Chapter 4 MANY PARTICLE SYSTEMS
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Chapter 4 Introduction to Many-Body Quantum Mechanics
Chapter 4 Introduction to many-body quantum mechanics 4.1 The complexity of the quantum many-body prob- lem After learning how to solve the 1-body Schr¨odinger equation, let us next generalize to more particles. If a single body quantum problem is described by a Hilbert space of dimension dim = d then N distinguishable quantum particles are described by theH H tensor product of N Hilbert spaces N (N) N ⊗ (4.1) H ≡ H ≡ H i=1 O with dimension dN . As a first example, a single spin-1/2 has a Hilbert space = C2 of dimension 2, but N spin-1/2 have a Hilbert space (N) = C2N of dimensionH 2N . Similarly, a single H particle in three dimensional space is described by a complex-valued wave function ψ(~x) of the position ~x of the particle, while N distinguishable particles are described by a complex-valued wave function ψ(~x1,...,~xN ) of the positions ~x1,...,~xN of the particles. Approximating the Hilbert space of the single particle by a finite basis set with d basis functions, the N-particle basisH approximated by the same finite basis set for single particles needs dN basis functions. This exponential scaling of the Hilbert space dimension with the number of particles is a big challenge. Even in the simplest case – a spin-1/2 with d = 2, the basis for N = 30 spins is already of of size 230 109. A single complex vector needs 16 GByte of memory and will not fit into the memory≈ of your personal computer anymore. -
Second Quantization
Chapter 1 Second Quantization 1.1 Creation and Annihilation Operators in Quan- tum Mechanics We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation a,a† = 1 (1.1) where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let α be a state which we will take to be an eigenvector of the Hermitian operators| ia†a with eigenvalue α which is a real number, a†a α = α α (1.2) | i | i Hence, α = α a†a α = a α 2 0 (1.3) h | | i k | ik ≥ where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a†a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C]= A [B, C] + [A, C] B (1.4) we get a†a,a = a (1.5) − † † † a a,a = a (1.6) 1 2 CHAPTER 1. SECOND QUANTIZATION i.e., a and a† are “eigen-operators” of a†a. Hence, a†a a = a a†a 1 (1.7) − † † † † a a a = a a a +1 (1.8) Consequently we find a†a a α = a a†a 1 α = (α 1) a α (1.9) | i − | i − | i Hence the state aα is an eigenstate of a†a with eigenvalue α 1, provided a α = 0. -
Orthogonalization of Fermion K-Body Operators and Representability
Orthogonalization of fermion k-Body operators and representability Bach, Volker Rauch, Robert April 10, 2019 Abstract The reduced k-particle density matrix of a density matrix on finite- dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of k- body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of k-body operators for all k. 1 Introduction 1.1 Motivation: Representability problems In quantum chemistry, molecules are usually modeled as non-relativistic many- fermion systems (Born-Oppenheimer approximation). More specifically, the Hilbert space of these systems is given by the fermion Fock space = f (h), where h is the (complex) Hilbert space of a single electron (e.g. h = LF2(R3)F C2), and the Hamiltonian H is usually a two-body operator or, more generally,⊗ a k-body operator on . A key physical quantity whose computation is an im- F arXiv:1807.05299v2 [math-ph] 9 Apr 2019 portant task is the ground state energy . E0(H) = inf ϕ(H) (1) ϕ∈S of the system, where ( )′ is a suitable set of states on ( ), where ( ) is the Banach space ofS⊆B boundedF operators on and ( )′ BitsF dual. A directB F evaluation of (1) is, however, practically impossibleF dueB F to the vast size of the state space . -
LYCEN 7112 Février 1972 Notice 92Ft on the Generalized Exchange
LYCEN 7112 Février 1972 notice 92ft On the Generalized Exchange Operators for SU(n) A. Partensky Institut de Physique Nucléaire, Université Claude Bernard de Lyon Institut National de Physique Nucléaire et de Physique des Particules 43, Bd du 11 novembre 1918, 69-Villeurbanne, France Abstract - By following the work of Biedenharn we have redefined the k-particles generalized exchange operators (g.e.o) and studied their properties. By a straightforward but cur-jersome calculation we have derived the expression, in terms of the SU(n) Casimir operators, of the ?., 3 or 4 particles g.e.o. acting on the A-particles states which span an irreducible representation of the grown SU(n). A striking and interesting result is that the eigenvalues of each of these g.e.o. do not depend on the n in SU(n) but only on the Young pattern associated with the irreducible representation considered. For a given g.e.o. , the eigenvalues corresponding to two conjugate Young patterns are the same except for the sign which depends on the parity of the g.e.o. considered. Three Appendices deal with some related problems and, more specifically, Appendix C 2 contains a method of obtaining the eigenvalues of Gel'fand invariants in a new and simple way. 1 . Introduction The interest attributed by physicists to the Lie groups has 3-4 been continuously growing since the pionner works of Weyl , Wigner and Racah • In particular, unitary groups have been used successfully in various domains of physics, although the physical reasons for their introduction have not always been completely explained. -
5.1 Two-Particle Systems
5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ). -
Fock Space Dynamics
TCM315 Fall 2020: Introduction to Open Quantum Systems Lecture 10: Fock space dynamics Course handouts are designed as a study aid and are not meant to replace the recommended textbooks. Handouts may contain typos and/or errors. The students are encouraged to verify the information contained within and to report any issue to the lecturer. CONTENTS Introduction 1 Composite systems of indistinguishable particles1 Permutation group S2 of 2-objects 2 Admissible states of three indistiguishable systems2 Parastatistics 4 The spin-statistics theorem 5 Fock space 5 Creation and annihilation of particles 6 Canonical commutation relations 6 Canonical anti-commutation relations 7 Explicit example for a Fock space with 2 distinguishable Fermion states7 Central oscillator of a linear boson system8 Decoupling of the one particle sector 9 References 9 INTRODUCTION The chapter 5 of [6] oers a conceptually very transparent introduction to indistinguishable particle kinematics in quantum mechanics. Particularly valuable is the brief but clear discussion of para-statistics. The last sections of the chapter are devoted to expounding, physics style, the second quantization formalism. Chapter I of [2] is a very clear and detailed presentation of second quantization formalism in the form that is needed in mathematical physics applications. Finally, the dynamics of a central oscillator in an external eld draws from chapter 11 of [4]. COMPOSITE SYSTEMS OF INDISTINGUISHABLE PARTICLES The composite system postulate tells us that the Hilbert space of a multi-partite system is the tensor product of the Hilbert spaces of the constituents. In other words, the composite system Hilbert space, is spanned by an orthonormal basis constructed by taking the tensor product of the elements of the orthonormal bases of the Hilbert spaces of the constituent systems. -
Phase Transitions and the Casimir Effect in Neutron Stars
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2017 Phase Transitions and the Casimir Effect in Neutron Stars William Patrick Moffitt University of Tennessee, Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Other Physics Commons Recommended Citation Moffitt, William Patrick, "Phase Transitions and the Casimir Effect in Neutron Stars. " Master's Thesis, University of Tennessee, 2017. https://trace.tennessee.edu/utk_gradthes/4956 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by William Patrick Moffitt entitled "Phaser T ansitions and the Casimir Effect in Neutron Stars." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Physics. Andrew W. Steiner, Major Professor We have read this thesis and recommend its acceptance: Marianne Breinig, Steve Johnston Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Phase Transitions and the Casimir Effect in Neutron Stars A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville William Patrick Moffitt December 2017 Abstract What lies at the core of a neutron star is still a highly debated topic, with both the composition and the physical interactions in question. -
Time-Reversal Symmetry Breaking Abelian Chiral Spin Liquid in Mott Phases of Three-Component Fermions on the Triangular Lattice
PHYSICAL REVIEW RESEARCH 2, 023098 (2020) Time-reversal symmetry breaking Abelian chiral spin liquid in Mott phases of three-component fermions on the triangular lattice C. Boos,1,2 C. J. Ganahl,3 M. Lajkó ,2 P. Nataf ,4 A. M. Läuchli ,3 K. Penc ,5 K. P. Schmidt,1 and F. Mila2 1Institute for Theoretical Physics, FAU Erlangen-Nürnberg, D-91058 Erlangen, Germany 2Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH 1015 Lausanne, Switzerland 3Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria 4Laboratoire de Physique et Modélisation des Milieux Condensés, Université Grenoble Alpes and CNRS, 25 avenue des Martyrs, 38042 Grenoble, France 5Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, H-1525 Budapest, P.O.B. 49, Hungary (Received 6 December 2019; revised manuscript received 20 March 2020; accepted 20 March 2020; published 29 April 2020) We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude −t (t > 0) and on-site interaction U. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform π flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers of t/U for general SU(N ) and arbitrary uniform charge flux per plaquette, which are subsequently studied using exact diagonalizations and variational Monte Carlo simulations for N = 3, as well as exact diagonalizations of the SU(3) Hubbard model on small clusters. -
2 Quantum Theory of Spin Waves
2 Quantum Theory of Spin Waves In Chapter 1, we discussed the angular momenta and magnetic moments of individual atoms and ions. When these atoms or ions are constituents of a solid, it is important to take into consideration the ways in which the angular momenta on different sites interact with one another. For simplicity, we will restrict our attention to the case when the angular momentum on each site is entirely due to spin. The elementary excitations of coupled spin systems in solids are called spin waves. In this chapter, we will introduce the quantum theory of these excita- tions at low temperatures. The two primary interaction mechanisms for spins are magnetic dipole–dipole coupling and a mechanism of quantum mechanical origin referred to as the exchange interaction. The dipolar interactions are of importance when the spin wavelength is very long compared to the spacing between spins, and the exchange interaction dominates when the spacing be- tween spins becomes significant on the scale of a wavelength. In this chapter, we focus on exchange-dominated spin waves, while dipolar spin waves are the primary topic of subsequent chapters. We begin this chapter with a quantum mechanical treatment of a sin- gle electron in a uniform field and follow it with the derivations of Zeeman energy and Larmor precession. We then consider one of the simplest exchange- coupled spin systems, molecular hydrogen. Exchange plays a crucial role in the existence of ordered spin systems. The ground state of H2 is a two-electron exchange-coupled system in an embryonic antiferromagnetic state. -
Quantum Physics in Non-Separable Hilbert Spaces
Quantum Physics in Non-Separable Hilbert Spaces John Earman Dept. HPS University of Pittsburgh In Mathematical Foundations of Quantum Mechanics (1932) von Neumann made separability of Hilbert space an axiom. Subsequent work in mathemat- ics (some of it by von Neumann himself) investigated non-separable Hilbert spaces, and mathematical physicists have sometimes made use of them. This note discusses some of the problems that arise in trying to treat quantum systems with non-separable spaces. Some of the problems are “merely tech- nical”but others point to interesting foundations issues for quantum theory, both in its abstract mathematical form and its applications to physical sys- tems. Nothing new or original is attempted here. Rather this note aims to bring into focus some issues that have for too long remained on the edge of consciousness for philosophers of physics. 1 Introduction A Hilbert space is separable if there is a countable dense set of vec- tors; equivalently,H admits a countable orthonormal basis. In Mathematical Foundations of QuantumH Mechanics (1932, 1955) von Neumann made sep- arability one of the axioms of his codification of the formalism of quantum mechanics. Working with a separable Hilbert space certainly simplifies mat- ters and provides for understandable realizations of the Hilbert space axioms: all infinite dimensional separable Hilbert spaces are the “same”: they are iso- 2 morphically isometric to LC(R), the space of square integrable complex val- 2 ued functions of R with inner product , := (x)(x)dx, , LC(R). h i 2 2 This Hilbert space in turn is isomorphically isometric to `C(N), the vector space of square summable sequences of complexR numbers with inner prod- S uct , := n1=1 xn yn, = (x1, x2, ...), = (y1, y2, ...) . -
Isometries, Fock Spaces, and Spectral Analysis of Schr¨Odinger Operators on Trees
Isometries, Fock Spaces, and Spectral Analysis of Schrodinger¨ Operators on Trees V. Georgescu and S. Golenia´ CNRS Departement´ de Mathematiques´ Universite´ de Cergy-Pontoise 2 avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex France [email protected] [email protected] June 15, 2004 Abstract We construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering the- ory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrodinger¨ operators on trees. We consider C ∗-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formu- las for the essential spectrum of these Hamiltonians. 1 Introduction The Laplace operator on a graph Γ acts on functions f : Γ C according to the relation ! (∆f)(x) = (f(y) f(x)); (1.1) − yX$x where y x means that x and y are connected by an edge. The spectral analysis $ and the scattering theory of the operators on `2(Γ) associated to expressions of the form L = ∆ + V , where V is a real function on Γ, is an interesting question which does not seem to have been much studied (we have in mind here only situations involving non trivial essential spectrum). 1 Our interest on these questions has been aroused by the work of C. Allard and R. Froese [All, AlF] devoted to the case when Γ is a binary tree: their main results are the construction of a conjugate operator for L under suitable conditions on the potential V and the proof of the Mourre estimate. -
Physics 461 / Quantum Mechanics I
Physics 461 / Quantum Mechanics I P.E. Parris Department of Physics University of Missouri-Rolla Rolla, Missouri 65409 January 10, 2005 CONTENTS 1 Introduction 7 1.1WhatisQuantumMechanics?......................... 7 1.1.1 WhatisMechanics?.......................... 7 1.1.2 PostulatesofClassicalMechanics:.................. 8 1.2TheDevelopmentofWaveMechanics..................... 10 1.3TheWaveMechanicsofSchrödinger..................... 13 1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . 13 1.3.2 Schrödinger’sMechanicsforConservativeSystems......... 16 1.3.3 The Principle of Superposition and Spectral Decomposition . 17 1.3.4 TheFreeParticle............................ 19 1.3.5 Superpositions of Plane Waves and the Fourier Transform . 23 1.4Appendix:TheDeltaFunction........................ 26 2 The Formalism of Quantum Mechanics 31 2.1 Postulate I: SpecificationoftheDynamicalState.............. 31 2.1.1 PropertiesofLinearVectorSpaces.................. 31 2.1.2 Additional Definitions......................... 33 2.1.3 ContinuousBasesandContinuousSets................ 34 2.1.4 InnerProducts............................. 35 2.1.5 ExpansionofaVectoronanOrthonormalBasis.......... 38 2.1.6 Calculation of Inner Products Using an Orthonormal Basis . 39 2.1.7 ThePositionRepresentation..................... 40 2.1.8 TheWavevectorRepresentation.................... 41 2.2PostulateII:ObservablesofQuantumMechanicalSystems......... 43 2.2.1 OperatorsandTheirProperties.................... 43 2.2.2 MultiplicativeOperators.......................