Quantum Mechanics and Paradigm Shifts
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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
The Philosophical Underpinnings of Educational Research
The Philosophical Underpinnings of Educational Research Lindsay Mack Abstract This article traces the underlying theoretical framework of educational research. It outlines the definitions of epistemology, ontology and paradigm and the origins, main tenets, and key thinkers of the 3 paradigms; positivist, interpetivist and critical. By closely analyzing each paradigm, the literature review focuses on the ontological and epistemological assumptions of each paradigm. Finally the author analyzes not only the paradigm’s weakness but also the author’s own construct of reality and knowledge which align with the critical paradigm. Key terms: Paradigm, Ontology, Epistemology, Positivism, Interpretivism The English Language Teaching (ELT) field has moved from an ad hoc field with amateurish research to a much more serious enterprise of professionalism. More teachers are conducting research to not only inform their teaching in the classroom but also to bridge the gap between the external researcher dictating policy and the teacher negotiating that policy with the practical demands of their classroom. I was a layperson, not an educational researcher. Determined to emancipate myself from my layperson identity, I began to analyze the different philosophical underpinnings of each paradigm, reading about the great thinkers’ theories and the evolution of social science research. Through this process I began to examine how I view the world, thus realizing my own construction of knowledge and social reality, which is actually quite loose and chaotic. Most importantly, I realized that I identify most with the critical paradigm assumptions and that my future desired role as an educational researcher is to affect change and challenge dominant social and political discourses in ELT. -
The Development of the Science of Superconductivity and Superfluidity
Universal Journal of Physics and Application 1(4): 392-407, 2013 DOI: 10.13189/ujpa.2013.010405 http://www.hrpub.org Superconductivity and Superfluidity-Part I: The development of the science of superconductivity and superfluidity in the 20th century Boris V.Vasiliev ∗Corresponding Author: [email protected] Copyright ⃝c 2013 Horizon Research Publishing All rights reserved. Abstract Currently there is a common belief that the explanation of superconductivity phenomenon lies in understanding the mechanism of the formation of electron pairs. Paired electrons, however, cannot form a super- conducting condensate spontaneously. These paired electrons perform disorderly zero-point oscillations and there are no force of attraction in their ensemble. In order to create a unified ensemble of particles, the pairs must order their zero-point fluctuations so that an attraction between the particles appears. As a result of this ordering of zero-point oscillations in the electron gas, superconductivity arises. This model of condensation of zero-point oscillations creates the possibility of being able to obtain estimates for the critical parameters of elementary super- conductors, which are in satisfactory agreement with the measured data. On the another hand, the phenomenon of superfluidity in He-4 and He-3 can be similarly explained, due to the ordering of zero-point fluctuations. It is therefore established that both related phenomena are based on the same physical mechanism. Keywords superconductivity superfluidity zero-point oscillations 1 Introduction 1.1 Superconductivity and public Superconductivity is a beautiful and unique natural phenomenon that was discovered in the early 20th century. Its unique nature comes from the fact that superconductivity is the result of quantum laws that act on a macroscopic ensemble of particles as a whole. -
The Quantum Epoché
Accepted Manuscript The quantum epoché Paavo Pylkkänen PII: S0079-6107(15)00127-3 DOI: 10.1016/j.pbiomolbio.2015.08.014 Reference: JPBM 1064 To appear in: Progress in Biophysics and Molecular Biology Please cite this article as: Pylkkänen, P., The quantum epoché, Progress in Biophysics and Molecular Biology (2015), doi: 10.1016/j.pbiomolbio.2015.08.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT The quantum epoché Paavo Pylkkänen Department of Philosophy, History, Culture and Art Studies & the Academy of Finland Center of Excellence in the Philosophy of the Social Sciences (TINT), P.O. Box 24, FI-00014 University of Helsinki, Finland. and Department of Cognitive Neuroscience and Philosophy, School of Biosciences, University of Skövde, P.O. Box 408, SE-541 28 Skövde, Sweden [email protected] Abstract. The theme of phenomenology and quantum physics is here tackled by examining some basic interpretational issues in quantum physics. One key issue in quantum theory from the very beginning has been whether it is possible to provide a quantum ontology of particles in motion in the same way as in classical physics, or whether we are restricted to stay within a more limited view of quantum systems, in terms of complementary but mutually exclusiveMANUSCRIPT phenomena. -
Analysis of Nonlinear Dynamics in a Classical Transmon Circuit
Analysis of Nonlinear Dynamics in a Classical Transmon Circuit Sasu Tuohino B. Sc. Thesis Department of Physical Sciences Theoretical Physics University of Oulu 2017 Contents 1 Introduction2 2 Classical network theory4 2.1 From electromagnetic fields to circuit elements.........4 2.2 Generalized flux and charge....................6 2.3 Node variables as degrees of freedom...............7 3 Hamiltonians for electric circuits8 3.1 LC Circuit and DC voltage source................8 3.2 Cooper-Pair Box.......................... 10 3.2.1 Josephson junction.................... 10 3.2.2 Dynamics of the Cooper-pair box............. 11 3.3 Transmon qubit.......................... 12 3.3.1 Cavity resonator...................... 12 3.3.2 Shunt capacitance CB .................. 12 3.3.3 Transmon Lagrangian................... 13 3.3.4 Matrix notation in the Legendre transformation..... 14 3.3.5 Hamiltonian of transmon................. 15 4 Classical dynamics of transmon qubit 16 4.1 Equations of motion for transmon................ 16 4.1.1 Relations with voltages.................. 17 4.1.2 Shunt resistances..................... 17 4.1.3 Linearized Josephson inductance............. 18 4.1.4 Relation with currents................... 18 4.2 Control and read-out signals................... 18 4.2.1 Transmission line model.................. 18 4.2.2 Equations of motion for coupled transmission line.... 20 4.3 Quantum notation......................... 22 5 Numerical solutions for equations of motion 23 5.1 Design parameters of the transmon................ 23 5.2 Resonance shift at nonlinear regime............... 24 6 Conclusions 27 1 Abstract The focus of this thesis is on classical dynamics of a transmon qubit. First we introduce the basic concepts of the classical circuit analysis and use this knowledge to derive the Lagrangians and Hamiltonians of an LC circuit, a Cooper-pair box, and ultimately we derive Hamiltonian for a transmon qubit. -
Study of Quantum Spin Correlations of Relativistic Electron Pairs
Study of quantum spin correlations of relativistic electron pairs Project status Nov. 2015 Jacek Ciborowski Marta Włodarczyk, Michał Drągowski, Artem Poliszczuk (UW) Joachim Enders, Yuliya Fritsche (TU Darmstadt) Warszawa 20.XI.2015 Quantum spin correlations In this exp: e1,e2 – electrons under study a, b - directions of spin projections (+- ½) 4 combinations for e1 and e2: ++, --, +-, -+ Probabilities: P++ , P+- , P-+ , P- - (ΣP=1) Correlation function : C = P++ + P-- - P-+ - P-+ Historical perspective • Einstein Podolsky Rosen (EPR) paradox (1935): QM is not a complete local realistic theory • Bohm & Aharonov formulation involving spin correlations (1957) • Bell inequalities (1964) a local realistic theory must obey a class of inequalities • practical approach to Bell’s inequalities: counting aacoincidences to measure correlations The EPR paradox Boris Podolsky Nathan Rosen Albert Einstein (1896-1966) (1909-1995) (1979-1955) A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999) Bohm’s version with the spin Two spin-1/2 fermions in a singlet state: E.g. if spin projection of 1 on Z axis is measured 1/2 spin projection of 2 must be -1/2 All projections should be elements of reality (QM predicts that only S2 and David Bohm (1917-1992) Sz can be determined) Hidden variables? ”Quantum Theory” (1951) Phys. Rev. 85(1952)166,180 The Bell inequalities J.S.Bell: Impossible to reconcile the concept of hidden variables with statistical predictions of QM If local realism quantum correlations -
Quantum Eraser
Quantum Eraser March 18, 2015 It was in 1935 that Albert Einstein, with his collaborators Boris Podolsky and Nathan Rosen, exploiting the bizarre property of quantum entanglement (not yet known under that name which was coined in Schrödinger (1935)), noted that QM demands that systems maintain a variety of ‘correlational properties’ amongst their parts no matter how far the parts might be sepa- rated from each other (see 1935, the source of what has become known as the EPR paradox). In itself there appears to be nothing strange in this; such cor- relational properties are common in classical physics no less than in ordinary experience. Consider two qualitatively identical billiard balls approaching each other with equal but opposite velocities. The total momentum is zero. After they collide and rebound, measurement of the velocity of one ball will naturally reveal the velocity of the other. But the EPR argument coupled this observation with the orthodox Copen- hagen interpretation of QM, which states that until a measurement of a par- ticular property is made on a system, that system cannot, in general, be said to possess any definite value of that property. It is easy to see that if dis- tant correlations are preserved through measurement processes that ‘bring into being’ the measured values there is a prima facie conflict between the Copenhagen interpretation and the relativistic stricture that no information can be transmitted faster than the speed of light. Suppose, for example, we have a system with some property which is anti-correlated (in real cases, this property could be spin). -
Required Readings
HPS/PHIL 93872 Spring 2006 Historical Foundations of the Quantum Theory Don Howard, Instructor Required Readings: Topic: Readings: Planck and black-body radiation. Martin Klein. “Planck, Entropy, and Quanta, 19011906.” The Natural Philosopher 1 (1963), 83-108. Martin Klein. “Einstein’s First Paper on Quanta.” The Natural Einstein and the photo-electric effect. Philosopher 2 (1963), 59-86. Max Jammer. “Regularities in Line Spectra”; “Bohr’s Theory The Bohr model of the atom and spectral of the Hydrogen Atom.” In The Conceptual Development of series. Quantum Mechanics. New York: McGraw-Hill, 1966, pp. 62- 88. The Bohr-Sommerfeld “old” quantum Max Jammer. “The Older Quantum Theory.” In The Conceptual theory; Einstein on transition Development of Quantum Mechanics. New York: McGraw-Hill, probabilities. 1966, pp. 89-156. The Bohr-Kramers-Slater theory. Max Jammer. “The Transition to Quantum Mechanics.” In The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966, pp. 157-195. Bose-Einstein statistics. Don Howard. “‘Nicht sein kann was nicht sein darf,’ or the Prehistory of EPR, 1909-1935: Einstein’s Early Worries about the Quantum Mechanics of Composite Systems.” In Sixty-Two Years of Uncertainty: Historical, Philosophical, and Physical Inquiries into the Foundations of Quantum Mechanics. Arthur Miller, ed. New York: Plenum, 1990, pp. 61-111. Max Jammer. “The Formation of Quantum Mechanics.” In The Schrödinger and wave mechanics; Conceptual Development of Quantum Mechanics. New York: Heisenberg and matrix mechanics. McGraw-Hill, 1966, pp. 196-280. James T. Cushing. “Early Attempts at Causal Theories: A De Broglie and the origins of pilot-wave Stillborn Program.” In Quantum Mechanics: Historical theory. -
Aleksei A. Abrikosov 1928–2017
Aleksei A. Abrikosov 1928–2017 A Biographical Memoir by M. R. Norman ©2018 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. ALEKSEI ALEKSEEVICH ABRIKOSOV June 25, 1928–March 29, 2017 Elected to the NAS, 2000 Shortly after the 2003 announcement that Aleksei Abrikosov had won the Nobel Prize in Physics, a number of colleagues took Alex to lunch at a nearby Italian restau- rant. During lunch, one of the Russian visitors exclaimed that Alex should get a second Nobel Prize, this time in Literature for his famous “AGD” book with Lev Gor’kov and Igor Dzyaloshinskii (Methods of Quantum Field Theory in Statistical Physics.) Somewhat taken aback, I looked closely at this individual and realized that he was deadly serious. Although I could imagine the reaction of the Nobel Literature committee to such a book (for a lay person, perhaps analogous to trying to read Finnegan’s Wake), I had to admit that my own copy of this book is quite dog-eared, having been put to good use over the By M. R. Norman years. In fact, you know you have made it in physics when your book gets a Dover edition. One of the most charming pictures I ever saw was a rare drawing in color that Alexei Tsvelik did (commissioned by Andrei Varlamov for Alex’s 50th birthday) that was proudly displayed in Alex’s home in Lemont, IL. It showed Alex with his fingers raised in a curled fashion as in the habit of medieval Popes. -
Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Emma Wikberg Project work, 4p Department of Physics Stockholm University 23rd March 2006 Abstract The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrödinger formulation. As will be seen in this project work, many "elementary" problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, start- ing from the well known quantum mechanics as formulated by Schrödinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time. Contents 1 Introduction and Outline 2 1.1 Introduction . 2 1.2 Outline . 2 2 Path Integrals from ordinary Quantum Mechanics 4 2.1 The Schrödinger equation and time evolution . 4 2.2 The propagator . 6 3 Equivalence to the Schrödinger Equation 8 3.1 From the Schrödinger equation to PI’s . 8 3.2 From PI’s to the Schrödinger equation . -
Consciousness, the Unconscious and Mathematical Modeling of Thinking
entropy Article On “Decisions and Revisions Which a Minute Will Reverse”: Consciousness, The Unconscious and Mathematical Modeling of Thinking Arkady Plotnitsky Literature, Theory and Cultural Studies Program, Philosophy and Literature Program, Purdue University, West Lafayette, IN 47907, USA; [email protected] Abstract: This article considers a partly philosophical question: What are the ontological and epistemological reasons for using quantum-like models or theories (models and theories based on the mathematical formalism of quantum theory) vs. classical-like ones (based on the mathematics of classical physics), in considering human thinking and decision making? This question is only partly philosophical because it also concerns the scientific understanding of the phenomena considered by the theories that use mathematical models of either type, just as in physics itself, where this question also arises as a physical question. This is because this question is in effect: What are the physical reasons for using, even if not requiring, these types of theories in considering quantum phenomena, which these theories predict fully in accord with the experiment? This is clearly also a physical, rather than only philosophical, question and so is, accordingly, the question of whether one needs classical-like or quantum-like theories or both (just as in physics we use both classical and quantum theories) in considering human thinking in psychology and related fields, such as decision Citation: Plotnitsky, A. On science. It comes as no surprise that many of these reasons are parallel to those that are responsible “Decisions and Revisions Which a for the use of QM and QFT in the case of quantum phenomena. -
Quantum Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction ..................................