again, ou undestanding of the quantum wold. The Signed Paticle Fomulation The signed paticle fomulation consists of a set of thee ules given below (which can be seen eithe as a physical intepetation of the Wigne equation o as a genealization of the Wigne Monte Calo method to an infinite domain and non-discetized phase-space). Rule I. Physical systems can be descibed by means of (vitual) Newtonian paticles, i.e. povided with a position x and a momentum p simultaneously, which cay a sign which can be positive o negative. Rule II. A signed paticle, evolving in a potential V = V (x), behaves as a field-less classical point-paticle which, duing the time inteval dt, ceates a new pai of signed paticles with a pobability γ ( x ( t) )dt whee V W ; + and ( x p) γ + ( x) = lim ( x M p) + p V ; 0 W M = is the positive pat of the quantity known as the Wigne kenel. If, at the moment of ceation, the paent paticle has sign s, position x and momentum p, the new paticles ae both located in x, have signs + s and s, and momenta p + p' and p p ' espectively, with p ' chosen andomly accoding to the (nomalized) pobability + V W γ ( x; p) ( x). Page 2 of 7

3 This ule can be simplified by saying that a paticle inteacts with an extenal potential by simply ceating a new pai of signed paticles andomly (although the andomness is given by an explicit mathematical expession) at the same position of the paent paticle. Rule III. Two paticles with opposite sign and same phase-space coodinates annihilate. The physical pictue offeed by this set of ules is athe peculia and diffeent than any othe mathematical fomulation of quantum mechanics. Quantum systems ae now descibed by means of ensembles of Newtonian field-less paticles which now cay a sign and inteact with an extenal potential by means of ceation and annihilation events only. When a pai of paticles is ceated, one is in an expeimentally eachable state (positive sign), and the othe in a noneachable state (negative sign) [3]. This new view point is elatively easy to gasp and allows the inclusion of quite complex effects in a natual way when it is time to simulate quantum systems in a ealistic context. This athe atypical pictue actually can ecove typical quantum phenomena histoically descibed by the standad appoach. Let us now see a few examples. The tunneling effect [2]. Tunneling is a typical quantum effect which cannot be explained in tems of classical mechanics. It epesents one of the foundational expeimental evidence fo the need of a quantum theoy. This expeiment shows that mateial paticles can tunnel though potential baies, even though the initial paticle enegy is classically not sufficient. Fig. 1 shows that such expeimental obsevation can be epoduced by the signed paticle fomulation. Entangled paticles [4]. It is a well-known fact that two entangled paticles can be simulated coectly only by a full many-body appoach. Indeed, methods such as the density functional theoy, while being vey successful fo non-stongly coelated system, completely fail to descibe entangled systems. Fig. 2 shows that the many-body vesion of the signed paticle fomulation can handle natually such kind of systems. The exclusion pinciple [5]. By pinching two electons with same enegy against each othe a lowe pobability is developed in the cental aea of the phase-space, peventing the two paticles to be in the same position with the same enegy. Fig. 3 poofs the pesence of the Pauli exclusion pinciple which, essentially, states that two paticles with the same spin cannot be in the same obital at the same time. Conclusions In this aticle, the signed paticle fomulation of quantum mechanics has been intoduced which consists of a set of thee ules that completely descibe the time evolution of quantum systems. Successful applications of the new appoach to quantum tunneling, entangled paticles and identical electons have been shown. Clealy, still a lot emains to be exploed fom both a theoetical and computational point of view. Fo instance, the definition of the function γ = γ ( x ) does not pevent the divegence of the seies and futhe mathematical investigation is needed. At the moment, fo pactical puposes, the seies is tuncated ove a finite phase-space. Futhemoe, the classical limit of the new theoy is still unde analysis. Despite of all, the autho thinks that, based on the pomising esults epoted in this and othe aticles, it is a vey exciting Page 3 of 7

4 time fo those scientists who want to use an altenative appoach to the standad fomulation of quantum mechanics. Computational Aspects The code utilized to simulate the examples epoted in this aticle is nano-achimedes [6], a GNU package available on-line unde GPL. The eade is stongly encouaged to download it in ode to duplicate the esults shown. This code is entiely developed in C and optimized to get the best pefomance fom the hadwae. It can un on both seial and paallel machines exploiting the OpenMP standad libay. The esults pesented in Figs. 2 and 3 have been obtained using the HPC cluste deployed at the Institute of Infomation and Communication Technologies of the Bulgaian Academy of Sciences. This cluste consists of two acks which contain HP Cluste Platfom Expess 7000 enclosues with 36 blades BL 280c with dual Intel Xeon 2.8 Ghz (total 576 coes), 24 GB RAM pe blade. Thee ae 8 stoage and management contolling nodes 8 HP DL 380 G6 with dual Intel 2.8 Ghz and 32 GB RAM. All these seves ae inteconnected via nonblocking DDR Infiniband inteconnect at 20Gbps line speed. The theoetical peak pefomance is 3.23 Tflops. Refeences and Notes 1. E. Wigne, On the quantum coection fo themodynamic equilibium, Phys. Rev. 40, 749 (1932). 2. J.M. Sellie, M. Nedjalkov, I. Dimov, An intoduction to applied quantum mechanics in the Wigne Monte Calo fomalism, Physics Repots 577, 1-34, (2015). 3. D. Leibfied, T. Pfau, C. Monoe, Shadows and mios: econstucting quantum states of atom motion, Physics Today, Apil (1998). 4. J.M. Sellie, I. Dimov, The many-body Wigne Monte Calo method fo time-dependent ab-initio quantum simulations, Jounal of Computational Physics 273, (2014). 5. J.M. Sellie, I. Dimov, On the simulation of indistinguishable Femions in the many-body Wigne fomalism, Jounal of Computational Physics 280, (2015) Acknowledgments The authos would like to thank Pof. M. Nedjalkov fo the vey fuitful convesations. This wok has been suppoted by the poject EC AComIn (FP7-REGPOT ). Page 4 of 7

5 Figues and Tables Figue 1. Quantum tunneling effect at diffeent times. A Gaussian wave packet tavels towads a potential baie (symbolized by a ed line). The dynamics is shown in the phase-space. Pat of the wave-packet is eflected (lowe pat of the plot) while the othe pat tunnels though the baie (uppe pat of the plot). Page 5 of 7

6 Figue 2. Simulation of two entangled Gaussian wave-packets. The educed quasi-distibution function is shown at time 20 fs. A otation in the phase-space of both packets and entanglement (oscillations between the packets) is clealy obsevable. Page 6 of 7

7 Figue 3. Simulation of two indistinguishable Femions (electons) squeezed one against the othe while having the same enegy. The fomation of a Femi (o exchange-coelation) hole at time 2.5 fs is clealy visible. This is a stong evidence of the pesence of the Pauli exclusion pinciple. Page 7 of 7