DOCSLIB.ORG

Violation of Heisenberg's Uncertainty Principle

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Violation of Heisenberg's Uncertainty Principle Violation of Heisenberg’s uncertainty principle Koji Nagata1 and Tadao Nakamura2 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea E-mail: ko mi na@yahoo.co.jp 2Department of Information and− Computer− Science, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: nakamura@pipelining.jp ( Dated: July 28, 2015) Recently, violation of Heisenberg’s uncertainty relation in spin measurements is discussed [J. Erhart et al., Nature Physics 8, 185 (2012)] and [G. Sulyok et al., Phys. Rev. A 88, 022110 (2013)]. We derive the optimal limitation of Heisenberg’s uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on). Some physical situation is that we would measure σx and σy, simultaneously. The optimality is certified by the Bloch sphere. We show that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific case. Thus, the above experiments show a violation of the Bloch sphere when we use 1 as measurement outcome. This conclusion agrees with recent researches [K. Nagata, Int. J. Theor.± Phys. 48, 3532 (2009)] and [K. Nagata et al., Int. J. Theor. Phys. 49, 162 (2010)]. PACS numbers: 03.65.Ta, 03.67.-a I. INTRODUCTION position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard [18] later that year The quantum theory (cf. [1—5]) gives accurate and at- and by Hermann Weyl [19] in 1928. times-remarkably accurate numerical predictions. Much Recently, Ozawa discusses universally valid reformu- experimental data has fit to quantum predictions for long lation of the Heisenberg uncertainty principle on noise time. and disturbance in measurement [20]. And experimen- As for the foundations of the quantum theory, Leggett- tal demonstration of a universally valid error-disturbance type non-local variables theory [6] is experimentally in- uncertainty relation in spin-measurements is discussed vestigated [7—9]. The experiments report that the quan- [21]. Violation of Heisenberg’s error-disturbance uncer- tum theory does not accept Leggett-type non-local vari- tainty relation in neutron spin measurements is also dis- ables interpretation. cussed [22]. As for the applications of the quantum theory, the im- In this paper, we derive the optimal limitation of plementation of a quantum algorithm to solve Deutsch’s Heisenberg’s uncertainty principle in a specific two-level problem [10] on a nuclear magnetic resonance quantum system (e.g., electron spin, photon polarizations, and so computer was reported firstly [11]. An implementation on). Some physical situation is that we would measure σx of the Deutsch-Jozsa algorithm on an ion-trap quantum and σy, simultaneously. The optimality is certified by the computer is also reported [12]. There are several at- Bloch sphere. We show that a violation of Heisenberg’s tempts to use single-photon two-qubit states for quan- uncertainty principle means a violation of the Bloch tum computing. Oliveira et al. implemented Deutsch’s sphere in the specific case. Especially, the Schr¨odinger algorithm with polarization and transverse spatial modes uncertainty relation is equivalent to the Bloch sphere re- of the electromagnetic field as qubits [13]. Single-photon lation in the physical situation. Thus, the experiments Bell states are prepared and measured [14]. Also the [21, 22] show a violation of the Bloch sphere when we decoherence-free implementation of Deutsch’s algorithm use 1 as measurement outcome. This conclusion agrees was reported by using such single-photon and by using with± recent researches [23, 24]. two logical qubits [15]. More recently, a one-way based experimental implementation of Deutsch’s algorithm was reported [16]. In quantum mechanics, the uncertainty principle is any II. THE SCHRODINGER¨ UNCERTAINTY of the variety of mathematical inequalities asserting a RELATION fundamental limit to the precision with which certain pairs of physical properties of a particle known as com- plementary variables, such as its position x and momen- In this section, we review the Schr¨odinger uncer- tum p, can be known simultaneously. For instance, in tainty relation. The derivation shown here incorpo- 1927, Werner Heisenberg stated that the more precisely rates and builds off of those shown in Robertson [25], the position of some particle is determined, the less pre- Schr¨odinger [26] and standard textbooks such as Grif- cisely its momentum can be known, and vice versa [17]. fiths [27]. The formal inequality relating the standard deviation of For any Hermitian operator Aˆ, based upon the defini- 2 tion of variance, we have and 2 ˆ ˆ ˆ ˆ σA = (A A )Ψ (A A )Ψ . (1) f g + g f = AˆBˆ Aˆ Bˆ + BˆAˆ Aˆ Bˆ − u | − u u | u | u u − u u u − u u We let f = (Aˆ Aˆ )Ψ and thus = A,ˆ Bˆ 2 Aˆ Bˆ , (12) | u | − u u u − u u σ2 = f f . (2) ˆ ˆ A | u where we have introduced the anticommutator, A, B = ˆ ˆ ˆ ˆ Similarly, for any other Hermitian operator Bˆ in the same AB + BA. state We now substitute the above two equations above back into Eq. (8) and get σ2 = (Bˆ Bˆ )Ψ (Bˆ Bˆ )Ψ = g g (3) B − u | − u u | u 2 1 ˆ ˆ ˆ ˆ 2 1 ˆ ˆ 2 for g = (Bˆ Bˆ )Ψ . f g = ( A, B A B ) + ( [A, B] ) . | u | − u u | | u| 2 u − u u 2i u The product of the two deviations can thus be ex- (13) pressed as Substituting the above into Eq. (6) we get the σ2 σ2 = f f g g . (4) A B | u | u Schr¨odinger uncertainty relation In order to relate the two vectors f and g , we use the | u | u Cauchy-Schwarz inequality [28] which is defined as 1 2 1 2 σAσB ( A,ˆ Bˆ Aˆ Bˆ ) + ( [A,ˆ Bˆ] ) . ≥ 2 u − u u 2i u f f g g f g 2, (5) R | u | u ≥ | | u| (14) and thus Eq. (4) can be written as σ2 σ2 f g 2. (6) A B ≥ | | u| III. MAIN RESULT Since f g is in general a complex number, we use the | u fact that the modulus squared of any complex number z is In this section, we present an example that the 2 defined as z = zz∗, where z∗ is the complex conjugate Schr¨odinger uncertainty relation is optimal. The opti- | | of z. The modulus squared can also be expressed as mality is certified by the Bloch sphere. In fact, a vio- lation of the Schr¨odinger uncertainty relation means a z 2 = (Re(z))2 + (Im(z))2 | | violation of the Bloch sphere in the specific case. We z + z z z = ( ∗ )2 + ( − ∗ )2. (7) derive the Schr¨odinger uncertainty relation by using the 2 2i Bloch sphere relation in the specific case. Let V (X) be 2 2 We let z = f g and z = g f and substitute these into the variance of X, i.e., (X) X . ∗ u − u the equation above| u to get | u Statement 1 f g + g f f g g f 2 2 2 1 2 2 2 f g = ( | u | u) + ( | u − | u ) . (8) V (σx)V (σy) ( [σx, σy] ) + σx σy . | | u| 2 2i ≥ 2i u u u 2 R The inner product f g is written out explicitly as (15) | u 2 2 2 f g = (Aˆ Aˆ )Ψ (Bˆ Bˆ )Ψ , (9) Proof. By using 1 σx σy σz , we have | u − u | − u u − u − u ≥ u ˆ ˆ and using the fact that A and B are Hermitian operators, V (σx)V (σy) we find 2 2 = (1 σx )(1 σy ) ˆ ˆ ˆ ˆ − 2u − 2 u 2 2 f g = Ψ (A A )(B B )Ψ = 1 σx σy + σx σy | u | − u − u u − u − u u u = Ψ (AˆBˆ Aˆ Bˆ Bˆ Aˆ + Aˆ Bˆ )Ψ σ 2 + σ 2 σ 2 | − u − u u u u z x y ˆ ˆ ˆ ˆ ≥ u u u = Ψ ABΨ Ψ A B Ψ 1 2 2 2 | u − | u u = ( [σx, σy] ) + σx σy . (16) Ψ Bˆ Aˆ Ψ + Ψ Aˆ Bˆ Ψ 2i u u u − | u u | u u u = AˆBˆ Aˆ Bˆ Aˆ Bˆ + Aˆ Bˆ Thus, u − u u − u u u u = AˆBˆ Aˆ Bˆ . (10) u − u u 1 2 2 2 V (σx)V (σy) ( [σx, σy] ) + σx σy . Similarly it can be shown that g f = BˆAˆ Aˆ Bˆ . ≥ 2i u u u | u u − u u 2 R For a pair of operators Aˆ and Bˆ, we may define their (17) commutator as [A,ˆ Bˆ] = AˆBˆ BˆAˆ. Thus we have − QED We define N as follows: f g g f = AˆBˆ Aˆ Bˆ BˆAˆ + Aˆ Bˆ | u − | u u − u u − u u u 1 2 2 2 = [A,ˆ Bˆ] (11) N := ( [σx, σy] ) + σx σy . (18) u 2i u u u 3 We define S as follows: outcome. This conclusion agrees with recent researches Schrodinger term [23, 24]. 1 2 1 2 S := ( σx, σy σx σy ) + ( [σx, σy] ) . )2 u − u N. u 2i u I IV. CONCLUSIONS (19) We discuss the relation between N and S in the following In conclusions, we have derived the optimal limitation statement. of Heisenberg’s uncertainty principle in a specific two- Statement 2 level system (e.g., electron spin, photon polarizations, and so on). Some physical situation has been that we N = S. (20) would measure σx and σy, simultaneously. The opti- mality has been certified by the Bloch sphere.
Load more

Recommended publications
  • Wien Peaks, Planck Distribution Function and Its Decomposition, The
    Applied Physics Research; Vol. 12, No. 4; 2020 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education How to Understand the Planck´s Oscillators? Wien Peaks, Planck Distribution Function and Its Decomposition, the Bohm Sheath Criterion, Plasma Coupling Constant, the Barrier of Determinacy, Hubble Cooling Constant. (24.04.2020) Jiří Stávek1 1 Bazovského 1228, 163 00 Prague, Czech republic Correspondence: Jiří Stávek, Bazovského 1228, 163 00 Prague, Czech republic. E-mail: stavek.jiri@seznam.cz Received: April 23, 2020 Accepted: June 26, 2020 Online Published: July 31, 2020 doi:10.5539/apr.v12n4p63 URL: http://dx.doi.org/10.5539/apr.v12n4p63 Abstract In our approach we have combined knowledge of Old Masters (working in this field before the year 1905), New Masters (working in this field after the year 1905) and Dissidents under the guidance of Louis de Broglie and David Bohm. Based on the great works of Wilhelm Wien and Max Planck we have presented a new look on the “Wien Peaks” and the Planck Distribution Function and proposed the “core-shell” model of the photon. There are known many “Wien Peaks” defined for different contexts. We have introduced a thermodynamic approach to define the Wien Photopic Peak at the wavelength λ = 555 nm and the Wien Scotopic Peak at the wavelength λ = 501 nm to document why Nature excellently optimized the human vision at those wavelengths. There could be discovered many more the so-called Wien Thermodynamic Peaks for other physical and chemical processes. We have attempted to describe the so-called Planck oscillators as coupled oscillations of geons and dyons.
    [Show full text]
  • Sir Arthur Eddington and the Foundations of Modern Physics
    Sir Arthur Eddington and the Foundations of Modern Physics Ian T. Durham Submitted for the degree of PhD 1 December 2004 University of St. Andrews School of Mathematics & Statistics St. Andrews, Fife, Scotland 1 Dedicated to Alyson Nate & Sadie for living through it all and loving me for being you Mom & Dad my heroes Larry & Alice Sharon for constant love and support for everything said and unsaid Maggie for making 13 a lucky number Gram D. Gram S. for always being interested for strength and good food Steve & Alice for making Texas worth visiting 2 Contents Preface … 4 Eddington’s Life and Worldview … 10 A Philosophical Analysis of Eddington’s Work … 23 The Roaring Twenties: Dawn of the New Quantum Theory … 52 Probability Leads to Uncertainty … 85 Filling in the Gaps … 116 Uniqueness … 151 Exclusion … 185 Numerical Considerations and Applications … 211 Clarity of Perception … 232 Appendix A: The Zoo Puzzle … 268 Appendix B: The Burying Ground at St. Giles … 274 Appendix C: A Dialogue Concerning the Nature of Exclusion and its Relation to Force … 278 References … 283 3 I Preface Albert Einstein’s theory of general relativity is perhaps the most significant development in the history of modern cosmology. It turned the entire field of cosmology into a quantitative science. In it, Einstein described gravity as being a consequence of the geometry of the universe. Though this precise point is still unsettled, it is undeniable that dimensionality plays a role in modern physics and in gravity itself. Following quickly on the heels of Einstein’s discovery, physicists attempted to link gravity to the only other fundamental force of nature known at that time: electromagnetism.
    [Show full text]
  • Some Study Materials on Quantum Mechanics Part of Paper C9t (Vu Physics Hons
    SOME STUDY MATERIALS ON QUANTUM MECHANICS PART OF PAPER C9T (VU PHYSICS HONS. CBCS 4TH SEMESTER) DEBASISH AICH VU CBCS Semester-IV 2019: Planck’s quantum, Planck’s constant and light as a collection of photons; Blackbody Radiation: Quantum theory of Light; Photo-electric effect and Compton scattering. De Broglie wavelength and matter waves; Davisson-Germer experiment. Wave description of particles by wave packets. Group and Phase velocities and relation between them. Two-Slit experiment with electrons. Probability. Wave amplitude and wave functions. Historical Note 1. Situation towards the end of the 19th century and the beginning of the 20th century 1.1 Advancement in Physics: Classical mechanics: Newtonian Mechanics (Principia 1687-1713-1726; Sir Isaac Newton, English, 1643-1727) > Lagrangian Formulation (1750s) (Joseph-Louis Lagrange, Italian-French, 1736-1813) > Hamiltonian Formulation (1833) (William Rowan Hamilton, Irish, 1805-1865) Electrodynamics: Maxwell’s (James Clerk Maxwell, Scottish, 1831-1879) Equations of Electromagnetic waves [1861]. [In present form by Oliver Heaviside (English), Josiah W Gibbs (American), Heinrich Hertz (German, 1857-1894) in 1884] Lorentz (Dutch, 1853-1928) Force Equation [1861 Maxwell > 1881J. J. Thomson1 (English) > 1884 Heaviside > 1895 Lorentz] Thermodynamics: Carnot Theorem (1824) [Nicolas Leonard Carnot, French 1796-1832], Maxwell-Boltzmann (Ludwig Eduard Boltzmann, German, 1844-1906) Statistics (1868). 1.2. Major unsolved Questions: Energy Distribution [푢(휈)푑휈 표푟 푢(휆)푑휆] of Blackbody Radiation. Photo electric effect: Experiment by Hertz in 1887. Stability of Rutherford’s (New Zealand-born British) atom (1911 Gold Foil Expt.). Existence of aether: Michelson (American) –Morley (American) Experiment (1887, at Western Reserve University, Ohio). Atomic Spectra: Balmer (Swiss Mathematician) series: Balmer formula (1885) 휆 = 퐵 = 퐵 Rydberg (Swedish Physicist) Formula (1888) 휈̅ = = − = 푅 − With 푅 = 1.09737309 × 10 푚 .
    [Show full text]
  • Unit – I - Physics for Engineers-Spha1101
    Basic Quantum Physics SCHOOL OF SCIENCE AND HUMANITIES DEPARTMENT OF PHYSICS UNIT – I - PHYSICS FOR ENGINEERS-SPHA1101 1 Basic Quantum Physics UNIT 1 BASIC QUANTUM PHYSICS 1.0 INTRODUCTION Quantum theory is the theoretical basis of modern physics that explains the behaviour of matter and energy on the scale of atoms and subatomic particles / waves where classical physics does not always apply due to wave-particle duality and the uncertainty principle. In 1900, physicist Max Planck presented his quantum theory to the German Physical Society. Planck had sought to discover the reason that radiation from a glowing body changes in color from red, to orange, and, finally, to blue as its temperature rises. The Development of Quantum Theory In 1900, Planck made the assumption that energy was made of individual units, or quanta. In 1905, Albert Einstein theorized that not just the energy, but the radiation itself was quantizedin the same manner. In 1924, Louis de Broglie proposed the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis, an example of wave–particle duality, and forms a central part of the theory of quantum mechanics. In 1927, Werner Heisenberg proposed uncertainty principle. It states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. The formal inequality relating 2 Basic Quantum Physics the standard deviation of position σx and the standard deviation of [3] momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928 ℎ ≥ 푥 푦 4 where ħ is the reduced Planck constant, h/(2π).
    [Show full text]
  • Uncertainty Principle 1 Uncertainty Principle
    Uncertainty principle 1 Uncertainty principle For other uses, see Uncertainty principle (disambiguation). Quantum mechanics Uncertainty principle • Introduction • Glossary • History • v • t [1] • e In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.[2] The formal inequality relating the standard deviation of position σ and x the standard deviation of momentum σ was derived by Earle Hesse Kennard later that year and by Hermann Weyl in p 1928: (ħ is the reduced Planck constant). The original heuristic argument that such a limit should exist was given by Heisenberg, after whom it is sometimes named the Heisenberg principle. This ascribes the uncertainty in the measurable quantities to the jolt-like disturbance triggered by the act of observation. Though widely repeated in textbooks, this physical argument is now known to be fundamentally misleading.[3] While the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenberg's argument; the formal mathematical result remains valid, however. Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.[4] It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects.
    [Show full text]
  • Contributors
    Contributors Massimiliano Badino: Centre d’Història de la Ciència, Facultat de Ciències, Universitat Autònoma de Barcelona, Cerdanyola del Valles, 08193 Bellaterra (Barcelona), Spain massimiliano.badino@uab.cat Michael Eckert: Forschungsinstitut Deutsches Museum, Museumsinsel 1, 80538 München, Germany m.eckert@deutsches-museum.de Clayton Gearhart: St. John’s University, Collegeville, MN 56321, USA cgearhart@csbsju.edu Domenico Giulini: Institut für Theoretische Physik, Leibniz Universität Hannover, Appel- straße 2, 30167 Hannover, Germany giulini@itp.uni-hannover.de Dieter Hoffmann: Max Planck Institute for the History of Science, Boltzmannstraße 22, 14195 Berlin, Germany dh@mpiwg-berlin.mpg.de Don Howard: Department of Philosophy, 100 Malloy Hall, University of Notre Dame, Notre Dame, Indiana 46556, USA dhoward1@nd.edu Michel Janssen: Program in the History of Science, Technology, and Medicine, University of Minnesota, Minneapolis, MN 55455, USA janss011@umn.edu Marta Jordi Taltavull: Max Planck Institute for the History of Science, Boltzmannstraße 22, 14195 Berlin, Germany mjordi@mpiwg-berlin.mpg.de David Kaiser: Program in Science, Technology, & Society and Department of Physics, Room E51–179, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cam- bridge, MA 02139, USA dikaiser@MIT.EDU Helge Kragh: Centre for Science Studies, Department of Physics and Astronomy, Aarhus University, Building 1520, 8000 Aarhus, Denmark helge.kragh@ivs.au.dk Charles Midwinter: Program in the History of Science, Technology, and Medicine, Univer- sity of Minnesota, Minneapolis, MN 55455, USA charles.midwinter@gmail.com 2 Contributors Jaume Navarro: University of the Basque Country and Ikerbasque (Basque Research Foun- dation), D10, Plaza Elhuyar 2, 20018, San Sebastian, Spain jaume.navarro@ehu.es Chapter 1 Pedagogy and Research.
    [Show full text]
  • Quantum Spectral Analysis: Frequency in Time Mario Mastriani
    Quantum spectral analysis: frequency in time Mario Mastriani To cite this version: Mario Mastriani. Quantum spectral analysis: frequency in time. 2018. hal-01655209v2 HAL Id: hal-01655209 https://hal.inria.fr/hal-01655209v2 Preprint submitted on 22 Jan 2018 (v2), last revised 26 Dec 2018 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum spectral analysis: frequency in time Mario Mastriani Quantum Communications Group, Merx Communications LLC, 2875 NE 191 st, suite 801, Aventura, FL 33180, USA mmastri@merxcomm.com ABSTRACT A quantum time-dependent spectrum analysis, or simply, a quantum spectral analysis (QSA) is presented in this work based on Schrödinger equation, which is a partial differential equation that describes how the quantum state of a non-relativistic physical system changes with time. In the classic world, it is named frequency in time (FIT), which is presented here in opposition and as a complement of traditional spectral analysis frequency-dependent based on Fourier theory. Besides, FIT is a metric, which assesses the impact of the flanks of a signal on its frequency spectrum, which is not taken into account by Fourier theory, let alone in real time.
    [Show full text]
  • Quantum Enhanced Microscopy, Equs Annual Workshop, Noosa, AUS, December 2016 - Poster Accompanied by Abstract
    QUANTUM ENHANCED MICROSCOPY Catxere Andrade Casacio B.Sc., B.Eng., M.Sc. A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2020 School of Mathematics and Physics, ARC Centre of Excellence for Engineered Quantum Systems (EQUS) Abstract In optical microscopy, as in any biological measurement performed with light, there is a compromise between increasing power to improve resolution and decreasing power to prevent photodamage to the biological structure. Quantum optics allows this compromise to be overcome, enabling improved precision without increased illumination intensity. This thesis demonstrates, for the first time, absolute quantum advantage in microscopy, with a specific focus on Stimulated Raman Scattering (SRS) microscopy. SRS microscopy is a powerful tool widely used in many fields, especially in biomedical sciences. It uses a two-photon process where the light is downshifted in frequency by being absorbed and subsequently inelastically scattered by the intrinsic molecular vibrations in the sample. Each element or molecule has a different response, allowing selective imaging of biological systems. SRS microscopy is particularly useful because it is label-free (no addition of dyes, beads or any implanted tracer), meaning that imaging contrast is achieved without modification of, or interference with, the specimen. Here I present our newly built state-of-the-art shot-noise limited SRS microscope. I also characterize the Raman microscope, describe the developped imaging system and show SRS spectroscopy and imaging of both polystyrene beads and living yeast cells. The quantisation of light introduces a fundamental uncertainty, or quantum noise, in both the amplitude and phase quadratures of an optical field.
    [Show full text]
  • Quantum Interpretations and World War Two an Analysis of the Interpretive Debate from the Twenties to the Sixties in Historical Context
    Quantum interpretations and world war two An analysis of the interpretive debate from the twenties to the sixties in historical context Edo van Veen October 2014 Supervisors: dr. Dean Rickles and prof. dr. Wim Beenakker Second corrector: dr. Michiel Seevinck Unit for the History and Philosophy of Science University of Sydney Front page image: Einstein and Oppenheimer, two generations of physicists. Wikimedia commons, image courtesy of US Govt. Defense Threat Reduction Agency 2 Introduction The theory of quantum mechanics has puzzled physicists since the early 20th century, when they discovered that the atomic and subatomic world must be described by a theory that is fundamentally different from classical mechanics. It has since been incredibly successful in allowing physicists to describe physical systems with an accuracy that has never been reached before and making pos- sible the development of technologies like semiconductor transistors and more recently quantum cryptography. The puzzling part, however, is still unresolved. Because after modern quan- tum mechanics was born in 1925, it was clear that the mathematical formalism of quantum mechanics describes measurement outcomes and not necessarily the underlying physical processes. This has left many physicists and philosophers with a deeply rooted desire to explore the relation between the formalism and its underlying processes, which has led to the development of different interpre- tations of quantum mechanics. The discussion of conceptual issues and interpretations finds place in a field that is called the foundations of quantum mechanics. It was highly popular until the second world war, after which the interest for the field reduced. Around this time, the scientific focus changed from foundational considerations and analytical methods to more pragmatic and numerical methods.
    [Show full text]
  • Violation of Heisenberg's Uncertainty Principle
    Open Access Library Journal Violation of Heisenberg’s Uncertainty Principle Koji Nagata1, Tadao Nakamura2 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea 2Department of Information and Computer Science, Keio University, Yokohama, Japan Email: ko_mi_na@yahoo.co.jp, nakamura@pipelining.jp Received 22 July 2015; accepted 9 August 2015; published 12 August 2015 Copyright © 2015 by authors and OALib. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Recently, violation of Heisenberg’s uncertainty relation in spin measurements is discussed [J. Er- hart et al., Nature Physics 8, 185 (2012)] and [G. Sulyok et al., Phys. Rev. A 88, 022110 (2013)]. We derive the optimal limitation of Heisenberg’s uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on). Some physical situation is that we would measure σ x and σ y , simultaneously. The optimality is certified by the Bloch sphere. We show that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific case. Thus, the above experiments show a violation of the Bloch sphere when we use ±1 as measurement outcome. This conclusion agrees with recent researches [K. Nagata, Int. J. Theor. Phys. 48, 3532 (2009)] and [K. Nagata et al., Int. J. Theor. Phys. 49, 162 (2010)]. Keywords Quantum Measurement Theory, Quantum Information Theory Subject Areas: Applied Physics 1. Introduction The quantum theory (cf. [1]-[5]) gives accurate and at-times-remarkably accurate numerical predictions. Much experimental data has fit to quantum predictions for long time.
    [Show full text]
  • Experimental Demonstrator of the Uncertainty Principle Degree Project in Engineering Physics, First Level (SA104X)
    Experimental Demonstrator of the Uncertainty Principle Degree Project in Engineering Physics, First Level (SA104X) Philip Ekfeldt Anders Pettersson pekfeldt@kth.se anderpet@kth.se May 21, 2013 Supervisors: Marcin Swillo and Gunnar Bj¨ork Examinator: M˚artenOlsson Dept. of Applied Physics Royal Institute of Technology Abstract The goal of this project was to create an intuitive and clear demonstrator of the defining properties of quantum mechanics using single slit diffraction of light, which has quantum mechanical properties because of light's wave- particle duality. In this report we will describe the process and thoughts behind our project of creating a portable demonstration of the uncertainty principle. By designing and building both a physical setup with a laser, a slit, mirrors, lenses, beam-splitters, attenuators and cameras, and developing software which displays images from the cameras in a clear user interface with calculations we hope that students from high schools and gymnasiums that visit Vetenskapens Hus at Alba Nova will learn something new while using the demonstrator. CONTENTS Philip Ekfeldt, Anders Pettersson Contents 1 Introduction 2 1.1 Objective . 2 2 Theory 3 2.1 The Uncertainty Principle . 3 2.2 Theoretical Value of Uncertainty . 3 2.3 Uncertainty of Laser Source . 5 3 Method 6 3.1 Optical Setup . 6 3.1.1 Projecting Images to the Cameras . 8 3.2 Software . 10 3.2.1 Description of the User Environment . 10 3.2.2 Calculating the Standard Deviation of the Position . 11 3.2.3 Calculating the Standard Deviation of the Momentum 12 4 Results 14 5 Discussion 14 5.1 Physical Design .
    [Show full text]
  • Arxiv:1705.00650V1 [Quant-Ph] 1 May 2017
    Classical light vs. nonclassical light: Characterizations and interesting applications Anirban Pathaka;,∗ Ajoy Ghatakb;y May 3, 2017 aJaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India bM.N. Saha Fellow, The National Academy of Sciences, India (NASI), D42, Hauz Khas, New Delhi, India Abstract We briefly review the ideas that have shaped modern optics and have led to various applications of light ranging from spectroscopy to astrophysics, and street lights to quantum communication. The review is primarily focused on the modern applications of classical light and nonclassical light. Specific attention has been given to the applications of squeezed, antibunched, and entangled states of radiation field. Applications of Fock states (especially single photon states) in the field of quantum communication are also discussed. 1 Introduction Once Poincare said “The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it; and he takes pleasure in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and life would not be worth living. I mean the intimate beauty which comes from the harmonious order of its parts and which a pure intelligence can grasp” (as quoted in Chapter 4 of [1]). It is the search of this harmony in nature that brought different theories of light. If we look back at Newton’s corpuscular theory of light, it would not be difficult to guess that the harmony of nature revealed through his 3 laws of motion and the law of gravity (all four of which are obeyed by particles), which together can explain almost every phenomena known at that time might have forced him to consider light also as corpuscular.
    [Show full text]