DOCSLIB.ORG

Notes on Quantum Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Quantum Mechanics Notes on Quantum Mechanics Finn Ravndal Institute of Physics University of Oslo, Norway e-mail: [email protected] v.4: December, 2006 2 Contents 1 Linear vector spaces 7 1.1 Realvectorspaces............................... 7 1.2 Rotations ................................... 9 1.3 Liegroups................................... 11 1.4 Changeofbasis ................................ 12 1.5 Complexvectorspaces ............................ 12 1.6 Unitarytransformations . 14 2 Basic quantum mechanics 17 2.1 Braandketvectors.............................. 17 2.2 OperatorsinHilbertspace . 19 2.3 Matrixrepresentations . 20 2.4 Adjointoperations .............................. 21 2.5 Eigenvectors and eigenvalues . ... 22 2.6 Thequantumpostulates ........................... 23 2.7 Ehrenfest’stheorem.............................. 25 2.8 Thetimeevolutionoperator . 25 2.9 Stationarystatesandenergybasis. .... 27 2.10Changeofbasis ................................ 28 2.11 Schr¨odinger and Heisenberg pictures . ...... 29 2.12TheLieformula................................ 31 2.13 Weyl and Cambell-Baker-Hausdorff formulas . ...... 32 3 Discrete quantum systems 35 3.1 Matrixmechanics............................... 35 3.2 Thehydrogenmolecule............................ 36 3 4 CONTENTS 3.3 Thegeneraltwo-statesystem . 38 3.4 Neutrinooscillations . 40 3.5 Reflection invariance and symmetry transformations . ......... 41 3.6 Latticetranslationinvariance . .... 43 3.7 Latticedynamicsandenergybands . .. 45 3.8 Three-dimensional lattices . ... 47 4 Schr¨odinger wave mechanics 51 4.1 Coordinatebasis ............................... 51 4.2 The Dirac δ-function ............................. 52 4.3 Continuous translations and momentum . ... 53 4.4 Operators in the coordinate representation . ....... 55 4.5 Themomentumrepresentation. 58 4.6 Threedimensions ............................... 58 4.7 Thewaveequation .............................. 60 4.8 Algebraic solution of the harmonic oscillator . ........ 61 4.9 Hermitepolynomials ............................. 66 4.10Coherentstates ................................ 67 5 Rotations and spherical symmetry 71 5.1 Cyclicmolecules................................ 71 5.2 Particleonacircle .............................. 73 5.3 Axialsymmetry................................ 74 5.4 Planarharmonicoscillator . .. 76 5.5 Sphericalrotationsandthegenerators . ..... 79 5.6 Quantizationofspin ............................. 83 5.7 Spin-1/2andthePaulimatrices . .. 86 5.8 Orbital angular momentum and spherical harmonics . ....... 89 5.9 Spherical symmetric potentials . .... 93 5.10Thesphericalwell............................... 96 5.11Hydrogenatoms................................ 97 5.12 RotationmatricesandEulerangles . 101 6 Approximation schemes 109 6.1 Thevariationalmethod. 109 CONTENTS 5 6.2 Rayleigh-Ritzscheme. .. .. 112 6.3 Staticperturbations. 113 6.4 Degeneracy ..................................116 6.5 Time-dependent perturbations . 118 6.6 Harmonicperturbations . 121 6.7 Rutherfordscattering. 122 6.8 Quantumpotentialscattering . 125 7 Electromagnetic interactions 129 7.1 Magneticmoments ..............................129 7.2 Vectorpotentialsandgaugeinvariance . 130 7.3 Gaugetransformationsinquantumtheory . 133 7.4 ThePauliequation ..............................134 7.5 Zeemaneffect .................................137 7.6 Absorptionandemissionoflight. 138 7.7 EinsteinAandBcoefficients. 142 7.8 Quantization of the electromagnetic field . ......144 7.9 Spontaneousemission. 151 7.10 Angulardistributions . 154 7.11 Magnetictransitions . 156 7.12 Photo-electriceffect. 158 7.13 Thomsonscattering. .. .. 160 7.14 Vaccum energies and the cosmological constant . .......163 7.15TheCasimirforce...............................165 8 Spin interactions 169 8.1 Relativistic velocity correction . ......169 8.2 Spin-orbitsplitting . .. .. 171 8.3 Additionofspins ...............................174 8.4 Finestructure.................................177 8.5 Anomalous Zeeman effect and Land´eg-factor . 179 8.6 Hyperfinesplitting ..............................182 8.7 Magnetic transitions in hyperfine doublet . ......186 9 Many-electron atoms 189 6 CONTENTS 9.1 Theheliumatom ...............................189 9.2 Electron spin and the Pauli principle . 193 9.3 Central-fieldapproximation . 196 9.4 TermsandtheHundrules . .. .. .. 200 9.5 Atomicmagneticmoments . 204 9.6 Hartreeapproximations. 205 10 Non-relativistic quantum fields 207 10.1 Identicalparticles . 207 10.2Fockspace...................................210 10.3 Quantumfieldoperators . 213 10.4 Inter-particleinteractions. ......217 10.5Fermions....................................220 10.6Particlesandholes .............................. 222 10.7Latticefieldtheory .............................. 224 Chapter 1 Linear vector spaces The most general formulation of quantum mechanics can be made within the framework of a linear vector space. This is a generalization of ideas we have about ordinary vectors in three-dimensional Euclidean space. We will need vector spaces of higher dimensions. In fact, ordinary quantum-mechanical wavefunctions will be found to be the components of a state vector in a vector space with infinite dimensions. Another generalization will be to consider vectors with components that are not real, but complex numbers. A new scalar product must then be defined. In order to discuss the components, we need to have a basis in the vector space. The values of the components depend on this choice of basis vectors and are related by linear transformations to the components in another basis. When such transformations conserve the lengths of vectors, they are said to be rotations in a real vector space and unitary transformations in a complex vector space. 1.1 Real vector spaces Let us consider a set of vectors x, y, z,... which has the basic property that that sum of any two vectorsV is a new vector{ in the} same set, x + y (1.1) ∈ V Summation is assumed to be commutative, x + y = y + x and associative, (x + y)+ z = x +(y + z). There must also be is null vector 0 such that for any vector x we will have x + 0 = x. The vector which adds to x to give the null vector, is called the inverse vector. These are the basic definitions of a linear vector space. Now to get any further, we must assume that we also have a set of scalars a, b, c, . which we first take to be ordinary, real numbers. Multiplying a vector with such{ a scalar,} we find a new vector, i.e. ax . The linear combination ax + by is therefore also a vector. Multiplying with zero∈ we V get the zero vector. Thus we can write the vector inverse to x to be x. − A vector v is said to be linearly independent of the vectors x, y, z,... when it cannot be written as a linear combination of these vectors. The maxi{ mum number} of linearly independent vectors is called the dimension N of the vector space which now can be 7 8 Chapter 1. Linear vector spaces denoted as N . Such a set is called a basis for the vector space. If this basis set is denoted by e , e ,...,V e , any vector x can therefore be written as the linear combination { 1 2 N } x = x1e1 + x2e2 + . + xN eN or N x = xnen (1.2) nX=1 The scalar coefficients xn are called the components of the vector with respect to this basis. It therefore follows that the components of the sum x + y of two vectors are just the sum xn + yn of their components. Obviously, the components of the null vector are all zero. We have now constructed a linear vector space over real numbers. Measurements in physics are represented by real numbers. For a mathematical construction like a vector space to be physically useful, we must therefore be able to assign sizes or lengths to all vectors in the space. This can be done by endowing the vector space with an inner or scalar product. In analogy with standard vector analysis, we denote the scalar product of a vector x with itself by x x which is then called its squared length and is a positive · number. Similarly, the scalar product of vector x with vector y is written as x y. When this product is zero, we say that the two vectors are normal or orthogonal to each· other. All scalar products are given in terms of the scalar products between the vectors in the basis set. It is most convenient to choose a very special set of such vectors where each is orthogonal to all the others and all have the same length which we take to be one. They are therefore unit vectors. In this orthonormal basis we thus have 1, m = n e e = δ = (1.3) m · n mn 0, m = n 6 where we one the right have introduced the Kroenecker δ-symbol. The scalar product of two vectors x and y is then N N x y = x y e e = x y (1.4) · m n m · n n n m,nX=1 nX=1 and similarly for the squared length of the vector x, N x 2 = x x = x2 (1.5) | | · n nX=1 The length of the null vector is therefore null. Multiplying the vector x with the basis vector en, we find the component of the vector in this direction, x = e x (1.6) n n · Inserting this back into the vector (1.2), we have N x = e e x n n · nX=1 1.2 Rotations 9 Since this is valid for all possible vectors x, we must have that the object N I = enen (1.7) nX=1 acts as a unit operator or identity in the sense that when it acts on any vector, it just gives the same vector back. In fact, we have just shown that I x = x and obviously also I e = e for any basis vector. That the sum in (1.7) is just· the identity, is usually · m m said to show that the basis vectors satisfy the completeness relation. The unit operator I is just one special operator on this vector space. A general operator S acting on a vector x gives a new vector x′, i.e. x′ = S x. We will only consider linear · operators defined by S (x + y) = S x + S y. It is trivial to see from the definition (1.7) that I is linear. Each· operator ·S will have· an inverse denoted by S−1 such that S S−1 = S−1 S = I. · · 1.2 Rotations A rotation of a vector x is a linear transformation of its components so that its length ′ N ′ stays the same. The transformed vector x = n=1 xnen will thus have components P e2 x' x’2 x2 x φ’ α φ x’1 x 1 e1 Figure 1.1: Rotation by an angle α of a two-dimensional vector x into the final vector x′. which in general can be written as N ′ xm = Rmnxn (1.8) nX=1 where the coefficients Rmn are real numbers.
Load more

Recommended publications
  • Theory of Angular Momentum Rotations About Different Axes Fail to Commute
    153 CHAPTER 3 3.1. Rotations and Angular Momentum Commutation Relations followed by a 90° rotation about the z-axis. The net results are different, as we can see from Figure 3.1. Our first basic task is to work out quantitatively the manner in which Theory of Angular Momentum rotations about different axes fail to commute. To this end, we first recall how to represent rotations in three dimensions by 3 X 3 real, orthogonal matrices. Consider a vector V with components Vx ' VY ' and When we rotate, the three components become some other set of numbers, V;, V;, and Vz'. The old and new components are related via a 3 X 3 orthogonal matrix R: V'X Vx V'y R Vy I, (3.1.1a) V'z RRT = RTR =1, (3.1.1b) where the superscript T stands for a transpose of a matrix. It is a property of orthogonal matrices that 2 2 /2 /2 /2 vVx + V2y + Vz =IVVx + Vy + Vz (3.1.2) is automatically satisfied. This chapter is concerned with a systematic treatment of angular momen- tum and related topics. The importance of angular momentum in modern z physics can hardly be overemphasized. A thorough understanding of angu- Z z lar momentum is essential in molecular, atomic, and nuclear spectroscopy; I I angular-momentum considerations play an important role in scattering and I I collision problems as well as in bound-state problems. Furthermore, angu- I lar-momentum concepts have important generalizations-isospin in nuclear physics, SU(3), SU(2)® U(l) in particle physics, and so forth.
    [Show full text]
  • 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].
    [Show full text]
  • {|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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • Gauge Theory
    Preprint typeset in JHEP style - HYPER VERSION 2018 Gauge Theory David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html [email protected] Contents 0. Introduction 1 1. Topics in Electromagnetism 3 1.1 Magnetic Monopoles 3 1.1.1 Dirac Quantisation 4 1.1.2 A Patchwork of Gauge Fields 6 1.1.3 Monopoles and Angular Momentum 8 1.2 The Theta Term 10 1.2.1 The Topological Insulator 11 1.2.2 A Mirage Monopole 14 1.2.3 The Witten Effect 16 1.2.4 Why θ is Periodic 18 1.2.5 Parity, Time-Reversal and θ = π 21 1.3 Further Reading 22 2. Yang-Mills Theory 26 2.1 Introducing Yang-Mills 26 2.1.1 The Action 29 2.1.2 Gauge Symmetry 31 2.1.3 Wilson Lines and Wilson Loops 33 2.2 The Theta Term 38 2.2.1 Canonical Quantisation of Yang-Mills 40 2.2.2 The Wavefunction and the Chern-Simons Functional 42 2.2.3 Analogies From Quantum Mechanics 47 2.3 Instantons 51 2.3.1 The Self-Dual Yang-Mills Equations 52 2.3.2 Tunnelling: Another Quantum Mechanics Analogy 56 2.3.3 Instanton Contributions to the Path Integral 58 2.4 The Flow to Strong Coupling 61 2.4.1 Anti-Screening and Paramagnetism 65 2.4.2 Computing the Beta Function 67 2.5 Electric Probes 74 2.5.1 Coulomb vs Confining 74 2.5.2 An Analogy: Flux Lines in a Superconductor 78 { 1 { 2.5.3 Wilson Loops Revisited 85 2.6 Magnetic Probes 88 2.6.1 't Hooft Lines 89 2.6.2 SU(N) vs SU(N)=ZN 92 2.6.3 What is the Gauge Group of the Standard Model? 97 2.7 Dynamical Matter 99 2.7.1 The Beta Function Revisited 100 2.7.2 The Infra-Red Phases of QCD-like Theories 102 2.7.3 The Higgs vs Confining Phase 105 2.8 't Hooft-Polyakov Monopoles 109 2.8.1 Monopole Solutions 112 2.8.2 The Witten Effect Again 114 2.9 Further Reading 115 3.
    [Show full text]
  • Entanglement of Dirac Bi-Spinor States Driven by Poincar\'E Classes Of
    Entanglement of Dirac bi-spinor states driven by Poincar´eclasses of SU(2) ⊗ SU(2) coupling potentials Victor A. S. V. Bittencourt∗ and Alex E. Bernardiniy Departamento de F´ısica, Universidade Federal de S~aoCarlos, PO Box 676, 13565-905, S~aoCarlos, SP, Brasil (Dated: October 8, 2018) Abstract A generalized description of entanglement and quantum correlation properties constraining in- ternal degrees of freedom of Dirac(-like) structures driven by arbitrary Poincar´eclasses of external field potentials is proposed. The role of (pseudo)scalar, (pseudo)vector and tensor interactions in producing/destroying intrinsic quantum correlations for SU(2) ⊗ SU(2) bi-spinor structures is discussed in terms of generic coupling constants. By using a suitable ansatz to obtain the Dirac Hamiltonian eigenspinor structure of time-independent solutions of the associated Liouville equa- tion, the quantum entanglement, via concurrence, and quantum correlations, via geometric discord, are computed for several combinations of well-defined Poincar´eclasses of Dirac potentials. Besides its inherent formal structure, our results set up a framework which can be enlarged as to include localization effects and to map quantum correlation effects into Dirac-like systems which describe low-energy excitations of graphene and trapped ions. PACS numbers: 03.65.-w,03.65.Pm, 03.67.Bg arXiv:1511.07047v3 [quant-ph] 19 Jan 2016 ∗Electronic address: [email protected] yElectronic address: [email protected] 1 I. INTRODUCTION The map of controllable physical systems onto the Dirac equation formal structure has been in the core of recent investigations which search for reproducing quantum relativistic effects, such as the zitterbewegung effect and the Klein tunneling/paradox, on table-top experiments [1{6].
    [Show full text]
  • 12 Magnetic Moments. Spin
    TFY4250/FY2045 Lecture notes 12 - Magnetic moments. Spin 1 Lecture notes 12 12 Magnetic moments. Spin The hypothesis about the electron spin (intrinsic angular momentum) was pub- lished already in 1925 by Uhlenbeck and Goudsmit, almost at the same time as the discovery by Heisenberg and Schr¨odingerof quantum mechanics and Pauli's formulation of the exclusion principle. The main experimental clues leading to the spin hypothesis were • the fine structure of optical spectra • the Zeeman effect • the Stern{Gerlach experiment All these effects involve the behaviour of a particle with spin and/or orbital angular momentum in a magnetic field. In section 12.1 in these notes, we begin by considering the magnetic moment due to the orbital motion of a particle, both classically and quantum-mechanically. We then review the experiment of Stern and Gerlach, which clearly reveals the intrinsic angular momentum (spin) of the electron. We also give a survey of the spins of other particles. 1 In section 12.2 we establish a formalism for spin 2 , based on the general discussion of angular momenta in Lecture notes 11. 12.1 Magnetic moments connected with orbital angular momentum and spin 12.1.a Classical magnetic moment (Hemmer p 178, 1.5 in B&J) First a little bit of classical electrodynamics. TFY4250/FY2045 Lecture notes 12 - Magnetic moments. Spin 2 In electromagnetic theory (see e.g. D.J. Griffiths, Introduction to Electrodynamics, chapter 6) one learns that an infinitesimal current loop, placed in a magnetic field, experiences a torque τ = µ × B (T12.1) and a force F = −r(−µ·B): (T12.2) Here µ is the magnetic (dipole) moment of the infinitesimal current loop.
    [Show full text]
  • 1 Problem 1. PROBLEMS from SAKURAI 4.1) Calculate the Three Lowest Energy Levels, Together with Their Degeneracies for the Follo
    1 SOLUTION SET 5 215B-QUANTUM MECHANICS, WINTER 2019 Problem 1. PROBLEMS FROM SAKURAI 4.1) Calculate the three lowest energy levels, together with their degeneracies for the following systems (assuming equal mass distinguishable particles) (a) Three non-interacting spin 1=2 particles in a one-dimensional box of length L The Hamiltonian for three non-interacting particles of mass m in a one-dimensional box of length L is 1 X3 H = p2 (1) 2m i i=1 Consequently the state completely factorizes and we can independently designate the spin and position eigenstates of each particle. The energy levels are independent of spin and given by π2~2 X E = 3n2 (2) ~n 2mL2 i i=1 The ground state has energy π2~2 E = 3 ; (3) (1;1;1) 2mL2 with no degeneracy in the position wave-function, but a 2-fold degeneracy in equal energy spin states for each of the three particles. Thus the ground state degeneracy is 8. The first excited states have energy π2~2 E = E = E = 3 ; (4) (2;1;1) (1;2;1) (1;1;2) mL2 with a 3-fold degeneracy in position wavefunctions and 8-fold degeneracy in spin giving a total degeneracy of 24. The second exited states have energy π2~2 E = E = E = 9 ; (5) (2;2;1) (1;2;2) (2;1;2) 2mL2 with a 3-fold degeneracy in position wavefunctions and 8-fold degeneracy in spin giving a total degeneracy of 24. (b) Four non-interacting spin 1=2 particles in a one-dimensional box of length L Four particles is exactly analogous.
    [Show full text]
  • Understanding Angular Momentum in Hadrons
    Understanding Angular Momentum in Hadrons Kunal Kathuria Charlottesville, VA A Thesis Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Physics UNIVERSITY OF VIRGINIA August, 2013 Declaration of Authorship I, Kunal Kathuria, declare that this thesis titled, `THESIS TITLE' and the work pre- sented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is my own work. I have acknowledged all main sources of help. Signed: Date: i "Humility means not being anxious to be honored by others." \Eloquence is truth concisely stated." [AC Bhaktivedanta Swami] If you can't explain it to a six year old, you don't understand it yourself. [Albert Einstein] At the heart of all knowledge lies simplicity of expression. UNIVERSITY OF VIRGINIA Abstract Department of Physics Doctor of Philosophy by Kunal Kathuria Charlottesville, VA The spin puzzle has been a relatively long-standing unresolved issue in hadron physics. We review and derive, using a wave-packet formalism, the first step preceding all spin sum rules: the connection of the angular momentum operator to the gravitomagnetic form factors of the energy-momentum tensor (EMT).
    [Show full text]