DOCSLIB.ORG

Short Time Quantum Propagator and Bohmian Trajectories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Short Time Quantum Propagator and Bohmian Trajectories Short Time Quantum Propagator and Bohmian Trajectories Maurice de Gosson∗ Universit¨at Wien, Fakult¨at f¨ur Mathematik, NuHAG Wien 1090 Basil Hiley University of London Birkbeck College, Theoretical Physics Unit London WC1E 7HX September 11, 2018 Abstract We begin by giving correct expressions for the short-time action; following the work of one of us and Makri–Miller. We use these es- timates to derive a correct expression modulo ∆t2 for the quantum propagator and we show that the quantum potential is negligible mod- ulo ∆t2 for a point source. We finally prove that this implies that the quantum motion is classical for very short times. Keywords: generating function, short-time action, quantum potential, Bohmian trajectories arXiv:1304.4771v1 [quant-ph] 17 Apr 2013 1 Introduction Explicit approximate expressions for the short-time action play an essential role in various aspects of quantum mechanics (for instance the Feynman path integral, or semi-classical mechanics), and so does the associated Van Vleck determinant. Unfortunately, as already observed by Makri and Miller [15, 16], the literature seems to be dominated by formulas which are wrong ∗Financed by the Austrian Research Agency FWF (Projektnummer P20442-N13). 1 even to the first order of approximation! Using the calculations of Makri and Miller, which were independently derived by one of us in [3] using a slightly different method, we show that the correct approximations lead to precise estimates for the short-time Bohmian quantum trajectories for an initially sharply located particle. We will see that these trajectories are classical to the second order in time, due to the vanishing of the quantum potential for small time intervals. In this paper we sidestep the philosophical and ontological debate around the “reality” of Bohm’s trajectories and rather focus on the mathematical issues. 2 Bohmian trajectories Consider a time-dependent Hamiltonian function n p2 H(x,p,t)= j + U(x,t) (1) =1 2mj Xj and the corresponding quantum operator n ~2 2 ~ ∂ H(x, i x,t)= − 2 + U(x,t). (2) − ∇ =1 2mj ∂xj Xj b The associated Schr¨odinger equation is ∂Ψ i~ = H(x, i~ ,t)Ψ , Ψ(x, 0)=Ψ0(x). (3) ∂t − ∇x Let us write Ψ in polarb form ReiΦ/~; here R = R(x,t) 0 and Φ= S(x,t) ~ ≥ are real functions. On inserting ReiS/ into Schr¨odinger’s equation and separating real and imaginary parts, one sees that the functions R and S satisfy, at the points (x,t) where R(x,t) > 0, the coupled system of non- linear partial differential equations n 2 n ∂S 1 ∂S ~2 ∂2R + + U(x,t) 2 = 0 (4) ∂t =1 2mj ∂xj − =1 2mjR ∂xj Xj Xj 2 n ∂R 1 ∂ 2 ∂Φ + R = 0. (5) ∂t =1 mj ∂xj ∂xj Xj 2 The crucial step now consists in recognizing the first equation as a Hamilton– Jacobi equation, and the second as a continuity equation. In fact, introduc- ing the quantum potential n 2 2 Ψ ~ ∂ R Q = 2 (6) − =1 2mjR ∂xj Xj (Bohm and Hiley [1]) and the velocity field Ψ 1 ∂Φ 1 ∂Φ v (x,t)= , ..., (7) m1 ∂x1 m ∂x n n the equations (4) and (5) become ∂Φ Ψ + H(x, Φ,t)+ Q (x,t) = 0 (8) ∂t ∇x ∂ρ Ψ 2 + div(ρv )=0 , ρ = R (9) ∂t The main postulate of the Bohmian theory of motion is that particles follow quantum trajectories, and that these trajectories are the solutions of the differential equations Ψ ~ 1 ∂Ψ x˙ j = Im . (10) mj Ψ ∂xj The phase space interpretation is that the Bohmian trajectories are de- termined by the equations Ψ Ψ 1 Ψ Ψ ∂U Ψ ∂Q Ψ x˙ j = pj ,p ˙j = (x ,t) (x ,t). (11) mj −∂xj − ∂xj It is immediate to check that these are just Hamilton’s equations for the Hamiltonian function n 2 Ψ p Ψ H (x,p,t)= j + U(x,t)+ Q (x,t) (12) =1 2mj Xj which can be viewed as a perturbation of the original Hamiltonian H by the quantum potential QΨ (see Holland [13, 14] for a detailed study of quantum trajectories in the context of Hamiltonian mechanics). The Bohmian equations of motion are a priori only defined when R = 0 6 (that is, outside the nodes of the wavefunction); this will be the case in our constructions since for sufficiently small times this condition will be satisfied by continuity if we assume that it is case at the initial time. An important feature of the quantum trajectories is that they cannot cross; thus there will be no conjugate points like those that complicate the usual Hamiltonian dynamics. 3 3 The Short-Time Propagator The solution Ψ of Schr¨odinger’s equation (3) can be written Ψ(x,t)= K(x,x0; t)Ψ0(x0)dx0 Z where the kernel K is the “quantum propagator”: K(x,x0; t)= x exp( iHt/~) x0 . h | − | i Schr¨odinger’s equation (3) is then equivalentb to ∂K i~ = H(x, i~ ,t)K , K(x,x0;0) = δ(x x0) (13) ∂t − ∇x − where δ is the Diracb distribution. Physically this equation describes an isotropic source of point-like particle emanating from the point x0 at initial time t0 = 0. We want to find an asymptotic formula for K for short time intervals ∆t. Referring to the usual literature, such approximations are given by expressions of the type 2 1 n/ i K(x,x0;∆t)= ρ(x,x0;∆t)exp S(x,x0;∆t) 2πi~ ~ p where S(x,x0;∆t) is the action along the classical trajectory from x0 to x in time ∆t and 2 ∂ S(x,x0;∆t) ρ(x,x0;∆t) = det − ∂x ∂x j k 1≤j,k≤n is the corresponding Van Vleck determinant. It is then common practice (especially in the Feynman path integral literature) to use the following “midpoint approximation” for the generating function S: n mj 2 1 S(x,x0;∆t) (xj x0) (U(x,t0)+ U(x0,t0))∆t (14) ≈ =1 2∆t − − 2 Xj or, worse, n mj 2 1 S(x,x0; t,t0) (xj x0) (U( 2 (x + x0),t0)∆t (15) ≈ =1 2∆t − − Xj However, as already pointed out by Makri and Miller [15, 16], these “approx- imations” are wrong; they fail to be correct even to first orderin∆t! In fact, 4 Makri and Miller, and one of us [3] have shown independently, and using different methods, that the correct asymptotic expression for the generating function is given by n mj 2 2 S(x,x0;∆t)= (xj x0) U(x,x0)∆t + O(∆t ) (16) =1 2∆t − − Xj e where U(x,x0, 0) is the average value of the potential over the straight line joining x0 at time t0 to x at time t with constant velocity: e 1 U(x,x0)= U(λx + (1 λ)x0, 0)dλ. (17) 0 − Z For instance whene 1 2 2 2 2 H(x,p)= (p + m ω x ) 2m is the one-dimensional harmonic oscillator formula (16) yields the correct expansion 2 m 2 mω 2 2 S(x,x0; t)= (x x0) (x + xx0 + x0)∆t + O(∆t ); (18) 2∆t − − 6 the latter can of course be deduced directly from the exact value mω 2 S(x,x0; t,t0)= ((x + x0) cos ω∆t 2xx0) (19) 2sin ω∆t − by expanding sin ω∆t and cos ω∆t for ∆t 0. This correct expression is → of course totally different from the erroneous approximations obtained by using the “rules” (14) or (15). Introducing the following notation, n 2 (xj x0) S(x,x0;∆t)= mj − U(x,x0)∆t, (20) =1 2∆t − Xj e e leads us to the Makri and Miller approximation (formula (17c) in [15]) for the short-time propagator: n/2 1 i 2 K(x,x0;∆t)= ρ(x,x0;∆t)exp S(x,x0;∆t) + O(∆t ) 2πi~ ~ p (21) e 5 where 2 ∂ S(x,x0;∆t) ρ(x,x0;∆t) = det . ∂x ∂x − j k !1 e ≤j,k≤n It turns out that this formula can be somewhat improved. The Van Vleck determinant ρ(x,x0;∆t) is explicitly given, taking formula (20) into account, by 1 ′′ ρ(x,x0;∆t) = det M U (x,x0)∆t −∆t − x,x0 where M is the mass matrix (the diagonal matrixe with positive entries the masses mj) and 2 ∂ U(x,x0) U ′′ = . x,x0 ∂x ∂x − j k !1 e ≤j,k≤n Writing e 1 ′′ 1 −1 ′′ 2 M U (x,x0)∆t = M[I × M U (x,x0)∆t ] −∆t − x,x0 −∆t n n − x,x0 1 2 e = M[I × + O(∆te)], −∆t n n we have by taking the determinant of both sides m1 mn 2 ρ(x,x0;∆t)= · · · det I × + O(∆t ) . (∆t)n n n 2 2 Noting that det In×n + O(∆t ) =1+ O(∆t ), we thus have m1 mn 2 ρ(x,x0;∆t)= · · · (1 + O(∆t )). (22) (∆t)n Writing m1 m ρ(∆t)= · · · n (23) (∆t)n which is just the Van Vleck density for the free particle Hamiltonian. We e thus have 2 ρ(x,x0;∆t)= ρ(∆t)+ O(∆t ) (24) and hence we can rewrite formula (21) as e n/2 1 i 2 K(x,x0;∆t)= ρ(∆t)exp S(x,x0;∆t) + O(∆t ). (25) 2πi~ ~ p We will see below that this formulae allows ane easy study of the quantum potential for K.
Load more

Recommended publications
  • Arxiv:0908.1787V1 [Quant-Ph] 13 Aug 2009 Ematical Underpinning
    August 13, 2009 18:17 Contemporary Physics QuantumPicturalismFinal Contemporary Physics Vol. 00, No. 00, February 2009, 1{32 RESEARCH ARTICLE Quantum picturalism Bob Coecke∗ Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD Oxford, UK (Received 01 02 2009; final version received XX YY ZZZZ) Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the `no-cloning theorem' is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of `quantum teleportation'? We claim that the quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is `low-level'. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation': it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols.
    [Show full text]
  • Dynamical Relaxation to Quantum Equilibrium
    Dynamical relaxation to quantum equilibrium Or, an account of Mike's attempt to write an entirely new computer code that doesn't do quantum Monte Carlo for the first time in years. ESDG, 10th February 2010 Mike Towler TCM Group, Cavendish Laboratory, University of Cambridge www.tcm.phy.cam.ac.uk/∼mdt26 and www.vallico.net/tti/tti.html [email protected] { Typeset by FoilTEX { 1 What I talked about a month ago (`Exchange, antisymmetry and Pauli repulsion', ESDG Jan 13th 2010) I showed that (1) the assumption that fermions are point particles with a continuous objective existence, and (2) the equations of non-relativistic QM, allow us to deduce: • ..that a mathematically well-defined ‘fifth force', non-local in character, appears to act on the particles and causes their trajectories to differ from the classical ones. • ..that this force appears to have its origin in an objectively-existing `wave field’ mathematically represented by the usual QM wave function. • ..that indistinguishability arguments are invalid under these assumptions; rather antisymmetrization implies the introduction of forces between particles. • ..the nature of spin. • ..that the action of the force prevents two fermions from coming into close proximity when `their spins are the same', and that in general, this mechanism prevents fermions from occupying the same quantum state. This is a readily understandable causal explanation for the Exclusion principle and for its otherwise inexplicable consequences such as `degeneracy pressure' in a white dwarf star. Furthermore, if assume antisymmetry of wave field not fundamental but develops naturally over the course of time, then can see character of reason for fermionic wave functions having symmetry behaviour they do.
    [Show full text]
  • Quantum Trajectories: Real Or Surreal?
    entropy Article Quantum Trajectories: Real or Surreal? Basil J. Hiley * and Peter Van Reeth * Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK * Correspondence: [email protected] (B.J.H.); [email protected] (P.V.R.) Received: 8 April 2018; Accepted: 2 May 2018; Published: 8 May 2018 Abstract: The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration. Keywords: Stern-Gerlach; trajectories; spin 1. Introduction The recent claims to have observed “photon trajectories” [1–3] calls for a re-examination of what we precisely mean by a “particle trajectory” in the quantum domain. Mahler et al. [2] applied the Bohm approach [4] based on the non-relativistic Schrödinger equation to interpret their results, claiming their empirical evidence supported this approach producing “trajectories” remarkably similar to those presented in Philippidis, Dewdney and Hiley [5]. However, the Schrödinger equation cannot be applied to photons because photons have zero rest mass and are relativistic “particles” which must be treated differently.
    [Show full text]
  • The Stueckelberg Wave Equation and the Anomalous Magnetic Moment of the Electron
    The Stueckelberg wave equation and the anomalous magnetic moment of the electron A. F. Bennett College of Earth, Ocean and Atmospheric Sciences Oregon State University 104 CEOAS Administration Building Corvallis, OR 97331-5503, USA E-mail: [email protected] Abstract. The parametrized relativistic quantum mechanics of Stueckelberg [Helv. Phys. Acta 15, 23 (1942)] represents time as an operator, and has been shown elsewhere to yield the recently observed phenomena of quantum interference in time, quantum diffraction in time and quantum entanglement in time. The Stueckelberg wave equation as extended to a spin–1/2 particle by Horwitz and Arshansky [J. Phys. A: Math. Gen. 15, L659 (1982)] is shown here to yield the electron g–factor g = 2(1+ α/2π), to leading order in the renormalized fine structure constant α, in agreement with the quantum electrodynamics of Schwinger [Phys. Rev., 73, 416L (1948)]. PACS numbers: 03.65.Nk, 03.65.Pm, 03.65.Sq Keywords: relativistic quantum mechanics, quantum coherence in time, anomalous magnetic moment 1. Introduction The relativistic quantum mechanics of Dirac [1, 2] represents position as an operator and time as a parameter. The Dirac wave functions can be normalized over space with respect to a Lorentz–invariant measure of volume [2, Ch 3], but cannot be meaningfully normalized over time. Thus the Dirac formalism offers no precise meaning for the expectation of time [3, 9.5], and offers no representations for the recently– § observed phenomena of quantum interference in time [4, 5], quantum diffraction in time [6, 7] and quantum entanglement in time [8]. Quantum interference patterns and diffraction patterns [9, Chs 1–3] are multi–lobed probability distribution functions for the eigenvalues of an Hermitian operator, which is typically position.
    [Show full text]
  • Bohmian Mechanics Versus Madelung Quantum Hydrodynamics
    Ann. Univ. Sofia, Fac. Phys. Special Edition (2012) 112-119 [arXiv 0904.0723] Bohmian mechanics versus Madelung quantum hydrodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, Bulgaria It is shown that the Bohmian mechanics and the Madelung quantum hy- drodynamics are different theories and the latter is a better ontological interpre- tation of quantum mechanics. A new stochastic interpretation of quantum me- chanics is proposed, which is the background of the Madelung quantum hydro- dynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum po- tential. It describes the quantum evolution of the probability density by a con- vective diffusion with imaginary transport coefficients. The Copenhagen interpretation of quantum mechanics is guilty for the quantum mys- tery and many strange phenomena such as the Schrödinger cat, parallel quantum and classical worlds, wave-particle duality, decoherence, etc. Many scientists have tried, however, to put the quantum mechanics back on ontological foundations. For instance, Bohm [1] proposed an al- ternative interpretation of quantum mechanics, which is able to overcome some puzzles of the Copenhagen interpretation. He developed further the de Broglie pilot-wave theory and, for this reason, the Bohmian mechanics is also known as the de Broglie-Bohm theory. At the time of inception of quantum mechanics Madelung [2] has demonstrated that the Schrödinger equa- tion can be transformed in hydrodynamic form. This so-called Madelung quantum hydrodynam- ics is a less elaborated theory and usually considered as a precursor of the Bohmian mechanics. The scope of the present paper is to show that these two theories are different and the Made- lung hydrodynamics is a better interpretation of quantum mechanics than the Bohmian me- chanics.
    [Show full text]
  • Infinite Potential Show Title: Generic Show Episode #101 Date: 3/17/20 Transcribed by Daily Transcription Tre
    INFINITE POTENTIAL SHOW TITLE: GENERIC SHOW EPISODE #101 DATE: 3/17/20 TRANSCRIBED BY DAILY TRANSCRIPTION_TRE 00:00:28 DAVID BOHM People talked about the world, and I said, "Where does it end?" and, uh, uh... MALE And what answer did they give? DAVID BOHM Well, they-I never, uh, I don't remember. It wasn't terribly convincing I suppose. [LAUGH] DR. DAVID C. It was, uh, a night, and we were SCHRUM walking under the stars, black sky, and he looked up to the stars, and he said, "Ordinarily, when we look to the sky and look to the stars, we think of stars as objects far out and vast spaces between them." He said, "There's another way we can look at it. We can look at the vacuum at the emptiness instead as a planum, as infinitely full rather than infinitely empty. And that the material objects themselves are like little bubbles, little vacancies in this vast sea." 00:01:25 DR. DAVID C. So David Bohm, in a sense, was SCHRUM using that view to have me look at the stars and to have a sense of the night sky all of a sudden in a different way, as one whole living organism and these little bits that we call matter as sort to just little holes in it. DR. DAVID C. He often mentioned just one other SCHRUM aspect of this, that this planum, in a cubic centimeter of the planum, there's more, uh, energy matter than in the entire visible Page 2 of 62 universe.
    [Show full text]
  • Pilot-Wave Theory, Bohmian Metaphysics, and the Foundations of Quantum Mechanics Lecture 8 Bohmian Metaphysics: the Implicate Order and Other Arcana
    Pilot-wave theory, Bohmian metaphysics, and the foundations of quantum mechanics Lecture 8 Bohmian metaphysics: the implicate order and other arcana Mike Towler TCM Group, Cavendish Laboratory, University of Cambridge www.tcm.phy.cam.ac.uk/∼mdt26 and www.vallico.net/tti/tti.html [email protected] – Typeset by FoilTEX – 1 Acknowledgements The material in this lecture is largely derived from books and articles by David Bohm, Basil Hiley, Paavo Pylkk¨annen, F. David Peat, Marcello Guarini, Jack Sarfatti, Lee Nichol, Andrew Whitaker, and Constantine Pagonis. The text of an interview between Simeon Alev and Peat is extensively quoted. Other sources used and many other interesting papers are listed on the course web page: www.tcm.phy.cam.ac.uk/∼mdt26/pilot waves.html MDT – Typeset by FoilTEX – 2 More philosophical preliminaries Positivism: Observed phenomena are all that require discussion or scientific analysis; consideration of other questions, such as what the underlying mechanism may be, or what ‘real entities’ produce the phenomena, is dismissed as meaningless. Truth begins in sense experience, but does not end there. Positivism fails to prove that there are not abstract ideas, laws, and principles, beyond particular observable facts and relationships and necessary principles, or that we cannot know them. Nor does it prove that material and corporeal things constitute the whole order of existing beings, and that our knowledge is limited to them. Positivism ignores all humanly significant and interesting problems, citing its refusal to engage in reflection; it gives to a particular methodology an absolutist status and can do this only because it has partly forgotten, partly repressed its knowledge of the roots of this methodology in human concerns.
    [Show full text]
  • Non-Locality in the Stochastic Interpretation of the Quantum Theory Annales De L’I
    ANNALES DE L’I. H. P., SECTION A D. BOHM Non-locality in the stochastic interpretation of the quantum theory Annales de l’I. H. P., section A, tome 49, no 3 (1988), p. 287-296 <http://www.numdam.org/item?id=AIHPA_1988__49_3_287_0> © Gauthier-Villars, 1988, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincare, Vol. 49, n° 3, 1988, 287 Physique theorique Non-Locality in the Stochastic Interpretation of the Quantum Theory D. BOHM Department of Physics, Birkbeck College (London University), Malet Street, London WC1E 7HX ABSTRACT. - This article reviews the stochastic interpretation of the quantum theory and shows in detail how it implies non-locality. RESUME . - Cet article presente une revue de 1’interpretation stochas- tique de la theorie quantique et montre en detail en quoi elle implique une non-localite. 1. INTRODUCTION It gives me great pleasure on the occasion of Jean-Pierre Vigier’s retire- ment to recall our long period of fruitful association and to say something about the stochastic interpretation of the quantum theory, in which we worked together in earlier days. Because of requirements of space, however, this article will be a condensation of my talk (*) which was in fact based on a much more extended article by Basil Hiley and me [1 ].
    [Show full text]
  • The Quantum Potential And''causal''trajectories For
    REFERENCE IC/88/161 \ ^' I 6 AUG INTERNATIONAL CENTRE FOR / Co THEORETICAL PHYSICS THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES A.O. Barut and M. Bo2ic INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/161 I. Introduction International Atomic Energy Agency During the last decade Bohm's causal trajectories based on the and concept of the so called "quantum potential" /I/ have been evaluated United Nations Educational Scientific and Cultural Organization in a number of cases, the double slit /2,3/, tunnelling through the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS barrier /4,5/, the Stern-Gerlach magnet /6/, neutron interferometer /8,9/, EPRB experiment /10/. In all these works the trajectories have been evaluated numerically for the time dependent wave packet of the Schrodinger equation. THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES * In the present paper we study explicitly the quantum potential and the associated trajectories for stationary states and for coherent states. We show how the quantum potential arises from the classical A.O. Barut ** and M. totii *** action. The classical action is a sum of two parts S ,= S^ + Sz> one International Centre for Theoretical Physics, Trieste, Italy. part Sj becomes the quantum action, the other part S2 is shifted to the quantum potential. We also show that the two definitions of causal ABSTRACT trajectories do not coincide for stationary states. We show for stationary states in a central potential that the quantum The content of our paper is as follows. Section II is devoted to action S is only a part of the classical action W and derive an expression for the "quantum potential" IL in terms of the other part.
    [Show full text]
  • Photon Wave Mechanics: a De Broglie - Bohm Approach
    PHOTON WAVE MECHANICS: A DE BROGLIE - BOHM APPROACH S. ESPOSITO Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Mostra d’Oltremare Pad. 20, I-80125 Napoli Italy e-mail: [email protected] Abstract. We compare the de Broglie - Bohm theory for non-relativistic, scalar matter particles with the Majorana-R¨omer theory of electrodynamics, pointing out the impressive common pecu- liarities and the role of the spin in both theories. A novel insight into photon wave mechanics is envisaged. 1. Introduction Modern Quantum Mechanics was born with the observation of Heisenberg [1] that in atomic (and subatomic) systems there are directly observable quantities, such as emission frequencies, intensities and so on, as well as non directly observable quantities such as, for example, the position coordinates of an electron in an atom at a given time instant. The later fruitful developments of the quantum formalism was then devoted to connect observable quantities between them without the use of a model, differently to what happened in the framework of old quantum mechanics where specific geometrical and mechanical models were investigated to deduce the values of the observable quantities from a substantially non observable underlying structure. We now know that quantum phenomena are completely described by a complex- valued state function ψ satisfying the Schr¨odinger equation. The probabilistic in- terpretation of it was first suggested by Born [2] and, in the light of Heisenberg uncertainty principle, is a pillar of quantum mechanics itself. All the known experiments show that the probabilistic interpretation of the wave function is indeed the correct one (see any textbook on quantum mechanics, for 2 S.
    [Show full text]
  • I Speculative Physics: the Ontology of Theory And
    Speculative Physics: the Ontology of Theory and Experiment in High Energy Particle Physics and Science Fiction by Clarissa Ai Ling Lee Graduate Program in Literature Duke University Date: _______________________ Approved: ___________________________ N. Katherine Hayles, Co-Supervisor ___________________________ Mark C. Kruse, Co-Supervisor ___________________________ Mark B. Hansen ___________________________ Andrew Janiak ___________________________ Timothy Lenoir Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate Program in Literature in the Graduate School of Duke University 2014 i v ABSTRACT Speculative Physics: the Ontology of Theory and Experiment in High Energy Particle Physics and Science Fiction by Clarissa Ai Ling Lee Graduate Program in Literature Duke University Date:_______________________ Approved: ___________________________ N. Katherine Hayles, Co-Supervisor ___________________________ Mark C. Kruse, Co-Supervisor ___________________________ Mark B. Hansen ___________________________ Andrew Janiak ___________________________ Timothy Lenoir An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate Program in Literature in the Graduate School of Duke University 2014 i v Copyright by Clarissa Ai Ling Lee 2014 Abstract The dissertation brings together approaches across the fields of physics, critical theory, literary studies, philosophy of physics, sociology of science, and history of science to synthesize a hybrid approach for instigating more rigorous and intense cross- disciplinary interrogations between the sciences and the humanities. I explore the concept of speculation in particle physics and science fiction to examine emergent critical approaches for working in the two areas of literature and physics (the latter through critical science studies), but with the expectation of contributing new insights to media theory, critical code studies, and also the science studies of science fiction.
    [Show full text]
  • Relativistic Quantum Field Theory - Implications for Gestalt Therapy (Brian O’Neil)
    Relativistic Quantum Field Theory - Implications for Gestalt Therapy (Brian O’Neil) It is timely to address how the terms ‘field’ and ‘field theory’ have developed in physics and Gestalt therapy and the inter-relation between the two. Gestalt therapy writers such as Parlett have noted advances in physics which have moved beyond the initial formulation of Maxwell’s electromagnetic field. This includes the quantum field, the relativitistic quantum field, and now post relativistic quantum fields through the work of Bohm and others. These advanced notions of field are beyond the original field theory taken up by Smuts, Lewin, Wertheimer and Gestalt therapy. The possible implication of this expanded field perception in physics for Gestalt therapy is outlined theoretically and clinically. Arjuna Said What are Nature and Self? What are the field and its Knower, knowledge and the object of knowledge? Teach me about them Krishna. Krisna: I am the Knower of the field in everybody Arjuna. genuine knowledge means knowing both the field and the Knower The Bhagavad Gita This paper is offered as a heuristic device to better understand the impact and application of field theory in Gestalt therapy and to refresh and enrich the dialogue between Gestalt therapy and other domains, specifically physics and biology. In particular, to what extent are the terms ‘field’ and field theory’ used as an epistemology (ie as a method of obtaining and validating knowledge) and as an ontology (ie expressing the nature of being), with the understanding that each term is not exclusive of the other. In the present state of physics there is an acceptance of ‘field’ as an ontological reality, while many Gestalt therapy writers relate to ‘field’ epistemologically as a metaphor or method.
    [Show full text]