DOCSLIB.ORG

Quantum Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Mathematics Quantum Mathematics by Peter J. Olver University of Minnesota Table of Contents 1. Introduction .................... 3 2. Hamiltonian Mechanics ............... 3 Hamiltonian Systems ................ 3 Poisson Brackets ................. 7 Symmetries and First Integrals ........... 9 Hamilton-Jacobi Theory .............. 11 3. Geometric Optics and Wave Mechanics ....... 25 The Wave Equation ................ 25 Maxwell’s Equations ................ 27 High Frequency Limit and Quantization ........ 29 4. Basics of Quantum Mechanics ........... 36 Phase space ....................36 Observables ....................37 Schr¨odinger’s Equation ............... 38 Spectrum .....................39 The Uncertainty Principle ............. 43 5. Linear Operators and Their Spectrum ....... 44 Hilbert Space and Linear Operators ......... 44 The Spectral Theorem ............... 46 Spectrum of Operators ............... 48 Dispersion and Stationary Phase ........... 55 6. One–Dimensional Problems ............. 59 The Harmonic Oscillator .............. 63 Scattering and Inverse Scattering .......... 65 Isopectral Flows ..................75 9/9/21 1 c 2021 Peter J. Olver 7. Symmetry in Quantum Mechanics ......... 77 Why is Symmetry Important? ............ 77 Group Representations ............... 79 Lie Algebra Representations ............. 81 Representations of the Orthogonal Group ....... 83 The Hydrogen Atom ................ 91 The Zeeman Effect ................. 96 Addition of Angular Momenta ............ 97 Intensities and Selection Rules ............104 Parity ......................106 Spin .......................107 Systems of Identical Particles ............113 Representation Theory of Finite Groups . .116 Representations of the Symmetric Group . .119 Symmetry Classes of Tensors ............122 Tensor Products ..................126 Plethysm .....................128 Orbital–Spin and Total Angular Momentum Coupling . 128 8. Relativistic Quantum Mechanics ..........132 The Lorentz Group .................133 The Dirac Equation ................136 Clifford Algebras ..................137 Clifford Groups and Spinors .............142 Representations of the Lorentz Group .. .. .. .146 Relativistic Spinors ................149 Lorentz–Invariant Equations ............150 Symmetries of the Dirac Equation ..........151 Relativistic Spin ..................152 Particle in an Electromagnetic Field .. .. .. .152 Solutions of the Dirac Equation ...........153 Spherically Symmetric Potential ...........154 Non-relativistic Limit ...............155 References .....................157 9/9/21 2 c 2021 Peter J. Olver 1. Introduction. These are the lecture notes that I prepared for a graduate level course on the basics of mathematical quantum mechanics that I gave in the academic year 1989–90 at the University of Minnesota. The original notes were typed in Microsoft Word and hence never distributed widely. After 15 years elapsed, I was finally inspired to convert them to TEX. Aside from minor editing, corrections, reordering of some of the material, and updating the references, they are the same as were distributed to the students in the class. The goal was to provide a broad overview of mathematical techniques used in quan- tum mechanics. While applications appear throughout, the emphasis is on the fundamental mathematical tools used in the physical theory. Techniques include Hamiltonian mechan- ics, quantization, operator theory, stationary phase, inverse scattering and solitons, group representation theory, Clifford algebras, and much more. I hope that you, as reader, will find them of use and as enjoyable as when I first prepared them and gave the course. 2. Hamiltonian Mechanics. References: [4], [16], [23], [25], [27], [40]. The fundamental insight of Schr¨odinger’s approach to quantum mechanics was that the quantum mechanical counterparts to the equations of classical mechanics are most easily found by writing the latter in Hamiltonian form. Indeed, Hamilton himself could, in direct analogy with the correspondence between geometric optics and wave optics — the prototype of wave/particle duality — have straightforwardly written down the equa- tions of “wave mechanics”, and thus anticipated quantum mechanics by almost a century. However, the lack of any physical motivation for taking this conceptual leap prevented such a mathematical advance occurring before its time. We thus begin our study with an overview of basic Hamiltonian mechanics. Hamiltonian Systems We consider a system of first order ordinary differential equations on the phase space 2n M = R , with canonical local coordinates (p, q)=(p1,...,pn, q1,...,qn). The system is called Hamiltonian if there is a smooth function H(p, q) such that it can be written in the form dp ∂H dq ∂H i = , i = , i =1,...,n. (2.1) dt − ∂qi dt ∂pi Typically the q’s represent the coordinates of the particles in the system, the p’s the corresponding momenta, and H is the physical energy. Essentially, Hamilton’s equations provide a convenient canonical form for writing the equations of conservative classical mechanics. A good example is the motion of masses in a potential force field. Here the q’s represent the positions of the masses. Let U(t, q) be the potential function, so the corresponding force 9/9/21 3 c 2021 Peter J. Olver field is its negative spatial gradient: F = U = ( ∂U/∂q1,..., ∂U/∂qn). Newton’s equations are −∇ − − 2 d qi ∂U mi 2 = , i =1,...,n, dt − ∂qi th where mi is the i mass. (For three-dimensional motion, n =3m, and the positions and masses are labelled accordingly.) To put these equations in Hamiltonian form, we let pi = mi qi, be the ith momentum, and the Newtonian dot notation is used for time derivatives: q = dq/dt. We define the Hamiltonian function n n 2 1 2 pi H = mi qi + U(t, q) = + U(t, q) 2 2mi iX=1 iX=1 to be the total energy (kinetic + potential). Then Hamilton’s equations (2.1) are dp ∂U dq p i = , i = i , i =1,...,n, dt − ∂qi dt mi and are clearly equivalent to Newton’s equations of motion. The system of differential equations governing the solutions to any (first order) vari- ational problem in one independent variable can always be put into Hamiltonian form. Indeed, consider the problem of minimizing b [q ] = L(t,q, q),dt q(a) = a, q(b) = b, (2.2) I Za where the Lagrangian L is a smooth function of t, qi, and qi = dqi/dt. Sufficiently smooth minima must satisfy the associated Euler-Lagrange equations d ∂L ∂L Ei(L) = + =0, i =1,...,n. (2.3) − dt ∂qi ∂qi To see this, suppose q(t) is a smooth minimizer of (2.2). Then for smooth perturbations h(t) satisfying h(a) = h(b) = 0 so as to preserve the boundary conditions, we must have d b n ∂L ∂L dh 0 = [q + εh] = h (t) + i dt dε I ∂q i ∂q dt ε=0 Za i=1 i i X b n ∂L d ∂L = h (t) dt, ∂q − dt ∂q i a i=1 i i Z X where we integrated the second set of summands by parts and used the boundary condi- tions on h to eliminate the boundary terms. Since the function h(t) is arbitrary, an easy argument (the so-called duBois–Reymond Lemma) shows that the quantities in brackets must vanish, implying (2.3). 9/9/21 4 c 2021 Peter J. Olver In general, we define the momenta via the Legendre transformation ∂L pi = , (2.4) ∂qi and impose the nondegeneracy assumption ∂2L det =0, (2.5) ∂qi∂qj ! 6 which, by the Implicit Function Theorem, allows us to locally solve (2.4) for q as a function of t, p, q: qi = ϕi(t, p, q). (2.6) Define the Hamiltonian function to be n H(t, p, q) = p q L(t,q, q), (2.7) i i − i=1 X where we use (2.6) to replace q. Then the Euler-Lagrange equations (2.3) are equivalent to Hamilton’s equations (2.1). The easiest way to see this is to use differentials: ∂H ∂H ∂H dH = dt + dqi + dpi. ∂t ∂qi ∂pi But, using (2.7), (2.4), ∂L ∂L ∂L ∂L ∂L dH = pidqi + qidpi dt dqi dqi = qidpi dt dqi. − ∂t − ∂qi − ∂qi − ∂t − ∂qi Equating the latter two expressions, we deduce that ∂H ∂L ∂H ∂H ∂L d ∂L = , = qi, = = = pi. ∂t − ∂t ∂pi ∂qi −∂qi −dt ∂qi − from which Hamilton’s equations (2.1) follow immediately. Example 2.1. In particle dynamics, the Lagrangian is the difference of kinetic and potential energy: n 1 L = m q2 U(t, q). 2 i i − i=1 X The Euler-Lagrange equations are just Newton’s laws F = ma: ∂U mi qi + =0, i =1,...,n. (2.8) ∂qi To place this system of second order ordinary differential equations in Hamiltonian form, in view of (2.4), set ∂L pi = = mi qi, ∂qi 9/9/21 5 c 2021 Peter J. Olver which is exactly the ith physical momentum. Thus, the Hamiltonian function (2.7) takes the form n n 1 p2 H(p, q) = m q2 + U(t, q) = i + U(t, q), 2 i i 2m i=1 i=1 i X X and is the total energy. As we have seen, Hamilton’s equations (2.1) are equivalent to Newton’s equations (2.8). Example 2.2. Geometric Optics: According to Fermat’s Principle, the light rays traveling between two points follows the path requiring the least time. If the light is traveling in a medium with index of refraction n(q), which is the ratio of the speed of light in the medium to the speed of light in vacuo, c, then its velocity is ds/dt = c/n. Therefore, taking units in which c = 1, the time to transit between two points A and B is B B t = dt = n(q) ds, ZA ZA where ds denotes the element of arc length on the path travelled by the light. Fermat says that we need to minimize (or at least find stationary values for) this functional. Now ds is not a very convenient parametrization, as the endpoints of the functional are no longer fixed. However, the integral does not depend on the particular parametrization of the path followed by the light ray, so that we can restrict attention to parametrized curves q(t) where the parameter t varies over a fixed interval [a,b].
Load more

Recommended publications
  • Strocchi's Quantum Mechanics
    Strocchi’s Quantum Mechanics: An alternative formulation to the dominant one? Antonino Drago ‒ Formerly at Naples University “Federico II”, Italy ‒ drago@un ina.it Abstract: At first glance, Strocchi’s formulation presents several characteri- stic features of a theory whose two choices are the alternative ones to the choices of the paradigmatic formulation: i) Its organization starts from not axioms, but an operative basis and it is aimed to solve a problem (i.e. the indeterminacy); moreover, it argues through both doubly negated proposi- tions and an ad absurdum proof; ii) It put, before the geometry, a polyno- mial algebra of bounded operators; which may pertain to constructive Mathematics. Eventually, it obtains the symmetries. However one has to solve several problems in order to accurately re-construct this formulation according to the two alternative choices. I conclude that rather than an al- ternative to the paradigmatic formulation, Strocchi’s represents a very inter- esting divergence from it. Keywords: Quantum Mechanics, C*-algebra approach, Strocchi’s formula- tion, Two dichotomies, Constructive Mathematics, Non-classical Logic 1. Strocchi’s Axiomatic of the paradigmatic formulation and his criticisms to it Segal (1947) has suggested a foundation of Quantum Mechanics (QM) on an algebraic approach of functional analysis; it is independent from the space-time variables or any other geometrical representation, as instead a Hilbert space is. By defining an algebra of the observables, it exploits Gelfand-Naimark theorem in order to faithfully represent this algebra into Hilbert space and hence to obtain the Schrödinger representation of QM. In the 70’s Emch (1984) has reiterated this formulation and improved it.
    [Show full text]
  • Quantum Information
    Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding .
    [Show full text]
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • A Representations of SU(2)
    A Representations of SU(2) In this appendix we provide details of the parameterization of the group SU(2) and differential forms on the group space. An arbitrary representation of the group SU(2) is given by the set of three generators Tk, which satisfy the Lie algebra [Ti,Tj ]=iεijkTk, with ε123 =1. The element of the group is given by the matrix U =exp{iT · ω} , (A.1) 1 where in the fundamental representation Tk = 2 σk, k =1, 2, 3, with 01 0 −i 10 σ = ,σ= ,σ= , 1 10 2 i 0 3 0 −1 standard Pauli matrices, which satisfy the relation σiσj = δij + iεijkσk . The vector ω has components ωk in a given coordinate frame. Geometrically, the matrices U are generators of spinor rotations in three- 3 dimensional space R and the parameters ωk are the corresponding angles of rotation. The Euler parameterization of an arbitrary matrix of SU(2) transformation is defined in terms of three angles θ, ϕ and ψ,as ϕ θ ψ iσ3 iσ2 iσ3 U(ϕ, θ, ψ)=Uz(ϕ)Uy(θ)Uz(ψ)=e 2 e 2 e 2 ϕ ψ ei 2 0 cos θ sin θ ei 2 0 = 2 2 −i ϕ θ θ − ψ 0 e 2 − sin cos 0 e i 2 2 2 i i θ 2 (ψ+ϕ) θ − 2 (ψ−ϕ) cos 2 e sin 2 e = i − − i . (A.2) − θ 2 (ψ ϕ) θ 2 (ψ+ϕ) sin 2 e cos 2 e Thus, the SU(2) group manifold is isomorphic to three-sphere S3.
    [Show full text]
  • Geometric Quantization of Chern Simons Gauge Theory
    J. DIFFERENTIAL GEOMETRY 33(1991) 787 902 GEOMETRIC QUANTIZATION OF CHERN SIMONS GAUGE THEORY SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN Abstract We present a new construction of the quantum Hubert space of Chern Simons gauge theory using methods which are natural from the three dimensional point of view. To show that the quantum Hubert space associated to a Riemann surface Σ is independent of the choice of com plex structure on Σ, we construct a natural projectively flat connection on the quantum Hubert bundle over Teichmuller space. This connec tion has been previously constructed in the context of two dimensional conformal field theory where it is interpreted as the stress energy tensor. Our construction thus gives a (2 + 1 ) dimensional derivation of the basic properties of (1 + 1) dimensional current algebra. To construct the con nection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hubert bundle may be ex pressed purely in terms of the intrinsic Kahler geometry of the quotient and the Quillen connection on a certain determinant line bundle. The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection. As an example, we treat the case when Σ has genus one explicitly. We also make some preliminary comments con cern ing the Hubert space structure. Introduction Several years ago, in examining the proof of a rather surprising result about von Neumann algebras, V. F. R. Jones [20] was led to the discovery of some unusual representations of the braid group from which invariants of links in S3 can be constructed.
    [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]
  • Multipartite Quantum States and Their Marginals
    Diss. ETH No. 22051 Multipartite Quantum States and their Marginals A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Michael Walter Diplom-Mathematiker arXiv:1410.6820v1 [quant-ph] 24 Oct 2014 Georg-August-Universität Göttingen born May 3, 1985 citizen of Germany accepted on the recommendation of Prof. Dr. Matthias Christandl, examiner Prof. Dr. Gian Michele Graf, co-examiner Prof. Dr. Aram Harrow, co-examiner 2014 Abstract Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions – such as Pauli’s exclusion principle or the monogamy of quantum entanglement – which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part of this thesis (Chapters 2–6) we focus on the one-body marginals of multipartite quantum states. Starting from a novel ge- ometric solution of the compatibility problem, we then turn towards the phenomenon of quantum entanglement. We find that the one-body marginals through their local eigenvalues can characterize the entan- glement of multipartite quantum states, and we propose the notion of an entanglement polytope for its systematic study.
    [Show full text]
  • Geometric Quantization
    GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bundle fibers are the momentum vectors. The solutions to Hamilton's equations (this is where the symplectic structure comes in) are the equations of motion of the physical system. The input is the total energy (the Hamiltonian function on the phase space), and the output is the Hamiltonian vector field, whose flow gives the time evolution of the system, say the motion of a particle acted on by certain forces. The Hamiltonian formalism also allows one to easily compute the values of any physical quantities (observables, functions on the phase space such as the Hamiltonian or the formula for angular momentum) using the Hamiltonian function and the symplectic structure. It turns out that various experiments showed that the Hamiltonian formalism of mechanics is sometimes inadequate, and the highly counter-intuitive quantum mechanical model turned out to produce more correct answers. Quantum mechanics is totally different from classical mechanics. In this model of reality, the position (or momentum) of a particle is an element of a Hilbert space, and an observable is an operator acting on the Hilbert space. The inner product on the Hilbert space provides structure that allows computations to be made, similar to the way the symplectic structure is a computational tool in Hamiltonian mechanics.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Some Background on Geometric Quantization
    SOME BACKGROUND ON GEOMETRIC QUANTIZATION NILAY KUMAR Contents 1. Introduction1 2. Classical versus quantum2 3. Geometric quantization4 References6 1. Introduction Let M be a compact oriented 3-manifold. Chern-Simons theory with gauge group G (that we will take to be compact, connected, and simply-connected) on M is the data of a principal G-bundle π : P ! M together with a Lagrangian density L : A ! Ω3(P ) on the space of connections A on P given by 2 LCS(A) = hA ^ F i + hA ^ [A ^ A]i: 3 Let us detail the notation used here. Recall first that a connection A 2 A is 1 ∗ a G-invariant g-valued one-form, i.e. A 2 Ω (P ; g) such that RgA = Adg−1 A, satisfying the additional condition that if ξ 2 g then A(ξP ) = ξ if ξP is the vector field associated to ξ. Notice that A , though not a vector space, is an affine space 1 modelled on Ω (M; P ×G g). The curvature F of a connection A is is the g-valued two-form given by F (v; w) = dA(vh; wh), where •h denotes projection onto the 1 horizontal distribution ker π∗. Finally, by h−; −i we denote an ad-invariant inner product on g. The Chern-Simons action is now given Z SSC(A) = LSC(A) M and the quantities of interest are expectation values of observables O : A ! R Z hOi = O(A)eiSSC(A)=~: A =G Here G is the group of automorphisms of P ! Σ, which acts by pullback on A { the physical states are unaffected by these gauge transformations, so we integrate over the quotient A =G to eliminate the redundancy.
    [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]