Theory of Angular Momentum Rotations About Different Axes Fail to Commute
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Introduction
Introduction This dissertation is a reading of chapters 16 (Introduction to Integer Liner Programming) and 19 (Totally Unimodular Matrices: fundamental properties and examples) in the book : Theory of Linear and Integer Programming, Alexander Schrijver, John Wiley & Sons © 1986. The chapter one is a collection of basic definitions (polyhedron, polyhedral cone, polytope etc.) and the statement of the decomposition theorem of polyhedra. Chapter two is “Introduction to Integer Linear Programming”. A finite set of vectors a1 at is a Hilbert basis if each integral vector b in cone { a1 at} is a non- ,… , ,… , negative integral combination of a1 at. We shall prove Hilbert basis theorem: Each ,… , rational polyhedral cone C is generated by an integral Hilbert basis. Further, an analogue of Caratheodory’s theorem is proved: If a system n a1x β1 amx βm has no integral solution then there are 2 or less constraints ƙ ,…, ƙ among above inequalities which already have no integral solution. Chapter three contains some basic result on totally unimodular matrices. The main theorem is due to Hoffman and Kruskal: Let A be an integral matrix then A is totally unimodular if and only if for each integral vector b the polyhedron x x 0 Ax b is integral. ʜ | ƚ ; ƙ ʝ Next, seven equivalent characterization of total unimodularity are proved. These characterizations are due to Hoffman & Kruskal, Ghouila-Houri, Camion and R.E.Gomory. Basic examples of totally unimodular matrices are incidence matrices of bipartite graphs & Directed graphs and Network matrices. We prove Konig-Egervary theorem for bipartite graph. 1 Chapter 1 Preliminaries Definition 1.1: (Polyhedron) A polyhedron P is the set of points that satisfies Μ finite number of linear inequalities i.e., P = {xɛ ő | ≤ b} where (A, b) is an Μ m (n + 1) matrix. -
Arxiv:2012.00197V2 [Quant-Ph] 7 Feb 2021
Unconventional supersymmetric quantum mechanics in spin systems 1, 1 1, Amin Naseri, ∗ Yutao Hu, and Wenchen Luo y 1School of Physics and Electronics, Central South University, Changsha, P. R. China 410083 It is shown that the eigenproblem of any 2 × 2 matrix Hamiltonian with discrete eigenvalues is involved with a supersymmetric quantum mechanics. The energy dependence of the superal- gebra marks the disparity between the deduced supersymmetry and the standard supersymmetric quantum mechanics. The components of an eigenspinor are superpartners|up to a SU(2) trans- formation|which allows to derive two reduced eigenproblems diagonalizing the Hamiltonian in the spin subspace. As a result, each component carries all information encoded in the eigenspinor. We p also discuss the generalization of the formalism to a system of a single spin- 2 coupled with external fields. The unconventional supersymmetry can be regarded as an extension of the Fulton-Gouterman transformation, which can be established for a two-level system coupled with multi oscillators dis- playing a mirror symmetry. The transformation is exploited recently to solve Rabi-type models. Correspondingly, we illustrate how the supersymmetric formalism can solve spin-boson models with no need to appeal a symmetry of the model. Furthermore, a pattern of entanglement between the components of an eigenstate of a many-spin system can be unveiled by exploiting the supersymmet- ric quantum mechanics associated with single spins which also recasts the eigenstate as a matrix product state. Examples of many-spin models are presented and solved by utilizing the formalism. I. INTRODUCTION sociated with single spins. This allows to reconstruct the eigenstate as a matrix product state (MPS) [2, 5]. -
{|Sz; ↑〉, |S Z; ↓〉} the Spin Operators Sx = (¯H
SU(2) • the spin returns to its original direction after time In the two dimensional space t = 2π/ωc. {|Sz; ↑i, |Sz; ↓i} • the wave vector returns to its original value after time t = 4π/ωc. the spin operators ¯h Matrix representation Sx = {(|Sz; ↑ihSz; ↓ |) + (|Sz; ↓ihSz; ↑ |)} 2 In the basis {|Sz; ↑i, |Sz; ↓i} the base vectors are i¯h represented as S = {−(|S ; ↑ihS ; ↓ |) + (|S ; ↓ihS ; ↑ |)} y 2 z z z z 1 0 ¯h |S ; ↑i 7→ ≡ χ |S ; ↓i 7→ ≡ χ S = {(|S ; ↑ihS ; ↑ |) − (|S ; ↓ihS ; ↓ |)} z 0 ↑ z 1 ↓ z 2 z z z z † † hSz; ↑ | 7→ (1, 0) ≡ χ↑ hSz; ↓ | 7→ (0, 1) ≡ χ↓, satisfy the angular momentum commutation relations so an arbitrary state vector is represented as [Sx,Sy] = i¯hSz + cyclic permutations. hSz; ↑ |αi Thus the smallest dimension where these commutation |αi 7→ hSz; ↓ |αi relations can be realized is 2. hα| 7→ (hα|S ; ↑i, hα|S ; ↓i). The state z z The column vector |αi = |Sz; ↑ihSz; ↑ |αi + |Sz; ↓ihSz; ↓ |αi hS ; ↑ |αi c behaves in the rotation χ = z ≡ ↑ hSz; ↓ |αi c↓ iS φ D (φ) = exp − z z ¯h is called the two component spinor like Pauli’s spin matrices Pauli’s spin matrices σi are defined via the relations iSzφ Dz(φ)|αi = exp − |αi ¯h ¯h −iφ/2 (Sk)ij ≡ (σk)ij, = e |Sz; ↑ihSz; ↑ |αi 2 iφ/2 +e |Sz; ↓ihSz; ↓ |αi. where the matrix elements are evaluated in the basis In particular: {|Sz; ↑i, |Sz; ↓i}. Dz(2π)|αi = −|αi. For example ¯h S = S = {(|S ; ↑ihS ; ↓ |) + (|S ; ↓ihS ; ↑ |)}, Spin precession 1 x 2 z z z z When the Hamiltonian is so H = ωcSz (S1)11 = (S1)22 = 0 the time evolution operator is ¯h (S1)12 = (S1)21 = , iS ω t 2 U(t, 0) = exp − z c = D (ω t). -
Qualification Exam: Quantum Mechanics
Qualification Exam: Quantum Mechanics Name: , QEID#43228029: July, 2019 Qualification Exam QEID#43228029 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1=2 particles interacting with one another and with an external uniform magnetic field B~ directed along the z-axis. The Hamiltonian is given by ~ ~ ~ ~ ~ H = −AS1 · S2 − µB(g1S1 + g2S2) · B where µB is the Bohr magneton, g1 and g2 are the g-factors, and A is a constant. 1. In the large field limit, what are the eigenvectors and eigenvalues of H in the "spin-space" { i.e. in the basis of eigenstates of S1z and S2z? 2. In the limit when jB~ j ! 0, what are the eigenvectors and eigenvalues of H in the same basis? 3. In the Intermediate regime, what are the eigenvectors and eigenvalues of H in the spin space? Show that you obtain the results of the previous two parts in the appropriate limits. Problem 2. 1983-Fall-QM-U-2 ID:QM-U-20 1. Show that, for an arbitrary normalized function j i, h jHj i > E0, where E0 is the lowest eigenvalue of H. 2. A particle of mass m moves in a potential 1 kx2; x ≤ 0 V (x) = 2 (1) +1; x < 0 Find the trial state of the lowest energy among those parameterized by σ 2 − x (x) = Axe 2σ2 : What does the first part tell you about E0? (Give your answers in terms of k, m, and ! = pk=m). Problem 3. 1983-Fall-QM-U-3 ID:QM-U-44 Consider two identical particles of spin zero, each having a mass m, that are con- strained to rotate in a plane with separation r. -
Quantum/Classical Interface: Fermion Spin
Quantum/Classical Interface: Fermion Spin W. E. Baylis, R. Cabrera, and D. Keselica Department of Physics, University of Windsor Although intrinsic spin is usually viewed as a purely quantum property with no classical ana- log, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise natu- rally in the Clifford’s geometric algebra of physical space and not only provide powerful tools for solving problems in classical electrodynamics, but also reproduce a number of quantum results. In particular, many properites of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, and its magnetic moment to Zitterbewegung and de Broglie waves. The relationship is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach resolves Pauli’s argument against treating time as an operator by recognizing phase factors as projected rotation operators. I. INTRODUCTION The intrinsic spin of elementary fermions like the electron is traditionally viewed as an essentially quantum property with no classical counterpart.[1, 2] Its two-valued nature, the fact that any measurement of the spin in an arbitrary direction gives a statistical distribution of either “spin up” or “spin down” and nothing in between, together with the commutation relation between orthogonal components, is responsible for many of its “nonclassical” properties. -
Perfect, Ideal and Balanced Matrices: a Survey
Perfect Ideal and Balanced Matrices Michele Conforti y Gerard Cornuejols Ajai Kap o or z and Kristina Vuskovic December Abstract In this pap er we survey results and op en problems on p erfect ideal and balanced matrices These matrices arise in connection with the set packing and set covering problems two integer programming mo dels with a wide range of applications We concentrate on some of the b eautiful p olyhedral results that have b een obtained in this area in the last thirty years This survey rst app eared in Ricerca Operativa Dipartimento di Matematica Pura ed Applicata Universitadi Padova Via Belzoni Padova Italy y Graduate School of Industrial administration Carnegie Mellon University Schenley Park Pittsburgh PA USA z Department of Mathematics University of Kentucky Lexington Ky USA This work was supp orted in part by NSF grant DMI and ONR grant N Introduction The integer programming mo dels known as set packing and set covering have a wide range of applications As examples the set packing mo del o ccurs in the phasing of trac lights Stoer in pattern recognition Lee Shan and Yang and the set covering mo del in scheduling crews for railways airlines buses Caprara Fischetti and Toth lo cation theory and vehicle routing Sometimes due to the sp ecial structure of the constraint matrix the natural linear programming relaxation yields an optimal solution that is integer thus solving the problem We investigate conditions under which this integrality prop erty holds Let A b e a matrix This matrix is perfect if the fractional -
Order Number 9807786 Effectiveness in Representations of Positive Definite Quadratic Forms Icaza Perez, Marfa In6s, Ph.D
Order Number 9807786 Effectiveness in representations of positive definite quadratic form s Icaza Perez, Marfa In6s, Ph.D. The Ohio State University, 1992 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 E ffectiveness in R epresentations o f P o s it iv e D e f i n it e Q u a d r a t ic F o r m s dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Maria Ines Icaza Perez, The Ohio State University 1992 Dissertation Committee: Approved by John S. Hsia Paul Ponomarev fJ Adviser Daniel B. Shapiro lepartment of Mathematics A cknowledgements I begin these acknowledgements by first expressing my gratitude to Professor John S. Hsia. Beyond accepting me as his student, he provided guidance and encourage ment which has proven invaluable throughout the course of this work. I thank Professors Paul Ponomarev and Daniel B. Shapiro for reading my thesis and for their helpful suggestions. I also express my appreciation to Professor Shapiro for giving me the opportunity to come to The Ohio State University and for helping me during the first years of my studies. My recognition goes to my undergraduate teacher Professor Ricardo Baeza for teaching me mathematics and the discipline of hard work. I thank my classmates from the Quadratic Forms Seminar especially, Y. Y. Shao for all the cheerful mathematical conversations we had during these months. These years were more enjoyable thanks to my friends from here and there. I thank them all. -
Matrices in Companion Rings, Smith Forms, and the Homology of 3
Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds Vanni Noferini∗and Gerald Williams† July 26, 2021 Abstract We study the Smith forms of matrices of the form f(Cg) where f(t),g(t) ∈ R[t], where R is an elementary divisor domain and Cg is the companion matrix of the (monic) polynomial g(t). Prominent examples of such matrices are circulant matrices, skew-circulant ma- trices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix f(Cg) to that of the matrix F (CG), where F, G are quotients of f(t),g(t) by some common divi- sor. This allows us to express the last non-zero determinantal divisor of f(Cg) as a resultant. A key tool is the observation that a matrix ring generated by Cg – the companion ring of g(t) – is isomorphic to the polynomial ring Qg = R[t]/<g(t) >. We relate several features of the Smith form of f(Cg) to the properties of the polynomial g(t) and the equivalence classes [f(t)] ∈ Qg. As an application we let f(t) be the Alexander polynomial of a torus knot and g(t) = tn − 1, and calculate the Smith form of the circulant matrix f(Cg). By appealing to results concerning cyclic branched covers of knots and cyclically pre- sented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r, s are coprime. Keywords: Smith form, elementary divisor domain, circulant, cyclically presented group, Brieskorn manifold, homology. -
1. the Hamiltonian, H = Α Ipi + Βm, Must Be Her- Mitian to Give Real
1. The hamiltonian, H = αipi + βm, must be her- yields mitian to give real eigenvalues. Thus, H = H† = † † † pi αi +β m. From non-relativistic quantum mechanics † † † αi pi + β m = aipi + βm (2) we know that pi = pi . In addition, [αi,pj ] = 0 because we impose that the α, β operators act on the spinor in- † † αi = αi, β = β. Thus α, β are hermitian operators. dices while pi act on the coordinates of the wave function itself. With these conditions we may proceed: To verify the other properties of the α, and β operators † † piαi + β m = αipi + βm (1) we compute 2 H = (αipi + βm) (αj pj + βm) 2 2 1 2 2 = αi pi + (αiαj + αj αi) pipj + (αiβ + βαi) m + β m , (3) 2 2 where for the second term i = j. Then imposing the Finally, making use of bi = 1 we can find the 26 2 2 2 relativistic energy relation, E = p + m , we obtain eigenvalues: biu = λu and bibiu = λbiu, hence u = λ u the anti-commutation relation | | and λ = 1. ± b ,b =2δ 1, (4) i j ij 2. We need to find the λ = +1/2 helicity eigenspinor { } ′ for an electron with momentum ~p = (p sin θ, 0, p cos θ). where b0 = β, bi = αi, and 1 n n idenitity matrix. ≡ × 1 Using the anti-commutation relation and the fact that The helicity operator is given by 2 σ pˆ, and the posi- 2 1 · bi = 1 we are able to find the trace of any bi tive eigenvalue 2 corresponds to u1 Dirac solution [see (1.5.98) in http://arXiv.org/abs/0906.1271]. -
A Classical Spinor Approach to the Quantum/Classical Interface
A Classical Spinor Approach to the Quantum/Classical Interface William E. Baylis and J. David Keselica Department of Physics, University of Windsor A promising approach to the quantum/classical interface is described. It is based on a formulation of relativistic classical mechanics in the Cli¤ord algebra of physical space. Spinors and projectors arise naturally and provide powerful tools for solving problems in classical electrodynamics. They also reproduce many quantum results, allowing insight into quantum processes. I. INTRODUCTION The quantum/classical (Q/C) interface has long been a subject area of interest. Not only should it shed light on quantum processes, it may also hold keys to unifying quantum theory with classical relativity. Traditional studies of the interface have largely concentrated on quantum systems in states of large quantum numbers and the relation of solutions to classical chaos,[1] to quantum states in decohering interactions with the environment,[2] or in continuum states, where semiclassical approximations are useful.[3] Our approach[4–7] is fundamentally di¤erent. We start with a description of classical dynamics using Cli¤ord’s geometric algebra of physical space (APS). An important tool in the algebra is the amplitude of the Lorentz transfor- mation that boosts the system from its rest frame to the lab. This enters as a spinor in a classical context, one that satis…es linear equations of evolution suggesting superposition and interference, and that bears a close relation to the quantum wave function. Although APS is the Cli¤ord algebra generated by a three-dimensional Euclidean space, it contains a four- dimensional linear space with a Minkowski spacetime metric that allows a covariant description of relativistic phe- nomena. -
Unimodular Modules Paul Camion1
Discrete Mathematics 306 (2006) 2355–2382 www.elsevier.com/locate/disc Unimodular modules Paul Camion1 CNRS, Universite Pierre et Marie Curie, Case Postale 189-Combinatoire, 4, Place Jussieu, 75252 Paris Cedex 05, France Received 12 July 2003; received in revised form 4 January 2006; accepted 9 January 2006 Available online 24 July 2006 Abstract The present publication is mainly a survey paper on the author’s contributions on the relations between graph theory and linear algebra. A system of axioms introduced by Ghouila-Houri allows one to generalize to an arbitrary Abelian group the notion of interval in a linearly ordered group and to state a theorem that generalizes two due to A.J. Hoffman. The first is on the feasibility of a system of inequalities and the other is Hoffman’s circulation theorem reported in the first Berge’s book on graph theory. Our point of view permitted us to prove classical results of linear programming in the more general setting of linearly ordered groups and rings. It also shed a new light on convex programming. © 2006 Published by Elsevier B.V. Keywords: Graph theory; Linear algebra; Abelian group; Ring; Modules; Generator; Linearly ordered group; Linearly ordered ring; Convexity; Polyhedron; Matroid 1. Introduction 1.1. Motivations We refer here to Berge’s book [3]. All results in the present paper have their origin in Section 15 of that book, translated into English from [1], and in Berge’s subsequent paper [2]. The aim of this paper, which is a rewriting of part of [14], is to gather together many of the results known already in 1965 and to set them in an algebraic context. -
Introduction to Lattices Instructor: Daniele Micciancio UCSD CSE
CSE 206A: Lattice Algorithms and Applications Winter 2010 1: Introduction to Lattices Instructor: Daniele Micciancio UCSD CSE Lattices are regular arrangements of points in Euclidean space. They naturally occur in many settings, like crystallography, sphere packings (stacking oranges), etc. They have many applications in computer science and mathematics, including the solution of inte- ger programming problems, diophantine approximation, cryptanalysis, the design of error correcting codes for multi antenna systems, and many more. Recently, lattices have also attracted much attention as a source of computational hardness for the design of secure cryptographic functions. This course oers an introduction to lattices. We will study the best currently known algorithms to solve the most important lattice problems, and how lattices are used in several representative applications. We begin with the denition of lattices and their most important mathematical properties. 1. Lattices Denition 1. A lattice is a discrete additive subgroup of Rn, i.e., it is a subset Λ ⊆ Rn satisfying the following properties: (subgroup) Λ is closed under addition and subtraction,1 (discrete) there is an > 0 such that any two distinct lattice points x 6= y 2 Λ are at distance at least kx − yk ≥ . Not every subgroup of Rn is a lattice. Example 1. Qn is a subgroup of Rn, but not a lattice, because it is not discrete. The simplest example of lattice is the set of all n-dimensional vectors with integer entries. Example 2. The set Zn is a lattice because integer vectors can be added and subtracted, and clearly the distance between any two integer vectors is at least 1.