Graduate Texts in Physics For further volumes: www.com/series/8431 Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and ad- vanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.
Series Editors Professor William T. Rhodes Department of Computer and Electrical Engineering and Computer Science Imaging Science and Technology Center Florida Atlantic University 777 Glades Road SE, Room 456 Boca Raton, FL 33431 USA wrhodes@fau. Eugene Stanley Center for Polymer Studies Department of Physics Boston University 590 Commonwealth Avenue, Room 204B Boston, MA 02215 USA hes@bu.edu Professor Richard Needs Cavendish Laboratory JJ Thomson Avenue Cambridge CB3 0HE UK rn11@cam.com Dirk Dubbers r Hans-Jürgen Stöckmann Quantum Physics: The Bottom-Up Approach From the Simple Two-Level System to Irreducible Representations www.com Dirk Dubbers Hans-Jürgen Stöckmann Fak. Physik und Astronomie Fachbereich Physik Physikalisches Institut Philipps-Universität Marburg Universität Heidelberg Marburg, Germany Heidelberg, Germany ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-642-31059-1 ISBN 978-3-642-31060-7 (eBook) DOI 10.1007/978-3-642-31060-7 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012950772 © Springer-Verlag Berlin Heidelberg 2013 Figure 11.8 is a contribution of the National Institute of Standards and Technology.2: Reprinted with permission, Copyright by the American Physical Society.
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While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.com Preface Quantum mechanics pervades many branches of science, from physics, material sci- ence, informatics, to chemistry and molecular biology. Many products of everyday life derive from discoveries based on quantum physics, like silicon chips, magnetic storage devices, lasers, medical imaging devices, as well as chemicals and biochem- icals.
Therefore, scientists and engineers in many fields need a good understanding of quantum theory, but often they are overwhelmed by the sheer volume of most standard textbooks on quantum physics. Our approach is to first limit discussion to the smallest systems in nature that still display the basic features of quantum theory. Hence this tutorial at first deals with systems in which only two quantum states are involved, subject to external perturbations. Such effective spin one-half systems are a valuable training ground to elucidate the subtleties of quantum theory and, indeed, the essence of quantum mechanics lies in the two-level system.
We present basic quantum calculations step by step in a simple notation and in sufficient detail because the practitioner usually has no time to lose when proceeding from one equation to the next. As a starting point we assume the reader to have taken introductory courses in quantum mechanics and linear algebra, and to be familiar with the Schrödinger equation and the essentials of angular momentum, which we recapitulate in Part I of the book. Part II covers essential topics of quantum physics based on the two-state approach, including subjects of high contemporary interest as quantum entangle- ment, quantum chaos, or geometric phases. In Part III, the results then are applied to various topics from atomic, condensed matter, and nuclear physics, and from quan- tum informatics.
We then proceed to the more general concepts of quantum theory. To this end, Part IV of this treatise restarts from first principles to develop the theory of angular momentum, spherical tensors, and irreducible representations. We derive a general- ized spin precession equation that covers the higher multipole interactions, and ap- ply the results to various topics in atomic and condensed matter physics. Chapters on multiple quantum transitions, dressed atom effects, spin relaxation and decoherence conclude the tutorial.com vi Preface The text is based on various lectures given by the authors on the two-state system, irreducible tensors, and quantum chaos, and is complemented with an illustrative set of basic experiments, many of them done in the authors’ respective laboratories.
In short, the aim of this tutorial is to provide the bachelor student as well as the practi- tioner with a compact text that lets them understand a wealth of quantum physics. Heidelberg, Germany Dirk Dubbers Marburg, Germany Hans-Jürgen Stöckmann www.com “Dürer meets Einstein, travelling.” Installation by Sabrina Hohmann, for an interpre- tation see http://www. With friendly per- mission of the artist. Foto: Wolf-Dieter Gericke www.com Contents Part I Prologue 1 Recollections from Elementary Quantum Physics.
8 Part II Two-State Quantum Systems 2 A Most Simple Two-Level System .1 Magnetic Moment and Spin .3 Stern–Gerlach Effect. 16 3 Quantum Theory in a Nutshell .1 Expectation Value of Energy .2 Expectation Value of Spin .1 Uncertainty of Spin .2 Uncertainty of Energy. 32 4 Experiments on Spin Precession .1 Muon Spin Precession .3 Spinor Rotation Through 720° .com x Contents 5 General Solution for the Two-Level System .2 Construction of the Eigenvectors .3 The Time Dependent Solution .1 Evolution of an Energy Eigenstate .2 Evolution of an Angular-Momentum Eigenstate. 46 6 Other Tools and Concepts .1 Time Evolution Operator .4 Pure States and Mixed States .5 The Density Matrix .6 Coherence and Interference .7 Dirac’s Bra-Ket Notation.
62 7 Diabolic Points, Geometric Phases, and Quantum Chaos .1 Level Crossings and Level Repulsions .1 The Field Dependence of Energy and of Polarization .2 Level Repulsion in a Spin- 21 Systems .3 Level Repulsion in a Spin-1 System .2 The Adiabatic Theorem .1 Derivation of the Berry Phase .2 Excursions in Magnetic-Field Space .3 Excursions in the Space of Shapes .4 The Aharonov–Bohm Effect. 84 8 The Coupling of Particles .1 Bosons and Fermions .2 The Coupling of Spins .3 Example: Hyperfine Structure. 90 9 “Spooky Action at a Distance”. 100 10 The Heisenberg Equation of Motion .2 Commutation Relations and Uncertainty Principle .3 The Bloch Equations .com Contents xi Part III Quantum Physics at Work 11 Spin Resonance .1 Basics of Spin Resonance .2 Methods of Spin Resonance .3 Applications of Spin Resonance.
124 12 Two-State Systems in Atomic and Molecular Physics .1 Photons as Two-State Systems .2 Optical Resonance Transitions .3 Optical Analogies of Spin Rotation and Spin Resonance .4 Particles in a Double Well .1 The NH3 Molecule .2 The Ammonia Maser .3 Bose–Einstein Condensate in a Double Trap. 136 13 Two-State Systems in Condensed Matter .1 Basics of Superconductivity .2 Josephson Junctions and Their Applications. 144 14 Two-State Systems in Nuclear and Particle Physics .2 Flavor and Color .1 Quantum Information Theory .2 Quantum Computing and Quantum Communication. 163 Part IV Multilevel Systems and Tensor Operators 16 Rotations and Angular Momentum .2 Properties of Angular Momentum .4 The Spherical Harmonics .5 The Rotation Matrices .com xii Contents 17 Irreducible Tensors .1 Scalars, Vectors, and Tensors .2 Properties of Irreducible Tensors .1 Definition of Irreducible Tensors .2 A More Practical Definition .3 The Coupling of Irreducible Tensors .1 The Coupling of Angular Momenta .2 General Tensor Coupling .3 Some Special Cases .4 The Wigner–Eckart Theorem.
190 18 Electromagnetic Multipole Interactions .1 Static Magnetic Interactions .2 Static Electric Interactions .1 Multipole Expansion of Electrostatic Energy .2 Electric Quadrupole Interaction .3 Selection Rules for Electromagnetic Transitions. 202 19 The Generalized Spin Precession Equation .1 The Density Matrix .1 Definition of the Density Matrix .2 The Liouville Equation of Motion .2 Some Preparative Steps .1 Normalized Irreducible Tensor Operators .2 A Bra-Ket Notation for Tensor Operators .3 The Irreducible Components of the Density Matrix .1 Definition of the Statistical Tensors .2 Simple Examples of Statistical Tensors .4 The Liouville Equation for the Statistical Tensors. 215 20 Reorientation in Static Electromagnetic Fields .1 Magnetic Dipole Precession .2 Electric Quadrupole Reorientation .3 Reorientation in Mixed Magnetic and Electric Fields .4 Time Average Results .5 Angular Distribution of Radiation .2 Anisotropic Photon Emission. 226 21 Reorientation in Time Dependent Fields .1 Radiofrequency Irradiation in a Magnetic Field .1 Density Operator in the Rotating Frame .2 Rotating Wave Approximation .com Contents xiii 21.3 Statistical Tensors in the Rotating Wave Approximation .4 Time Average Results .2 Multiple Quantum Transitions .1 Dressed Atoms and the Floquet Theorem .2 Dressed Atoms and Second Quantization .3 A Dressed Neutron Experiment .4 Outlook on Nonclassical Photon Interactions.
238 22 Relaxation and Decoherence .2 The Perturbative Approach .3 The Stochastic Approach .1 The Orthogonality of the Irreducible Tensor Operators .2 The Reduced Matrix Element j ||T̂L (j )||j .3 The Reduced Matrix Element (L||T̂L2 (j )||L1 ) .6 Clebsch–Gordan Coefficients .7 Normalized Irreducible Tensor Operators .8 Coefficients of the Generalized Precession Equation .9 Transforming away Part of an Interaction from the Liouville Equation .com Part I Prologue The authors assume that the reader of this treatise has taken an introductory course on quantum mechanics. This Prologue summarizes the essentials of such a course.com Chapter 1 Recollections from Elementary Quantum Physics Abstract We recall the prerequisites that we assume the reader to be familiar with, namely the Schrödinger equation in its time dependent and time independent form, the uncertainty relations, and the basic properties of angular momentum. Introductory courses on quantum physics discuss the one-dimensional Schrödinger equation for the wave function Ψ (x, t) of a particle of mass M moving in a poten- tial V ∂Ψ 2 ∂ 2 Ψ i =− + V Ψ.1) ∂t 2M ∂x 2 Therein = h/2π is the reduced Planck constant. The function Ψ is understood as a probability amplitude whose absolute square |Ψ (x, t)|2 = Ψ ∗ (x, t)Ψ (x, t) gives the probability density for finding the particle at time t at position x.
This probability density is insensitive to a phase factor eiϕ. With the Hamilton operator 2 ∂ 2 H=− + V, (1.2) 2M ∂x 2 the Schrödinger equation reads i Ψ̇ = − HΨ.3) The dot denotes the time derivative. In this text, we print operators and matrices in nonitalic type, like H, p, or σ, just to remind the reader that a simple letter may represent a mathematical object more complicated than a number or a function. Ordinary vectors in three-dimensional space are written in bold italic type, like x or B.