Graduate Texts in Physics Rainer Dick Advanced Quantum Mechanics Materials and Photons Third Edition Tai ngay!!! Ban co the xoa dong chu nay!!! Graduate Texts in Physics Series Editors Kurt H Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA ` Jean-Marc Di Meglio, Matie`re et Systemes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan William T Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-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 More information about this series at http://www.springer.com/series/8431 Rainer Dick Advanced Quantum Mechanics Materials and Photons Third Edition Rainer Dick Department of Physics University of Saskatchewan Saskatoon, SK, Canada ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-030-57869-5 ISBN 978-3-030-57870-1 (eBook) https://doi.org/10.1007/978-3-030-57870-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2016, 2020 This work is subject to copyright All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Cover illustration: © Rost-9D / Getty Images / iStock This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Quantum mechanics was invented in an era of intense and seminal scientific research between 1900 and 1928 (and in many regards continues to be developed and expanded) because neither the properties of atoms and electrons nor the spectrum of radiation from heat sources could be explained by the classical theories of mechanics, electrodynamics, and thermodynamics It was a major intellectual achievement and a breakthrough of curiosity-driven fundamental research which formed quantum theory into one of the pillars of our present understanding of the fundamental laws of nature The properties and behavior of every elementary particle are governed by the laws of quantum theory However, the rule of quantum mechanics is not limited to atomic and subatomic scales, but also affects macroscopic systems in a direct and profound manner The electric and thermal conductivity properties of materials are determined by quantum effects, and the electromagnetic spectrum emitted by a star is primarily determined by the quantum properties of photons It is therefore not surprising that quantum mechanics permeates all areas of research in advanced modern physics and materials science, and training in quantum mechanics plays a prominent role in the curriculum of every major physics or chemistry department The ubiquity of quantum effects in materials implies that quantum mechanics also evolved into a major tool for advanced technological research The construction of the first nuclear reactor in Chicago in 1942 and the development of nuclear technology could not have happened without a proper understanding of the quantum properties of particles and nuclei However, the real breakthrough for a wide recognition of the relevance of quantum effects in technology occurred with the invention of the transistor in 1948 and the ensuing rapid development of semiconductor electronics This proved once and for all the importance of quantum mechanics for the applied sciences and engineering, only 22 years after the publication of the Schrödinger equation! Electronic devices like transistors rely heavily on the quantum mechanical emergence of energy bands in materials, which can be considered as a consequence of combination of many atomic orbitals or as a consequence of delocalized electron states probing a lattice structure Today the rapid developments of spintronics, photonics, and nanotechnology provide continuing testimony to the technological relevance of quantum mechanics v vi Preface As a consequence, every physicist, chemist, and electrical engineer nowadays has to learn aspects of quantum mechanics, and we are witnessing a time when also mechanical and aerospace engineers are advised to take at least a second-year course, due to the importance of quantum mechanics for elasticity and stability properties of materials Furthermore, quantum information appears to become increasingly relevant for computer science and information technology, and a whole new area of quantum technology will likely follow in the wake of this development Therefore, it seems safe to posit that within the next two generations, second- and third-year quantum mechanics courses will become as abundant and important in the curricula of science and engineering colleges as first- and second-year calculus courses Quantum mechanics continues to play a dominant role in particle physics and atomic physics—after all, the standard model of particle physics is a quantum theory, and the spectra and stability of atoms cannot be explained without quantum mechanics However, most scientists and engineers use quantum mechanics in advanced materials research Furthermore, the dominant interaction mechanisms in materials (beyond the nuclear level) are electromagnetic, and many experimental techniques in materials science are based on photon probes The introduction to quantum mechanics in the present book takes this into account by including aspects of condensed matter theory and the theory of photons at earlier stages and to a larger extent than other quantum mechanics texts Quantum properties of materials provide neat and very interesting illustrations of time-independent and timedependent perturbation theory, and many students are better motivated to master the concepts of quantum mechanics when they are aware of the direct relevance for modern technology A focus on the quantum mechanics of photons and materials is also perfectly suited to prepare students for future developments in quantum information technology, where entanglement of photons or spins, decoherence, and time evolution operators will be key concepts Indeed, the rapid advancement of experimental quantum physics, nanoscience, and quantum technology warrants regular updates of our courses on quantum theory Therefore, besides containing more than 50 additional end of chapter problems, the third edition also features a discussion of chiral spin-momentum locking through Rashba spin–orbit coupling and the resulting Edelstein effects in Problem 22.31, as well as the new Chap 19 on epistemic and ontic interpretations of quantum states Other special features of the discussion of quantum mechanics in this book concern attention to relevant mathematical aspects which otherwise can only be found in journal articles or mathematical monographs Special appendices include a mathematically rigorous discussion of the completeness of Sturm–Liouville eigenfunctions in one spatial dimension, an evaluation of the Baker–Campbell–Hausdorff formula to higher orders, and a discussion of logarithms of matrices Quantum mechanics has an extremely rich and beautiful mathematical structure The growing prominence of quantum mechanics in the applied sciences and engineering has already reinvigorated increased research efforts on its mathematical aspects Both students who study quantum mechanics for the sake of its numerous applications Preface vii and mathematically inclined students with a primary interest in the formal structure of the theory should therefore find this book interesting This book emerged from a quantum mechanics course which I had introduced at the University of Saskatchewan in 2001 It should be suitable for both advanced undergraduate and introductory graduate courses on the subject To make advanced quantum mechanics accessible to wider audiences which might not have been exposed to standard second- and third-year courses on atomic physics, analytical mechanics, and electrodynamics, important aspects of these topics are briefly, but concisely introduced in special chapters and appendices The success and relevance of quantum mechanics has reached far beyond the realms of physics research, and physicists have a duty to disseminate the knowledge of quantum mechanics as widely as possible Saskatoon, SK, Canada Rainer Dick Contents The Need for Quantum Mechanics 1.1 Electromagnetic Spectra and Discrete Energy Levels 1.2 Blackbody Radiation and Planck’s Law 1.3 Blackbody Spectra and Photon Fluxes 1.4 The Photoelectric Effect 1.5 Wave-Particle Duality 1.6 Why Schrödinger’s Equation? 1.7 Interpretation of Schrödinger’s Wave Function 1.8 Problems 1 15 15 17 19 23 Self-Adjoint Operators and Eigenfunction Expansions 2.1 The δ Function and Fourier Transforms 2.2 Self-Adjoint Operators and Completeness of Eigenstates 2.3 Problems 25 25 30 35 Simple Model Systems 3.1 Barriers in Quantum Mechanics 3.2 Box Approximations for Quantum Wells, Quantum Wires and Quantum Dots 3.3 The Attractive δ Function Potential 3.4 Evolution of Free Schrödinger Wave Packets 3.5 Problems 37 37 44 48 51 57 Notions from Linear Algebra and Bra-Ket Notation 4.1 Notions from Linear Algebra 4.2 Bra-ket Notation in Quantum Mechanics 4.3 The Adjoint Schrödinger Equation and the Virial Theorem 4.4 Problems 63 64 74 79 82 Formal Developments 5.1 Uncertainty Relations 5.2 Frequency Representation of States 5.3 Dimensions of States 87 87 92 95 ix x Contents 5.4 5.5 5.6 Gradients and Laplace Operators in General Coordinate Systems 96 Separation of Differential Equations 100 Problems 103 Harmonic Oscillators and Coherent States 6.1 Basic Aspects of Harmonic Oscillators 6.2 Solution of the Harmonic Oscillator by the Operator Method 6.3 Construction of the x-Representation of the Eigenstates 6.4 Lemmata for Exponentials of Operators 6.5 Coherent States 6.6 Problems 105 105 106 109 112 115 123 Central Forces in Quantum Mechanics 7.1 Separation of Center of Mass Motion and Relative Motion 7.2 The Concept of Symmetry Groups 7.3 Operators for Kinetic Energy and Angular Momentum 7.4 Matrix Representations of the Rotation Group 7.5 Construction of the Spherical Harmonic Functions 7.6 Basic Features of Motion in Central Potentials 7.7 Free Spherical Waves: The Free Particle with Sharp Mz , M 7.8 Bound Energy Eigenstates of the Hydrogen Atom 7.9 Spherical Coulomb Waves 7.10 Problems 129 129 132 134 136 141 146 147 152 162 166 Spin and Addition of Angular Momentum Type Operators 8.1 Spin and Magnetic Dipole Interactions 8.2 Transformation of Scalar, Spinor, and Vector Wave Functions Under Rotations 8.3 Addition of Angular Momentum Like Quantities 8.4 Problems 175 176 10 179 181 187 Stationary Perturbations in Quantum Mechanics 9.1 Time-Independent Perturbation Theory Without Degeneracies 9.2 Time-Independent Perturbation Theory With Degenerate Energy Levels 9.3 Problems 189 189 Quantum Aspects of Materials I 10.1 Bloch’s Theorem 10.2 Wannier States 10.3 Time-Dependent Wannier States 10.4 The Kronig-Penney Model 10.5 kp Perturbation Theory and Effective Mass 10.6 Problems 203 203 207 210 212 217 218 195 200 22.3 The Dirac Equation 603 and Eq (22.100) is uσ,s (k)u+ ¯ σ,s (k) = 2hω(k)1 (22.116) σ,s The x representations of the spinor momentum eigenstates are x, a|k, σ, s = exp(ik · x) a uσ,s (k), √ 4π π h¯ ω(k) (22.117) and using Eqs (22.114) and (22.116) we can easily verify the relations k, σ, s|k , σ , s = δσ,σ δs,s δ(k − k ), (22.118) x, a|x , a = δa,a δ(x − x ) (22.119) Charge Operators and Quantization of the Dirac Field We can apply the results from Sect 16.2 to calculate the energy and momentum operator for the Dirac field The free Dirac Lagrangian % μ & L = c ihγ ¯ ∂μ − mc (22.120) yields the positive definite normal ordered Hamiltonian H = d x c(x, t) (mc − ihγ ¯ · ∇) (x, t) = d k h¯ ω(k) ! " bs+ (k)bs (k) + ds+ (k)ds (k) , (22.121) s∈{↓,↑} but only if we assume anti-commutation properties of the ds and ds+ operators The normal ordered momentum operator is then h¯ d x + (x, t) ∇(x, t) i ! " = d k hk bs+ (k)bs (k) + ds+ (k)ds (k) ¯ P = s∈{↓,↑} The electromagnetic current density (22.74) yields the charge operator Q=q d x + (x, t)(x, t) (22.122) 604 22 Relativistic Quantum Fields =q d 3k ! " bs+ (k)bs (k) − ds+ (k)ds (k) (22.123) s∈{↓,↑} The normalization in Eq (22.94) has been chosen such that the quantization condition {α (x, t), β + (x , t)} = δαβ δ(x − x ) (22.124) for the components of (x) yields {b(k, s), b+ (k , s )} = δss δ(k − k ), {d(k, s), d + (k , s )} = δss δ(k − k ), with the other anti-commutators vanishing Equations (22.121)–(22.123) then imply that the operator b+ (k, s) creates a fermion of mass m, momentum h¯ k and charge q, while d + (k, s) creates a particle with the same mass and momentum, but opposite charge −q For an explanation of the spin labels of the spinors u(k, ± 12 ), we notice that the spin operators corresponding to the rotation generators Mi = − iLi = ij k Mj k (22.125) are both in the Dirac and in the Weyl representation given by h¯ h¯ ih¯ Si = ij k Sj k = ij k γj γk = σi , σi (22.126) see Appendix H for an explanation of generators of Lorentz boosts and rotations for Dirac spinors Equation (22.126) implies that the rest frame spinors u(0, ± 12 ) transform under rotations around the z axis as spinors with z-component of spin h¯ s = ±h/2 ¯ For an explanation of the spin labels of the spinors v(p, ± 12 ), we have to look at charge conjugation Both in the Dirac and the Weyl representation of γ matrices we have γμ∗ = γ2 γμ γ2 (22.127) Therefore complex conjugation of the Dirac equation [iγ μ ∂μ + qγ μ Aμ (x) − m](x) = 0, (22.128) followed by multiplication with iγ2 from the left yields [iγ μ ∂μ − qγ μ Aμ (x) − m] c (x) = (22.129) 22.4 The Energy-Momentum Tensor for Quantum Electrodynamics 605 with the charge conjugate field c (x) = iγ2 ∗ (x) (22.130) v c (k, 12 ) = iγ2 v ∗ (k, 12 ) = u(k, 12 ) (22.131) In particular, we have and v c (k, − 12 ) = iγ2 v ∗ (k, − 12 ) = − u(k, − 12 ), (22.132) i.e the negative energy spinors for charge q, momentum h¯ k and spin projection hs ¯ correspond to positive energy spinors for charge −q, momentum h¯ k and spin projection h¯ s Please note that Eq (22.130) uses the property (22.127), which holds in the Dirac and Weyl representations, and in all representations which are related to the Dirac and Weyl representations through real orthogonal transformations, γμ = Rγμ R T , RR T = (recall that the Dirac and Weyl bases are related through an orthogonal transformation (22.70)) If we switch to any other representation of γ matrices with a unitary transformation γμ = U · γμ · U + , then Eq (22.130) generalizes to c (x) = − iC ∗ · ∗ (x), (22.133) with C = U ∗ γ2 U + , C −1 = C ∗ = − U γ2 U T , (22.134) see Problem 22.21 22.4 The Energy-Momentum Tensor for Quantum Electrodynamics We use the symmetrized form of the QED Lagrangian (22.72), ¯ ↔ μ ih ∂μ + qAμ − mc − Fμν F μν L = c γ 4μ0 This yields according to (16.31) a conserved energy-momentum tensor μ ν = ημ ν L − ∂μ ∂L ∂(∂ν ) − ∂μ ∂L ∂L − ∂μ Aλ ∂(∂ν ) ∂(∂ν Aλ ) (22.135) 606 22 Relativistic Quantum Fields = ημ − ν ¯ ↔ λ ih κλ ∂λ + qAλ − mc − Fκλ F c γ 4μ0 ↔ ih¯ cγ ν ∂μ + ∂μ Aλ F νλ μ0 (22.136) According to the results of Sect 16.2, this yields on-shell conserved charges, i.e we can use the equations of motion to simplify this expression The Dirac equation then implies μ ν = − ↔ ih¯ 1 cγ ν ∂μ + ∂μ Aλ F νλ − ημ ν Fκλ F κλ μ0 4μ0 (22.137) We can also add the identically conserved improvement term − % & 1 ∂λ Aμ F νλ = − ∂λ Aμ F νλ − Aμ ∂λ F νλ μ0 μ0 μ0 =− ∂λ Aμ F νλ − qcAμ γ ν , μ0 (22.138) where Maxwell’s equations ∂μ F μν = − μ0 qcγ ν have been used This yields the gauge invariant tensor tμ ν = μ ν − + ↔ % & ih¯ ∂λ Aμ F νλ = − cγ ν ∂μ − qcγ ν Aμ μ0 1 Fμλ F νλ − ημ ν Fκλ F κλ μ0 4μ0 (22.139) However, we can go one step further and replace tμ ν with a symmetric energymomentum tensor The divergence of the spinor term in tμ ν is ∂ν ↔ ih¯ γ ν ∂μ + qγ ν Aμ = − qFμν γ ν , (22.140) where again the Dirac equation was used The symmetrization of tμ ν also involves the commutators of γ matrices, Sμν = i + [γμ , γν ] = γ0 · Sμν · γ0 (22.141) Since we can write a product always as a sum of an anti-commutator and a commutator, we have γμ · γν = − ημν − 2iSμν , (22.142) 22.4 The Energy-Momentum Tensor for Quantum Electrodynamics 607 and the commutators also satisfy7 ημα γβ − ημβ γα + i[Sαβ , γμ ] = (22.143) Equations (22.141)–(22.143) together with % & μ ν h¯ ∂ = ihγ ¯ ∂μ mc − qγ Aν ! % μ & " ! " μν ν = ihq ¯ ∂μ A + 2iS ∂μ (Aν ) + mc mc − qγ Aν (22.144) imply also ∂ν ↔ ih¯ γμ ∂ ν + qγμ Aν = − qFμν γ ν (22.145) Therefore the local conservation law ∂ν Tμ ν = also holds for the symmetrized energy-momentum tensor Tμ ν ↔ c ih¯ ν ↔ ih¯ ν ν ν γ ∂μ + γμ ∂ + qγ Aμ + qγμ A =− 2 + 1 Fμλ F νλ − ημ ν Fκλ F κλ μ0 4μ0 (22.146) This yields in particular the Hamiltonian density ih¯ ↔ H = − T0 = c ∂0 + qA0 + ih¯ ↔ = c mc − γ · ∇ − qγ · A + + 0 2 E + B 2μ0 0 2 E + B , 2μ0 (22.147) and the momentum density with components Pi = Ti /c, ih¯ ↔ + h¯ ↔ ∇ − qA + γ ∂0 + qA0 + 0 E × B P= 2i 2 (22.148) Elimination of the time derivatives using the Dirac equation yields P = + The h¯ ↔ ∇ − qA + 0 E × B + ∇ × ( + · S · ) 2i (22.149) commutators Sμν provide the spinor representation of the generators of Lorentz transformations Furthermore, Eq (22.143) is the invariance of the γ matrices under Lorentz transformations, see Appendix H 608 22 Relativistic Quantum Fields The spin contribution P S = ∇×( + ·S ·)/2 with the vector of 4×4 spin matrices S = ihγ ¯ × γ /4 (22.126) appears here as an additional contribution compared to the orbital momentum density P O = P − P S that follows directly from the tensor (22.139) The spin term in the momentum density (22.149) generates the spin contribution in the total angular momentum density J = x × P = M + S from S = x × P S → + · S · if the symmetric energy-momentum tensor is used in the calculation of angular momentum This is explained in Problem 22.17c, see in particular Eqs (22.283)–(22.287) Energy and Momentum in QED in Coulomb Gauge In materials science it is convenient to explicitly disentangle the contributions from Coulomb and photon terms in Coulomb gauge ∇ · A = We split the electric field components in Coulomb gauge according to E = − ∇, E⊥ = − ∂A ∂t (22.150) The equation for the electrostatic potential decouples from the vector potential in Coulomb gauge, = − q + , 0 (22.151) and is solved by (x, t) = q 4π 0 d 3x + (x , t)(x , t) |x − x | (22.152) Furthermore, the two components (22.150) of the electric field are orthogonal in Coulomb gauge, d x E (x, t) · E ⊥ (x, t) = d k E (k, t) · E ⊥ (− k, t) =− d x (x, t) ∂ ∇ · A(x, t) = 0, ∂t and the contribution from E to the Hamiltonian is 0 0 d x E 2 (x, t) = − d x (x, t)(x, t) 2 = d x (x, t) (x, t) HC = (22.153) 22.4 The Energy-Momentum Tensor for Quantum Electrodynamics = q2 d 3x d 3x ss 609 s+ (x, t)s+ (x , t)s (x , t)s (x, t) , 8π 0 |x − x | (22.154) where the summation is over 4-spinor indices The presentation of the ordering of the field operators was conventionally chosen as the correct ordering in the nonrelativistic limit, cf (18.99), but (22.154) must actually be normal ordered such that the particle and anti-particle creation operators bs+ (k) and ds+ (k) appear leftmost in the Coulomb term in the forms b+ d + db, d + d + dd, etc Substituting the mode expansions ∼ b + d + and normal ordering therefore leads to the attractive Coulomb terms between particles and their anti-particles The resulting Hamiltonian in Coulomb gauge therefore has the form H = d x c(x, t) [mc − γ · (ih∇ ¯ + qA(x, t))] (x, t) 0 E ⊥ (x, t) + B (x, t) 2μ0 s+ (x, t)s+ (x , t)s (x , t)s (x, t) d 3x d 3x + q2 8π 0 |x − x | + (22.155) ss This Hamiltonian yields the corresponding Dirac equation in the Heisenberg form ih¯ ∂ (x, t) = [(x, t), H ] ∂t (22.156) if canonical anti-commutation relations are used for the spinor field The Coulomb gauge wave equation (18.17) with the relativistic current density j (22.74) follows in the form ih¯ ∂ A(x, t) = [A(x, t), H ], ∂t ∂2 A(x, t) = [H, [A(x, t), H ]] ∂t h¯ (22.157) if the commutation relations (18.50) and (18.53) are used This confirms the canonical relations between Heisenberg, Schrödinger and Dirac pictures, and the consistency of Coulomb gauge quantization with the transverse δ function (18.41) also in the fully relativistic theory It also implies appearance of the Dirac picture time evolution operator in the scattering matrix in the now familiar form The momentum operator in Coulomb gauge follows from (22.149) and d x 0 E × B = − as d x 0 A = d x A = q d x + A 610 22 Relativistic Quantum Fields P = ¯ +h ∇ + 0 E ⊥ × B , d x i (22.158) where boundary terms at infinity were discarded 22.5 The Non-relativistic Limit of the Dirac Equation The Dirac basis (22.67) for the γ -matrices is convenient for the non-relativistic limit Splitting off the time dependence due to the rest mass term mc2 mc2 ψ(x, t) t = t exp − i (x, t) = ϒ(x, t) exp − i φ(x, t) h¯ h¯ (22.159) in the Dirac equation (22.71) yields the equations (ih∂ ¯ t − q)ψ + cσ · (ih∇ ¯ + qA)φ = 0, (22.160) (ih∂ ¯ + qA)ψ = ¯ t − q + 2mc )φ + cσ · (ih∇ (22.161) This yields in the non-relativistic regime φ − σ · (ih∇ ¯ + qA)ψ 2mc (22.162) and substitution into the equation for ψ yields Pauli’s equation8 ih∂ ¯ tψ = − q h¯ (h∇ σ · Bψ + qψ ¯ − iqA) ψ − 2m 2m (22.163) The spin matrices for spin-1/2 Schrödinger fields are the upper block matrices in the spin matrices (22.126) for the full Dirac fields, S = hσ ¯ /2, see also Sect 8.1 and in particular Eq (8.17) If the external magnetic field B is approximately constant over the extension of the wave function ψ(x, t) we can use A(x, t) = Pauli B(t) × x (22.164) [131] actually only studied the time-independent Schrödinger equation with the Pauli term in the Hamiltonian, and although he mentions Schrödinger in the beginning, he seems to be more comfortable with Heisenberg’s matrix mechanics in the paper 22.5 The Non-relativistic Limit of the Dirac Equation 611 Substitution of the vector potential in Eq (22.163) then yields the following linear terms in B in the Hamiltonian on the right-hand side, i % & q q q h¯ (B × x) · ∇ − B · S = − B · M + 2S 2m m 2m & % q μB =− B · M + 2S e h¯ (22.165) Here μB = eh/2m is the Bohr magneton, and we used the shorthand notation ¯ −ihx × ∇ → M for the x representation of the angular momentum operator Recall ¯ that this operator is actually given by M = x × p = − ih¯ d x |xx × ∇x| (22.166) Equation (22.165) shows that the Dirac equation explains the double strength magnetic coupling of spin as compared to orbital angular momentum (often denoted as the magneto-mechanical anomaly of the electron or the anomalous magnetic moment of the electron) The corresponding electromagnetic currents in the nonrelativistic regime are = qψ + ψ, % & j = cq ψ + σ φ + φ + σ ψ & q % + + =− ψ σ ⊗ σ · (ih∇ − qψ + A) · σ ⊗ σ ψ , ¯ + qA)ψ − (ih∇ψ ¯ 2m where σ ⊗ σ is the three-dimensional tensor with the (2 × 2)-matrix entries σ i · σ j (we can think of it as a (3 × 3)-matrix containing (2 × 2)-matrices as entries) Substitution of σ ⊗ σ = + iei ⊗ ej εij k σ k (22.167) yields j= & q % + + ψ · h∇ψ − h∇ψ · ψ − 2iqψ + Aψ + j s , ¯ ¯ 2im (22.168) with a spin term js = % & q h¯ ∇ × ψ +σ ψ 2m (22.169) However, this term does not accumulate or diminish charges in any volume, ∇·j s = 0, and can therefore be neglected in the calculation of electric currents 612 22 Relativistic Quantum Fields The non-relativistic approximations for the Lagrange density L, the energy density H and the momentum density P are L= ∂ ∂ q h¯ + ih¯ ψ + · ψ − ψ + · ψ − qψ + ψ + ψ σ · Bψ ∂t ∂t 2m + 1 + − qψ + A) · (ih∇ψ + qAψ) − Fμν F μν , (ih∇ψ ¯ ¯ 2m 4μ0 q h¯ + + (h∇ψ ψ σ · Bψ + iqψ + A) · (h∇ψ − iqAψ) − ¯ ¯ 2m 2m 0 B , + E2 + 2μ0 & h¯ % + ψ · ∇ψ − ∇ψ + · ψ − qψ + Aψ + 0 E × B P= 2i (22.170) H= (22.171) (22.172) The Hamiltonian and momentum operators in Coulomb gauge are H = + d x − ψ (x, t)[h∇ ¯ − iqA(x, t)] ψ(x, t) 2m q h¯ + 0 B (x, t) − E ⊥ (x, t) + ψ (x, t)σ · B(x, t)ψ(x, t) 2μ0 2m ψs+ (x, t)ψs+ (x , t)ψs (x , t)ψs (x, t) d 3x d 3x (22.173) + q2 8π 0 |x − x | + ss =1 and (cf Eq (22.158)) P = ¯ +h ∇ψ + 0 E ⊥ × B d x ψ i (22.174) It is interesting to note that if we write the current density (22.168) as j =J + % & % & q h¯ q ∇ × ψ + σ ψ = J + μB ∇ × ψ + σ ψ 2m e (22.175) we can write Ampère’s law with Maxwell’s correction term as ∂ q ∇ × B − μ0 μB ψ + σ ψ = ∇ × B class = μ0 J + μ0 0 E, e ∂t (22.176) i.e the “spin density" S(x, t) = h¯ + ψ (x, t)σ ψ(x, t) (22.177) 22.5 The Non-relativistic Limit of the Dirac Equation 613 adds a spin magnetic field to the magnetic field B class which is generated by orbital currents J and time-dependent electric fields E, B(x, t) = B class (x, t) + q 2q μ0 μB S(x, t) = B class (x, t) + μ0 S(x, t) eh¯ m Higher Order Terms and Spin-Orbit Coupling We will discuss higher order terms in the framework of relativistic quantum mechanics, i.e our basic quantum operators are x and p etc., but not quantum fields This also entails a semi-classical approximation for the electromagnetic fields and potentials For the discussion of higher order terms, we write the Dirac equation in Schrödinger form, ih¯ d |ϒ(t) = H (t)|ϒ(t), dt (22.178) with the Hamilton operator H (t) = (γ − 1)mc2 + q(x, t) + c α · [p − qA(x, t)] (22.179) The operator α is α = γ 0γ , α i ab = a|α i |b = γ ac γ i cb , (22.180) and x, a|ϒ(t) = ϒa (x, t) is the a-th component of the 4-spinor ϒ (22.159) in x representation We continue to use the Dirac basis (22.67) of γ matrices in this section, such that as a matrix valued vector α is given by α= σ σ (22.181) The part of the Hamiltonian (22.179) which mixes the upper and lower components of the 4-spinor ϒ is K(t) = c α · [p − qA(x, t)] (22.182) Operators which mix upper and lower 2-spinors in 4-spinors are also denoted as odd terms in the Hamiltonian We can remove the odd contribution K(t) by using the anti-hermitian operator 614 22 Relativistic Quantum Fields γ0 γ · [p − qA(x, t)], K(t) = 2mc 2mc T (t) = (22.183) [T (t), γ mc2 ] = −K(t), (22.184) which implies subtraction of K(t) from the new transformed Hamiltonian exp[T (t)]H (t) exp[−T (t)] However, we also have to take into account that the transformed state |ϒT (t) = exp[T (t)]|ϒ(t) satisfies the equation ih¯ d |ϒT (t) = exp[T (t)]H (t) exp[−T (t)]|ϒ(t) dt d exp[T (t)] exp[−T (t)]|ϒ(t) + ih¯ dt (22.185) Therefore the transformed Hamiltonian is actually HT (t) = exp[T (t)]H (t) exp[−T (t)] − ih¯ exp[T (t)] = ∞ ∞ n n [T (t), H (t)] − ih¯ [T (t), d/dt] n! n! n=0 = n=0 ∞ ∞ n n−1 [T (t), H (t)] + ih¯ [ T (t), dT (t)/dt] n! n! n=0 = d exp[−T (t)] dt ∞ n=0 n=1 ∞ n iq h¯ n−1 ˙ [T (t), H (t)] − [ T (t), γ · A(t)] (22.186) n! 2mc n! n=1 We also wish to expand the Hamiltonian up to terms of order (E/mc2 )3 , where E contains contributions from the kinetic energy of the particle and from its interactions with the electromagnetic fields Equation (22.184) implies HT (t) = (γ − 1)mc2 + q(t) + mc2 n [T (t), γ ] n! n=2 + n=1 − n iq h¯ ˙ γ · A(t) [T (t), q(t) + K(t)] − n! 2mc n iq h¯ E ˙ [T (t), γ · A(t)] +O (22.187) 2mc (n + 1)! mc2 n=1 The relevant commutators are