DOCSLIB.ORG

Non-Relativistic Quantum Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Non-Relativistic Quantum Mechanics NON-RELATIVISTIC QUANTUM MECHANICS MICHAEL DICKSON Abstract. This chapter is a discussion of the philosophical and foundational is- sues that arise in non-relativistic quantum theory. After introducing the formalism of the theory, I consider: characterizations of the quantum formalism, empirical content, uncertainty, the measurement problem, and non-locality. In each case, the main point is to give the reader some introductory understanding of some of the major issues and recent ideas. Keywords: quantum theory, quantum mechanics, measurement problem, uncer- tainty, nonlocality Contents 1. The Theory 3 1.1. The Thought Behind Starting with Formalism 4 1.2. The Standard Formalism 4 1 1.3. Simple Example: A Spin- 2 Particle 19 1.4. Dirac Notation 25 1.5. Transformations 27 1.6. Preview of Philosophical Issues 37 2. Whence the Kinematical Formalism? 39 2.1. From Propositions to Hilbert Space 40 2.2. From States to Hilbert Space 42 2.3. Hardy’s Axioms 51 3. Empirical Content 53 3.1. Measurement 53 3.2. The Issue of Empirical Content in Terms of POVMs 59 3.3. Symmetries 60 3.4. Reference Frames 66 3.5. A Group-Theoretic Characterization of Empirical Content 70 Author’s address: Department of Philosophy, University of South Carolina, Columbia, SC 29208, USA. Thanks to Jeremy Butterfield and John Earman for their comments and suggestions. Thanks also to the participants of a workshop at the University of Pittsburgh in November, 2004, for helpful comments. 2 4. Uncertainty 73 4.1. Canonical Commutation Relations 73 4.2. The Uncertainty Relations 75 4.3. Interpretation of the Uncertainty Relations 79 4.4. The Einstein-Podolsky-Rosen Argument 82 5. The ‘Measurement Problem’ 89 5.1. The Basic Problem 89 5.2. Measurement 90 5.3. Generality of the Problem 91 5.4. Non-Solutions 94 5.5. Interpretations 103 6. Non-locality 118 6.1. No-Go Theorems 119 6.2. Reactions to the Theorems 127 7. Mathematical Appendix 136 7.1. Hilbert Spaces 136 7.2. Operators 139 7.3. The Hilbert Space C2 141 7.4. Posets and Lattices 142 7.5. Topology and Measure 143 7.6. Groups 144 References 147 This article is an introduction to some of the most important philosophical and foun- dational issues that arise from or concern non-relativistic quantum theory. The chap- ter has six main sections. The first introduces the theory, including some of the important mathematical results required to formulate and address many of the philo- sophical and foundational issues. This section is the longest, and most important, for it will begin to give the careful reader the background needed to understand and evaluate much of the vast literature on non-relativistic quantum theory. And that literature is indeed vast—there is no way that it can even be summarized in a chapter of this length. Instead, in the five subsequent sections, I will consider some of the more important: foundational characterizations of the formalism of quantum theory, empirical content, quantum uncertainty, the measurement problem, and non-locality. There are many other issues one could discuss, and some recent movements that merit 3 consideration. Alas, we will not have time for them. A careful reading of the material here is a start, however, towards understanding these other issues. Of these five issues, the first two are somewhat less discussed, especially in the Anglo-American philosophical literature. Those sections are therefore longer, relative to the final three, than some readers might expect. This fact is not meant to imply anything about the relative importance of the issues, but is an attempt to redress a relative lack of coverage in certain circles. Much of the material presented here—especially from §4,§5, and §6—is largely my review of standard material that can be found in many places. I have therefore chosen not to provide extensive bibliographic information. Indeed, I have kept bibliographic references to a minimum. This article is thus not intended to be a compendium of work in the field, much less an extensive annotated bibliography. The reader is encouraged to seek additional resources to fill out the brief accounts given here. Such resources are numerous. The final section is a brief mathematical appendix, reviewing essential definitions and results, mostly from the theory of Hilbert spaces and groups. It may serve one of two purposes, depending on the reader: a brief reminder of concepts learned elsewhere; or a prompt to learn the concepts elsewhere. It is unlikely that a reader who is completely unfamiliar with these concepts will absorb them just from what is said here. Reference is made to the relevant subsections of this appendix at the appropriate places in the text. While at some points I have made some effort at rigor, for the most part, the discussion here is only partly rigorous, with the occasional attempt made to point towards what further would be required for complete rigor. The reader is, again, encouraged to consult the literature for mathematical details, and in any case is encouraged to bear in mind that much of the discussion here is not intended to be entirely mathematically rigorous, while, I hope, also not being misleading. 1. The Theory This section is an introduction to the formalism of quantum theory. After a brief justification of this approach (§1.1), I will introduce the major elements of the formal- ism (§1.2), followed by a simple, but important, example (§1.3). I will then introduce the commonly used ‘Dirac’ notation (§1.4), and conclude by considering the role of transformations (groups) in the theory (§1.5), including dynamical transformations (equations of motion) and finally (§1.6) a brief preview of the philosophical issues to 4 come. More than subsequent sections, this section will rely heavily on the material from the mathematical appendix (§7), with references where appropriate. 1.1. The Thought Behind Starting with Formalism. Why begin an account of a physical theory with its formalism? Why not begin, instead, with its basic physical insights, or fundamental physical principles? One problem with such an approach here is that, in the case of quantum theory, there is not much significant agreement about what the basic physical insights, or fundamental physical principles, are. Some argue that the collapse postulate (to be discussed later) is at the heart of the theory. Others argue that it must be excised from the theory. Some argue that the theory is fundamentally indeterministic, while others argue that we can make sense of it only in terms of an underlying determinism. Some argue that the familiar notion of a ‘particle’ with a definite location is a casualty of the theory, while others argue that the theory makes sense only if one takes such a notion as fundamental. Now, advocates of these different views tend also, it is true, to advocate different formulations of the theory, but they will not suggest that formulations other than their preferred one are wrong, only that they, perhaps, emphasize the wrong points. (Indeed, there is no disputing that the standard formalism—the one presented here— is empirically successful; advocates of different views will ultimately have to account for that success in their own terms.) Hence, while the choice of a single formalism at the start of our discussion might slant our point of view somewhat, it will, unlike the choice of basic physical insights or fundamental physical principles, not prejudge the central issues. 1.2. The Standard Formalism. I begin with a very brief sketch of a common un- derstanding of the formalism, which I shall flesh out and generalize subsequently. (The reader is not expected to have a deep understanding of any aspect of the formalism merely as a result of reading this subsection.) 1.2.1. Hilbert Space. The formalism of quantum mechanics is normally understood in terms of the theory of Hilbert spaces (§7.1). A Hilbert space is a vector space (§7.1.1) with an inner product (§7.1.3) that is also complete with respect to the norm (§7.1.4) defined by this inner product. A standard example is the space, `2, of (modulus)- square-summable sequences of complex numbers. In this space, the inner product of P∞ ∗ two vectors, (x1, x2,...) and (y1, y2,...) is n=1 xnyn. Another standard example is the space, L2(RN ), of (modulus)-square-integrable, Lebesque-measurable, complex- valued functions on RN , where we identify two functions (i.e., they represent the same vector) if (and only if) they differ only on a set of Lebesque measure (§7.5.4) zero. 5 Here the inner product of two vectors, f(x) and g(x), is R f ∗(x)g(x)dx (where f(x) and g(x) are arbitrary representatives from their respective equivalence classes).1 1.2.2. Observables. The ‘observables’ of the theory—the physical quantities, or prop- erties, whose value or presence one can, in principle at least, measure, or ‘observe’— are normally taken to be represented by the self-adjoint operators (§7.2.1, §7.2.3) on the Hilbert space. (The nature of the representation—that is, which operators represent which observables—can depend on the physical situation being described.) Via the spectral theorem (discussed below), one can identify each observable with a spectral family of projection operators, the observable being given, essentially, by a map from Borel sets (§7.5.5) of possible values of the observable to elements in the spectral family. This subsection reviews these ideas briefly. 1.2.2.1. Positive Operator Valued Measures. It is often useful to adopt a broader notion of an observable, as a ‘positive-operator-valued measure’ (POVM).
Load more

Recommended publications
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • The Concept of Quantum State : New Views on Old Phenomena Michel Paty
    The concept of quantum state : new views on old phenomena Michel Paty To cite this version: Michel Paty. The concept of quantum state : new views on old phenomena. Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner. Revisiting the Founda- tions of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, p. 451-478, 2003. halshs-00189410 HAL Id: halshs-00189410 https://halshs.archives-ouvertes.fr/halshs-00189410 Submitted on 20 Nov 2007 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. « The concept of quantum state: new views on old phenomena », in Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner (eds.), Revisiting the Foundations of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, 451-478. , 2003 The concept of quantum state : new views on old phenomena par Michel PATY* ABSTRACT. Recent developments in the area of the knowledge of quantum systems have led to consider as physical facts statements that appeared formerly to be more related to interpretation, with free options.
    [Show full text]
  • Arxiv:2002.00255V3 [Quant-Ph] 13 Feb 2021 Lem (BVP)
    A Path Integral approach to Quantum Fluid Dynamics Sagnik Ghosh Indian Institute of Science Education and Research, Pune-411008, India Swapan K Ghosh UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina, Santacruz, Mumbai-400098, India ∗ (Dated: February 16, 2021) In this work we develop an alternative approach for solution of Quantum Trajectories using the Path Integral method. The state-of-the-art technique in the field is to solve a set of non-linear, coupled partial differential equations (PDEs) simultaneously. We opt for a fundamentally different route. We first derive a general closed form expression for the Path Integral propagator valid for any general potential as a functional of the corresponding classical path. The method is exact and is applicable in many dimensions as well as multi-particle cases. This, then, is used to compute the Quantum Potential (QP), which, in turn, can generate the Quantum Trajectories. For cases, where closed form solution is not possible, the problem is formally boiled down to solving the classical path as a boundary value problem. The work formally bridges the Path Integral approach with Quantum Fluid Dynamics. As a model application to illustrate the method, we work out a toy model viz. the double-well potential, where the boundary value problem for the classical path has been computed perturbatively, but the Quantum part is left exact. Using this we delve into seeking insight in one of the long standing debates with regard to Quantum Tunneling. Keywords: Path Integral, Quantum Fluid Dynamics, Analytical Solution, Quantum Potential, Quantum Tunneling, Quantum Trajectories Submitted to: J.
    [Show full text]
  • Appendix: the Interaction Picture
    Appendix: The Interaction Picture The technique of expressing operators and quantum states in the so-called interaction picture is of great importance in treating quantum-mechanical problems in a perturbative fashion. It plays an important role, for example, in the derivation of the Born–Markov master equation for decoherence detailed in Sect. 4.2.2. We shall review the basics of the interaction-picture approach in the following. We begin by considering a Hamiltonian of the form Hˆ = Hˆ0 + V.ˆ (A.1) Here Hˆ0 denotes the part of the Hamiltonian that describes the free (un- perturbed) evolution of a system S, whereas Vˆ is some added external per- turbation. In applications of the interaction-picture formalism, Vˆ is typically assumed to be weak in comparison with Hˆ0, and the idea is then to deter- mine the approximate dynamics of the system given this assumption. In the following, however, we shall proceed with exact calculations only and make no assumption about the relative strengths of the two components Hˆ0 and Vˆ . From standard quantum theory, we know that the expectation value of an operator observable Aˆ(t) is given by the trace rule [see (2.17)], & ' & ' ˆ ˆ Aˆ(t) =Tr Aˆ(t)ˆρ(t) =Tr Aˆ(t)e−iHtρˆ(0)eiHt . (A.2) For reasons that will immediately become obvious, let us rewrite this expres- sion as & ' ˆ ˆ ˆ ˆ ˆ ˆ Aˆ(t) =Tr eiH0tAˆ(t)e−iH0t eiH0te−iHtρˆ(0)eiHte−iH0t . (A.3) ˆ In inserting the time-evolution operators e±iH0t at the beginning and the end of the expression in square brackets, we have made use of the fact that the trace is cyclic under permutations of the arguments, i.e., that Tr AˆBˆCˆ ··· =Tr BˆCˆ ···Aˆ , etc.
    [Show full text]
  • Quantum Field Theory II
    Quantum Field Theory II T. Nguyen PHY 391 Independent Study Term Paper Prof. S.G. Rajeev University of Rochester April 20, 2018 1 Introduction The purpose of this independent study is to familiarize ourselves with the fundamental concepts of quantum field theory. In this paper, I discuss how the theory incorporates the interactions between quantum systems. With the success of the free Klein-Gordon field theory, ultimately we want to introduce the field interactions into our picture. Any interaction will modify the Klein-Gordon equation and thus its Lagrange density. Consider, for example, when the field interacts with an external source J(x). The Klein-Gordon equation and its Lagrange density are µ 2 (@µ@ + m )φ(x) = J(x); 1 m2 L = @ φ∂µφ − φ2 + Jφ. 2 µ 2 In a relativistic quantum field theory, we want to describe the self-interaction of the field and the interaction with other dynamical fields. This paper is based on my lecture for the Kapitza Society. The second section will discuss the concept of symmetry and conservation. The third section will discuss the self-interacting φ4 theory. Lastly, in the fourth section, I will briefly introduce the interaction (Dirac) picture and solve for the time evolution operator. 2 Noether's Theorem 2.1 Symmetries and Conservation Laws In 1918, the mathematician Emmy Noether published what is now known as the first Noether's Theorem. The theorem informally states that for every continuous symmetry there exists a corresponding conservation law. For example, the conservation of energy is a result of a time symmetry, the conservation of linear momentum is a result of a plane symmetry, and the conservation of angular momentum is a result of a spherical symmetry.
    [Show full text]
  • Entanglement of the Uncertainty Principle
    Open Journal of Mathematics and Physics | Volume 2, Article 127, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/9jhwx | published: 26 Jul 2020 | https://ojmp.org EX [original insight] Diamond Open Access Entanglement of the Uncertainty Principle Open Quantum Collaboration∗† August 15, 2020 Abstract We propose that position and momentum in the uncertainty principle are quantum entangled states. keywords: entanglement, uncertainty principle, quantum gravity The most updated version of this paper is available at https://osf.io/9jhwx/download Introduction 1. [1] 2. Logic leads us to conclude that quantum gravity–the theory of quan- tum spacetime–is the description of entanglement of the quantum states of spacetime and of the extra dimensions (if they really ex- ist) [2–7]. ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Quantum Collaboration 1 Quantum Gravity 3. quantum gravity quantum spacetime entanglement of spacetime 4. Quantum spaceti=me might be in a sup=erposition of the extra dimen- sions [8]. Notation 5. UP Uncertainty Principle 6. UP= the quantum state of the uncertainty principle ∣ ⟩ = Discussion on the notation 7. The universe is mathematical. 8. The uncertainty principle is part of our universe, thus, it is described by mathematics. 9. A physical theory must itself be described within its own notation. 10. Thereupon, we introduce UP . ∣ ⟩ Entanglement 11. UP α1 x1p1 α2 x2p2 α3 x3p3 ... 12. ∣xi ⟩u=ncer∣tainty⟩ +in po∣ sitio⟩n+ ∣ ⟩ + 13. pi = uncertainty in momentum 14. xi=pi xi pi xi pi ∣ ⟩ = ∣ ⟩ ∣ ⟩ = ∣ ⟩ ⊗ ∣ ⟩ 2 Final Remarks 15.
    [Show full text]
  • Path Integral for the Hydrogen Atom
    Path Integral for the Hydrogen Atom Solutions in two and three dimensions Vägintegral för Väteatomen Lösningar i två och tre dimensioner Anders Svensson Faculty of Health, Science and Technology Physics, Bachelor Degree Project 15 ECTS Credits Supervisor: Jürgen Fuchs Examiner: Marcus Berg June 2016 Abstract The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude. Sammanfattning V¨agintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen fr˚anklassisk meka- nik. Feynmans v¨agintegral kan ses som en summa ¨over alla m¨ojligav¨agaren partikel kan ta mellan tv˚a givna ¨andpunkterA och B, d¨arvarje v¨agbidrar till summan med en fasfaktor inneh˚allandeden klas- siska verkan f¨orv¨agen.Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln g˚arfr˚anA till B.
    [Show full text]
  • Two-State Systems
    1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A.
    [Show full text]
  • Interaction Representation
    Interaction Representation So far we have discussed two ways of handling time dependence. The most familiar is the Schr¨odingerrepresentation, where operators are constant, and states move in time. The Heisenberg representation does just the opposite; states are constant in time and operators move. The answer for any given physical quantity is of course independent of which is used. Here we discuss a third way of describing the time dependence, introduced by Dirac, known as the interaction representation, although it could equally well be called the Dirac representation. In this representation, both operators and states move in time. The interaction representation is particularly useful in problems involving time dependent external forces or potentials acting on a system. It also provides a route to the whole apparatus of quantum field theory and Feynman diagrams. This approach to field theory was pioneered by Dyson in the the 1950's. Here we will study the interaction representation in quantum mechanics, but develop the formalism in enough detail that the road to field theory can be followed. We consider an ordinary non-relativistic system where the Hamiltonian is broken up into two parts. It is almost always true that one of these two parts is going to be handled in perturbation theory, that is order by order. Writing our Hamiltonian we have H = H0 + V (1) Here H0 is the part of the Hamiltonian we have under control. It may not be particularly simple. For example it could be the full Hamiltonian of an atom or molecule. Neverthe- less, we assume we know all we need to know about the eigenstates of H0: The V term in H is the part we do not have control over.
    [Show full text]
  • Structure of a Spin ½ B
    Structure of a spin ½ B. C. Sanctuary Department of Chemistry, McGill University Montreal Quebec H3H 1N3 Canada Abstract. The non-hermitian states that lead to separation of the four Bell states are examined. In the absence of interactions, a new quantum state of spin magnitude 1/√2 is predicted. Properties of these states show that an isolated spin is a resonance state with zero net angular momentum, consistent with a point particle, and each resonance corresponds to a degenerate but well defined structure. By averaging and de-coherence these structures are shown to form ensembles which are consistent with the usual quantum description of a spin. Keywords: Bell states, Bell’s Inequalities, spin theory, quantum theory, statistical interpretation, entanglement, non-hermitian states. PACS: Quantum statistical mechanics, 05.30. Quantum Mechanics, 03.65. Entanglement and quantum non-locality, 03.65.Ud 1. INTRODUCTION In spite of its tremendous success in describing the properties of microscopic systems, the debate over whether quantum mechanics is the most fundamental theory has been going on since its inception1. The basic property of quantum mechanics that belies it as the most fundamental theory is its statistical nature2. The history is well known: EPR3 showed that two non-commuting operators are simultaneously elements of physical reality, thereby concluding that quantum mechanics is incomplete, albeit they assumed locality. Bohr4 replied with complementarity, arguing for completeness; thirty years later Bell5, questioned the locality assumption6, and the conclusion drawn that any deeper theory than quantum mechanics must be non-local. In subsequent years the idea that entangled states can persist to space-like separations became well accepted7.
    [Show full text]
  • Advanced Quantum Theory AMATH473/673, PHYS454
    Advanced Quantum Theory AMATH473/673, PHYS454 Achim Kempf Department of Applied Mathematics University of Waterloo Canada c Achim Kempf, September 2017 (Please do not copy: textbook in progress) 2 Contents 1 A brief history of quantum theory 9 1.1 The classical period . 9 1.2 Planck and the \Ultraviolet Catastrophe" . 9 1.3 Discovery of h ................................ 10 1.4 Mounting evidence for the fundamental importance of h . 11 1.5 The discovery of quantum theory . 11 1.6 Relativistic quantum mechanics . 13 1.7 Quantum field theory . 14 1.8 Beyond quantum field theory? . 16 1.9 Experiment and theory . 18 2 Classical mechanics in Hamiltonian form 21 2.1 Newton's laws for classical mechanics cannot be upgraded . 21 2.2 Levels of abstraction . 22 2.3 Classical mechanics in Hamiltonian formulation . 23 2.3.1 The energy function H contains all information . 23 2.3.2 The Poisson bracket . 25 2.3.3 The Hamilton equations . 27 2.3.4 Symmetries and Conservation laws . 29 2.3.5 A representation of the Poisson bracket . 31 2.4 Summary: The laws of classical mechanics . 32 2.5 Classical field theory . 33 3 Quantum mechanics in Hamiltonian form 35 3.1 Reconsidering the nature of observables . 36 3.2 The canonical commutation relations . 37 3.3 From the Hamiltonian to the equations of motion . 40 3.4 From the Hamiltonian to predictions of numbers . 44 3.4.1 Linear maps . 44 3.4.2 Choices of representation . 45 3.4.3 A matrix representation . 46 3 4 CONTENTS 3.4.4 Example: Solving the equations of motion for a free particle with matrix-valued functions .
    [Show full text]
  • Hamilton Equations, Commutator, and Energy Conservation †
    quantum reports Article Hamilton Equations, Commutator, and Energy Conservation † Weng Cho Chew 1,* , Aiyin Y. Liu 2 , Carlos Salazar-Lazaro 3 , Dong-Yeop Na 1 and Wei E. I. Sha 4 1 College of Engineering, Purdue University, West Lafayette, IN 47907, USA; [email protected] 2 College of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 3 Physics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 4 College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] * Correspondence: [email protected] † Based on the talk presented at the 40th Progress In Electromagnetics Research Symposium (PIERS, Toyama, Japan, 1–4 August 2018). Received: 12 September 2019; Accepted: 3 December 2019; Published: 9 December 2019 Abstract: We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations. Keywords: quantum mechanics; commutator relations; Heisenberg picture 1. Introduction In quantum theory, a classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator.
    [Show full text]