On a Simple Derivation of the Effect of Repeated. Measurements on Quantum Unstable Systems by Using. the Regularized Incomplete -Function

Transcription

1 Adv. Studies Theo. Phys., Vol. 6, 0, no. 5, 07-3 On a Simple Deivation of the Effect of Repeated Measuements on Quantum Unstable Systems by Using the Regulaized Incomplete -Function Elio Conte Depatment of Neuological Sciences; Depatment of Phamacology and human physiology-tires, Cente of Innovative Technologies fo Signal Detection and Pocessing, Univesity of Bai, Italy School of Advanced Intenational Studies on Applied Theoetical and non Linea Methodologies of Physics-Bai, Italy elio.conte@fastwebnet.it Abstact: a simple deivation of the effect induced fom epeated measuements on quantum unstable systems is obtained by using the egulaized incomplete - function. Keywods: quantum mechanics, quantum Zeno effect. Regulaized incomplete - function Intoduction The quantum Zeno effect is a name that was intoduced by Geoge Sudashan and Baidyanath Misa in 977 when they analyzed the physical condition that an unstable quantum system (as example the decay of a adioactive souce ), if obseved continuously, will neve decay []. The meaning of the tem has since expanded, leading to a moe technical definition in which time evolution can be suppessed not only by measuement. Actually, the quantum Zeno effect seems to be the suppession of the unitay time evolution caused by quantum decoheence in quantum systems povided by a vaiety of souces: measuement, inteactions with the envionment, stochastic fields, and so on []. As an outgowth of study of the quantum Zeno effect, it has become clea

2 08 E. Conte that applying a seies of sufficiently stong and fast pulses with appopiate symmety can also decouple a system fom its decoheing envionment [3] Afte the initial appoach of Geoge Sudashan and Baidyanath Misa a lot of phenomena has been discussed in liteatue and elating such quantum baic effect. In ode to cove all of these phenomena (including the oiginal effect of suppession of quantum decay), the Zeno effect can be defined as a class of phenomena in which some tansition is suppessed by an inteaction. It allows the intepetation of the esulting state in the tems of a tansition that did not yet happen and tansition has aleady occued, o by the statement that the evolution of a quantum system is halted if the state of the system is continuously measued by a macoscopic device to check whethe the system is still in its initial state.[3] Closely elated (and sometimes not distinguished fom the quantum Zeno effec is the so called watchdog effect, in which the time evolution of a system is affected by its continuous coupling to the envionment.[4,5] The aim of the pesent pape is to pesent a vey simple but consistent deivation of the effect induced on unstable quantum systems using epeated obsevations. In ou pevious studies we attempted to conside the possible advances in the application of quantum mechanics in neuoscience and psychology. As outlined in paticula by Stapp, human cognition may be diectly involved in aising of such so much extaodinay quantum effect. Theoetical Elaboation Conside an unstable quantum system descibed by the usual quantum supeposition of states c c (.) with ( undecayed c( p (.) decayed pobability fo the system to be in the undecayed state, and c p (.3) pobability fo the system to be in the decayed state, and p p (.4) ( Let us admit that we pefom (n) obsevations, being thus (n) an intege value.. Conside the tivial expession p p n (.5) The fist tem in the expansion of this fomula gives the pobability to find the system undecayed afte (n) obsevations at a time t, the second tem epesents that one decay has occued, and the (+) tem gives the pobability that decays have occued.

3 Quantum unstable systems 09 We have that n n y p np ( p... n C p ( p ) = ( n ni i n C i p ( p( ) (.6) i0 The inteesting and unexpected featue of the elaboation is that we ae inteested now to calculate the deivation of the calculated y ( function espect to p (. We obtain that dy( n ( n ) nci p ( p( (.7) dp The integation of the (.3) gives p n y( ( n ) nc p ( p( ) dp This is the incomplete -function. The complete function is 0 (.8) p q ( p )!( q )! ( p, q) p ( p( ) dp (.9) ( p q )! 0 so that, being ( n )!! ( n, ) (.0) n! and n! ( n ) nc (.) ( n )!! we have finally that in the computation of the coesponding suviving function, when suviving no moe than decays, is given fom the egulaized incomplete - function x ( n, ) y( (.) ( n, ) with x p and y ( becoming thus a function of n and of. We have obtained an elegant deivation on the manne in which obsevations affect the decay of the consideed quantum unstable system by using the egulaized -function. Inteestingly we may emembe hee that the egulaized incomplete beta function is the cumulative distibution function of the Beta distibution, and is elated to the cumulative distibution function of a andom vaiable X fom a binomial distibution, whee the "pobability of success" is p and the sample size is n: To complete the exposition we intend to give biefly some comment in the field of applied physics. In the last decade we have advanced a numbe of esults in the diect application of quantum mechanics at cognitive level. In detail, we have exploed the possible logical oigin of quantum mechanics, consequently its possible ole in

4 0 E. Conte explaining human cognitive pefomance, and we have also given a numbe of expeimental confimations evidencing that quantum intefeence aises duing peception- cognition in humans [ 6, 3 ]. It seems that an ulteio evidence is eached by finding the induced effect of epeated measuements on unstable quantum systems. In fact, the quantum Zeno effect (with its own contovesies elated to the poblem of measuemen is becoming a cental concept in the exploation of contovesial theoies of quantum mind consciousness within the discipline of cognitive science. In his book Mindful Univese (007), Heny Stapp [ 4 ] claims that the mind holds the bain in a supeposition of states using the quantum Zeno effect. He advances that this phenomenon is the pincipal method by which the conscious can effect change, a possible solution to the mind-body dichotomy. Refeences ) E.C.G. Sudashan, B. Misa, (977). The Zeno s paadox in quantum theoy, Jounal of Mathematical Physics 8 (4): (977) ) Nakanishi T., Yamane K., M. Kitano M.: Absoption-fee optical contol of spin systems: the quantum Zeno effect in optical pumping Phys. Rev. A 65, (00), ) Panov A.D.(00). "Quantum Zeno effect in spontaneous decay with distant detecto". Physics Lettes A 8: 9. (00); axiv:quant-ph/ ) Patha Ghose (999). Testing Quantum Mechanics on New Gound. Cambidge Univesity Pess. p. 07. ISBN ) Auletta G. (000). Foundations and Intepetation of Quantum Mechanics. Wold Scientific. (000),pages 34. ISBN [6] Conte E. An investigation on the basic conceptual foundations of quantum mechanics by using Cliffod algeba, Advanced Studies Theoetical Physics, 5,n., (0), [7]Conte E. A efomulation of von Neumann-spostulates on quantum measuement by using two theoems in Cliffod Algeba, Int. Joun. Theo. Phys., (00) 49: DOI > 0.007&s0773/009/039/z [8] Conte E.,. Khennikov A.Y.,Todaello O., Fedeici A., Mendolicchio L., Zbilut J.P., Mind states follow quantum mechanics duing peception and cognition of ambiguous figues, Jounal of Open Systems and Infomation Dynamics, 6, (009), -7

5 Quantum unstable systems [9] Conte E.,Todaello O., Lateza V.,Khennilov A.Y., Mendolicchio L.,Fedeici A., A Peliminay Expeimental Veification of Violation of Bell inequality in a Quantum Model of Jung Theoy of Pesonality Fomulated with Cliffod Algeba, Jounal of Consciousness Exploation and Reseach,, 7, (00) , 00 [0] Conte E., Testing quantum consciousness, Neuoquantology, 6,,(008) 6-39, 008. [] Conte E., Todaello O., Khennikov A.Y., Fedeici A.,Zbilut J.P,A peliminay expeimental veification on the possibility of Bell inequality violation in mental states, Neuoquatology, vol.6, issue 3,(008) 4-. [] Conte E., On the logical oigins of quantum mechanics demonstated by using Cliffod algeba: a poof that quantum intefeence aises in a Cliffod algebaic fomulation of quantum mechanics, Electonic Jounal of theoetical physics, EJTP 8, n.5, 09-6, 0 [3]Conte E. What is the Reason to Use Cliffod Algeba in Quantum Cognition: It Fom Qubit : On the Possibility that the Amino Acids can Discen Between Two Quantum Spin States., Neuoquantology, Vol.0,3,(0) [4] Conte E. A bief note of time evolution of quantum wave function and of quantum pobabilities duing peception and cognition of human subjects, Neuoquantology, Vol. /,3 (009) [5]Conte E. On the logical oigins of quantum mechanics demonstated by using Cliffod algeba, Neuoquantology, 9,, 3-75, 0 [6] Conte E., On the possibility that we think in a quantum pobabilistic manne, Neuoquatontology, Vol.8,4 (00), [7] Conte E. Khennikon A, Fedeici A, Todaello A. Lateza V, May we veify on existing dispesdion fee ensembles by application of quantum mechanics in ecxpeiments at peceptive-cognitive level?, Neuoquantology, Vol.0, (0),4-9 [8] Conte E., Khennikov A.Y., Todaello O., Fedeici A., Zbilut J.P. On the existence of quantum wave function and quantum intefeence effects in mental states : an expeimental confimation duing peception and cognition in humans, Neuoquatology, 7, (009),04-0 [9] Conte E., Piei GP, Fedeici A., Mendolicchio L., Zbilut J.P., A model of biological neuon with teminal chaos and quantum like featues, Chaos, Solitons and Factals, 30, , 006

6 E. Conte [0] E. Conte E., O. Todaello O., V. Lateza L., A. Khennilov A.Y., L. Mendolicchio L., A Fedeici A., A Peliminay Expeimental Veification of Violation of Bell inequality in a Quantum Model of Jung Theoy of Pesonality Fomulated with Cliffod Algeba, Jounal of Consciousness Exploation and Reseach,, 7, (00), [] Conte E.,Todaello O.Fedeici A.,Vitello F.,Lopane M.,Khennikov A.Y.,Zbilut J.P. Some emaks on an expeiment suggesting quantum-like behaviou of cognitive entities and fomulation of an abstact quantum mechanical fomalism to descibe cognitive entity and Its dynamics, Chaos, Solitons and Factals, 3,(007), []Conte E.,Santacoce N., Fedeici A, Possibile quantum model of consciousness intefaced with a non Lipschits chaotic dynamics Pat I, Intenational Jounal of Consciousness exploation and eseach, Vol3,n.8,(0),905-9 [3] Conte E.,Santacoce N., Fedeici A, Possibile quantum model of consciousness intefaced with a non Lipschits chaotic dynamics Pat II, Intenational Jounal of Consciousness exploation and eseach, Vol3,n.8,(0),9-937 Futhe eadings: E. Conte. On etocollapse in quantum mechanics, Lettee al Nuovo Cimento, vol. 3 (), , 98 Conte E., A pedictive model of collapse-etocollapse of quantum mechanics, Lettee al Nuovo Cimento, vol.3 (9), 86-88, 98 [ Conte E., Biological obsevables via quantum mechanics, Cibenetica, vol. XXVI (), 6-74, 983 [Conte E., Exploation of biological function by quantum mechanics using biquatenions, in Le Concept of Oganization en Cybenetique, Association Intenational de Cybenetique, Namu (Belgique), 6-3, 983 Conte E., A poposed expeimental test to veify the existence of the biological obsevables, Cibenetica, vol. XXX, 4, 5-8, 987 Conte E., An example of wave packet eduction using biquatenions, Physics Essays, vol. 6, 4, -7, 993 Conte E., Wave function collapse in quantum mechanics, Physics Essays, vol. 7, , 994 Conte E., A citicism of Fine s solution to the quantum measuement poblem: a new epistemology fo quantum mechanics, Physics Essays, vol. 9,, -3, 996

7 Quantum unstable systems 3 Conte E., Khennikov A.Y., Zbilut J.P., The tansition fom ontic potentiality to actualization of states in quantum mechanical appoach to eality: The Poof of a Mathematical Theoem to Suppot It, axiv:quant-ph/ Conte E., Khennikov A.,Zbilut J.P., The Tansition fom ontic potentiality to actualization of states in quantum mechanical appoach to eality: a poof of a mathematical theoem suppoting it, Axiv.og quant-ph/ Conte E., G.P Piei G.P., L. Mendolicchio L., A. Khennikov A., J.P. Zbilut JP, On some detailed examples of quantum like stuctues containing quantum potential states in the sphee of the biological dynamics, Axiv.og physics/ [4]Schwatz J.M., H.P. Stapp H.P. Beauegad M.(005),Quantum physics in neuoscience and psychology: a neuophysical model of mind-bain inteaction, Philosophical Tansactions of the Royal Society of London B Volume 360, Numbe 458 / June 9, (005) Pages Received: August, 0