Tai ngay!!! Ban co the xoa dong c From Classical to Quantum Mechanics This book provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics Part I covers the basic material that is necessary to an understanding of the transition from classical to wave mechanics Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton–Jacobi equation; the Cauchy problem for the wave equation, the Helmholtz equation and eikonal approximation; and introductions to spin, perturbation theory and scattering theory The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics The Weyl programme provides a geometric framework for a rigorous formulation of canonical quantization, as well as powerful tools for the analysis of problems of current interest in quantum physics In the chapters devoted to harmonic oscillators and angular momentum operators, the emphasis is on algebraic and group-theoretical methods Quantum entanglement, hidden-variable theories and the Bell inequalities are also discussed Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses It is self-contained and includes problems to advance the reader’s understanding Giampiero Esposito received his PhD from the University of Cambridge in 1991 and has been INFN Research Fellow at Naples University since November 1993 His research is devoted to gravitational physics and quantum theory His main contributions are to the boundary conditions in quantum field theory and quantum gravity via functional integrals Giuseppe Marmo has been Professor of Theoretical Physics at Naples University since 1986, where he is teaching the first undergraduate course in quantum mechanics His research interests are in the geometry of classical and quantum dynamical systems, deformation quantization, algebraic structures in physics, and constrained and integrable systems George Sudarshan has been Professor of Physics at the Department of Physics of the University of Texas at Austin since 1969 His research has revolutionized the understanding of classical and quantum dynamics He has been nominated for the Nobel Prize six times and has received many awards, including the Bose Medal in 1977 i ii FROM CLASSICAL TO QUANTUM MECHANICS An Introduction to the Formalism, Foundations and Applications Giampiero Esposito, Giuseppe Marmo INFN, Sezione di Napoli and Dipartimento di Scienze Fisiche, Universit` a Federico II di Napoli George Sudarshan Department of Physics, University of Texas, Austin iii Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521833240 © G Esposito, G Marmo and E C G Sudarshan 2004 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2004 - - ---- eBook (NetLibrary) --- eBook (NetLibrary) - - ---- hardback --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate For Michela, Patrizia, Bhamathi, and Margherita, Giuseppina, Nidia v vi Contents Preface Acknowledgments Part I From classical to wave mechanics page xiii xvi 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 Experimental foundations of quantum theory The need for a quantum theory Our path towards quantum theory Photoelectric effect Compton effect Interference experiments Atomic spectra and the Bohr hypotheses The experiment of Franck and Hertz Wave-like behaviour and the Bragg experiment The experiment of Davisson and Germer Position and velocity of an electron Problems Appendix 1.A The phase 1-form 3 11 17 22 26 27 33 37 41 41 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Classical dynamics Poisson brackets Symplectic geometry Generating functions of canonical transformations Hamilton and Hamilton–Jacobi equations The Hamilton principal function The characteristic function Hamilton equations associated with metric tensors Introduction to geometrical optics Problems Appendix 2.A Vector fields 43 44 45 49 59 61 64 66 68 73 74 vii viii Contents Appendix Appendix Appendix Appendix 2.B Lie algebras and basic group theory 2.C Some basic geometrical operations 2.D Space–time 2.E From Newton to Euler–Lagrange 76 80 83 83 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Wave equations The wave equation Cauchy problem for the wave equation Fundamental solutions Symmetries of wave equations Wave packets Fourier analysis and dispersion relations Geometrical optics from the wave equation Phase and group velocity The Helmholtz equation Eikonal approximation for the scalar wave equation Problems 86 86 88 90 91 92 92 99 100 104 105 114 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Wave mechanics From classical to wave mechanics Uncertainty relations for position and momentum Transformation properties of wave functions Green kernel of the Schrăodinger equation Example of isometric non-unitary operator Boundary conditions Harmonic oscillator JWKB solutions of the Schră odinger equation From wave mechanics to Bohr–Sommerfeld Problems Appendix 4.A Glossary of functional analysis Appendix 4.B JWKB approximation Appendix 4.C Asymptotic expansions 115 115 128 131 136 142 144 151 155 162 167 167 172 174 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Applications of wave mechanics Reflection and transmission Step-like potential; tunnelling eect Linear potential The Schrăodinger equation in a central potential Hydrogen atom Introduction to angular momentum Homomorphism between SU(2) and SO(3) Energy bands with periodic potentials Problems 176 176 180 186 191 196 201 211 217 220 Contents ix Appendix 5.A Stationary phase method Appendix 5.B Bessel functions 221 223 6.1 6.2 6.3 6.4 6.5 6.6 Introduction to spin Stern–Gerlach experiment and electron spin Wave functions with spin The Pauli equation Solutions of the Pauli equation Landau levels Problems Appendix 6.A Lagrangian of a charged particle Appendix 6.B Charged particle in a monopole field 226 226 230 233 235 239 241 242 242 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 Perturbation theory Approximate methods for stationary states Very close levels Anharmonic oscillator Occurrence of degeneracy Stark effect Zeeman effect Variational method Time-dependent formalism Limiting cases of time-dependent theory The nature of perturbative series More about singular perturbations Problems Appendix 7.A Convergence in the strong resolvent sense 244 244 250 252 255 259 263 266 269 274 280 284 293 295 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 Scattering theory Aims and problems of scattering theory Integral equation for scattering problems The Born series and potentials of the Rollnik class Partial wave expansion The Levinson theorem Scattering from singular potentials Resonances Separable potential model Bound states in the completeness relationship Excitable potential model Unitarity of the Mă oller operator Quantum decay and survival amplitude Problems 297 297 302 305 307 310 314 317 320 323 324 327 328 335 11.2 Two-dimensional harmonic oscillator J− |n1 , n2 = J+ J− |n1 , n2 = n1 (n2 + 1)|n1 − 1, n2 + 1, (11.2.9) n1 (n2 + 1)J+ |n1 − 1, n2 + 1 = n1 (n2 + 1)|n1 , n2 , J− J+ |n1 , n2 = 407 (11.2.10) (n1 + 1)n2 J− |n1 + 1, n2 − 1 = (n1 + 1)n2 |n1 , n2 , 1 n1 (n2 + 1) − n2 (n1 + 1) |n1 , n2 1 = n1 − n2 |n1 , n2 (11.2.11) J3 |n1 , n2 = (11.2.12) Thus, after defining j≡ 1 n1 + n , m≡ 1 n − n2 , (11.2.13) where j(j + 1) are the eigenvalues of J , the ket vectors for this twodimensional system can be written as |j, m, and represent eigenstates of the J and J3 operators, according to the rules of section 11.1 They can be obtained from the ground state as follows: (a† )j+m (a†2 )j−m |0 |j, m = (j + m)!(j − m)! (11.2.14) 11.2.1 Introduction of different bases One can also consider the operators A± ≡ √ a1 ∓ ia2 , (11.2.15) A†± ≡ √ a†1 ± ia†2 , (11.2.16) which satisfy the commutation relations Am , An = A†m , A†n = 0, Am , A†n = δm,n (11.2.17) (11.2.18) If we introduce ‘quanta of type + or −’, the operators A+ and A†+ turn out to be annihilation and creation operators of quanta of type +, while A− and A†− are annihilation and creation operators of quanta of type − 408 Angular momentum operators The introduction of the operators N ≡ N+ + N− (11.2.19) L ≡ N+ − N− , (11.2.20) and which form a complete set of commuting observables, makes it possible to consider a different basis of eigenstates Let us point out that the following commutation relations hold: L, A†± = ±A†± , L, A± = ∓A± (11.2.21) (11.2.22) Thus, on the eigenstates of L, the operators A†+ and A− raise by one the eigenvalues, while A†− and A+ lower by one unit the eigenvalues Various interpretations of this property are possible In the theory of charged fields, where the field is described by a set of isotropic oscillators in two dimensions, N+ is the number of particles with positive charge, and N− is the number of particles with negative charge The operator L represents (up to a constant) the total charge According to this interpretation, the operator A†+ creates a positive charge while A− absorbs a negative charge, and they both increase the charge by one unit By analogy, the operators A†− and A+ are viewed as reducing the charge by one unit In the theory of lattice vibrations the displacements of the lattice are equally represented by a set of isotropic two-dimensional oscillators, and the oscillation quanta are called phonons The representation in terms of phonons of type and provides a classification in stationary waves The use of phonons of type + and − corresponds instead to progressive waves that propagate in a given direction or in the opposite direction Similarly, in three dimensions one can define the operators A1 ≡ √ ax − iay , (11.2.23) A0 ≡ az , (11.2.24) A−1 ≡ √ ax + iay , (11.2.25) which, jointly with their Hermitian conjugates, obey canonical commutation relations analogous to (11.2.17) and (11.2.18), but bearing in mind that m and n range now from to These operators can again be 11.3 Rotations of angular momentum operators 409 interpreted as annihilation and creation operators of quanta of type −1, 0, The corresponding number operators are N−1 ≡ A†−1 A−1 , (11.2.26) N0 ≡ A†0 A0 , (11.2.27) N1 ≡ A†1 A1 , (11.2.28) which form a maximal set of commuting observables The Hamiltonian therefore reads as ¯hω, (11.2.29) H = N1 + N0 + N−1 + and the full number operator is N = N1 + N0 + N−1 (11.2.30) To each triple of eigenvalues (n−1 , n0 , n1 ) there corresponds an eigenvector for the three observables given by |n1 , n0 , n−1 = (n1 !n0 !n−1 !)−1/2 A†1 and one has n1 A†0 n0 A†−1 n−1 |0, (11.2.31) ¯ ω|n1 , n0 , n−1 h H|n1 , n0 , n−1 = n1 + n0 + n−1 + (11.2.32) The eigenvectors obtained in this way are not eigenvectors of L2 , but are eigenvectors of Lz , which takes the form h, Lz = (N1 − N−1 )¯ (11.2.33) and hence the eigenvalues of Lz are m ≡ n1 − n−1 The whole triple of angular momentum operators can be obtained from (Lx , Ly , Lz ) = L = A† SA, where S denotes the × spin-1 matrices the equation L 11.3 Rotations of angular momentum operators The theoretical investigation of interferometers shows the existence of a further deep link between the formalism of annihilation and creation operators and that for angular momentum (Yurke et al 1986) Here we are interested in lossless devices with two input ports and two output ports Let a1 and a2 be the annihilation operators for two light beams, e.g the two light beams entering a beamsplitter, or the two light beams leaving such a device The corresponding creation operators are denoted by a†1 and a†2 All of these operators obey the commutation relations , aj = ∀i, j = 1, 2, (11.3.1) 410 Angular momentum operators a†i , a†j = ∀i, j = 1, 2, (11.3.2) , a†j = δij 1I (11.3.3) One can now define the operators (cf Eqs (11.2.5)–(11.2.7)) Jx ≡ 1 † a1 a2 + a†2 a1 , Jy ≡ − (11.3.4) i † a1 a2 − a†2 a1 , (11.3.5) 1 † a1 a1 − a†2 a2 (11.3.6) Moreover, a ‘number’ operator can also be defined as in (11.2.6) One can check, by repeated application of (11.3.1)–(11.3.3), that the following formulae hold: Jz ≡ Jx , Jy = iJz , J ≡ Jy , Jz = iJx , Jx2 + Jy2 + Jz2 N = Jz , Jx = iJy , (11.3.7) N +1 (11.3.8) In the course of deriving these identities, besides the many cancellations that occur, one has to re-express any term such as a†i as a†i + 1I, using (11.3.3) One thus finds that (11.3.4)–(11.3.6) are indeed angular momentum operators Let a1 and a2 be the annihilation operators for incoming light, while b1 and b2 denote the annihilation operators for the outgoing light The scattering matrix for our interferometric device is defined by the equation (Yurke et al 1986) b1 b2 = U11 U21 U12 U22 a1 a2 (11.3.9) If the commutation relations (11.3.1)–(11.3.3) are required to hold for both the ingoing and outgoing annihilation and creation operators, one finds that U U † = U † U = 1I, (11.3.10) i.e the scattering matrix should be a unitary matrix At a deeper level, the matrix U preserves both the symplectic and the complex structure, and hence it can only be unitary For example, a beamsplitter for which the scattering matrix takes the form cos α2 −i sin α2 U= , (11.3.11) −i sin α2 cos α2 11.3 Rotations of angular momentum operators leads to Jx Jy = cos α sin α Jz out 411 Jx − sin α Jy , cos α Jz in (11.3.12) by virtue of Eqs (11.3.4)–(11.3.6), (11.3.9) and (11.3.11) Equation (11.3.12) shows that the abstract angular momentum operators are rotated by an angle α about the x-axis This result can be re-expressed in the form Jx Jx Jy = eiαJx Jy e−iαJx (11.3.13) Jz out Jz in This corresponds to a Heisenberg-like picture However, one can also work in a Schră odinger-like picture, where Jx , Jy and Jz remain fixed, while the state vector, after interacting with the beamsplitter, is turned into |out = e−iαJx |in (11.3.14) With our notation, |in is the state vector for light before it interacts with the beamsplitter Another relevant example of scattering matrix is given by U= cos β2 − sin β2 sin β2 cos β2 , (11.3.15) which leads to a change of the angular momentum operators according to Jx cos β sin β Jx Jy Jy = (11.3.16) − sin β cos β Jz out Jz in This means that a rotation by an angle β about the y-axis is performed: Jx Jx Jy = eiβJy Jy e−iβJy Jz out Jz in (11.3.17) In the corresponding Schră odinger picture, the state vector for light is turned into |out = e−iβJy |in (11.3.18) Lastly, if the light beams incur the phase shifts γ1 and γ2 respectively, the scattering matrix reads U =e i γ1 +γ2 ei γ1 −γ2 e−i γ −γ 2 , (11.3.19) 412 Angular momentum operators so that a rotation by an angle γ2 − γ1 about the z-axis is performed: Jx Jx Jy = ei(γ2 −γ1 )Jz Jy e−i(γ2 −γ1 )Jz Jz out Jz in (11.3.20) In the Schră odinger picture, one writes |out = ei (γ1 +γ2 )N e−i(γ2 −γ1 )Jz |in (11.3.21) These fairly simple equations express a deep property: a lossless device with two input ports and two output ports may be viewed as the process of measuring rotations of angular momentum operators These (abstract) operators are defined in terms of annihilation and creation operators for incoming and outgoing light beams 11.4 Clebsch–Gordan coefficients and the Regge map Recall, from section 6.2, that the mathematical problem of adding angular momentum operators finds its motivation in the physical problem of studying particles possessing both orbital and spin angular momentum Moreover, multiparticle states involving two spins and one relative angular momentum may also be relevant Here we are interested in a more advanced treatment Of course, we start with two angular momentum operators, JA and JB , which commute: JA,k , JA,l = i¯ hεklp JA,p , JB,k , JB,l = i¯ hεklp JB,p , JA,k , JB,l = 0, ∀k, l = 1, 2, 3, (11.4.1) (11.4.2) (11.4.3) so that the total angular momentum J ≡ JA + JB (11.4.4) generates a representation of rotations which is the direct product of the representations generated by JA and JB : U (R) = UA (R)UB (R) (11.4.5) By hypothesis, UA (R) and UB (R) define two irreducible representations D(j1 ) and D(j2 ) : UA (R) = D(j1 ) (R) ⊗ 1I, (11.4.6) UB (R) = 1I ⊗ D(j2 ) (R) (11.4.7) 11.4 Clebsch–Gordan coefficients and the Regge map 413 The representation R → U (R) = D(j1 ) (R) ⊗ D(j2 ) (R) is reducible, and the problem arises to decompose it into irreducible representations 2 2Thecorresponding complete set of commuting observables is JA , JB , J , Jz The operators U (R) act on the (2j1 + 1)(2j2 + 1)dimensional space, linearly generated by the simultaneous eigenvectors , J2 , J of JA B A,z , JB,z , i.e |j1 , m1 ⊗ |j2 , m2 The operator Jz ≡ JA,z + JB,z is diagonal in this basis, with eigenvalues m = m1 + m2 Each eigenvalue m has its own degeneracy Only the eigenvector |j1 , j1 ; j2 , j2 has the eigenvalue m = j1 + j2 , whereas there exist two eigenvectors with eigenvalue m = j1 + j2 − 1, three eigenvectors with eigenvalue m = j1 + j2 − 2, , up to m = |j1 − j2 |, for which there are two min(j1 , j2 ) eigenvectors This is the maximal degeneracy, which remains constant until m reaches the negative value −|j1 − j2 | For lower values of m, the degeneracy starts decreasing until m reaches the lowest value, −j1 − j2 , for which only one eigenvector exists: |j1 , −j1 ; j2 , −j2 Our task is now to develop a technique to express the common eigenvec2 , J , J , J in terms of the common eigenvectors of J , J tors of JA z B A A,z , JB and JB,z For this purpose, we denote the former by |j1 , j2 ; j, m, and the latter by |j1 , m1 ; j2 , m2 = |j1 , m1 ⊗ |j2 , m2 = |j1 , m1 |j2 , m2 , and we point out that these two sets of eigenvectors are related by an orthogonal transformation (Srinivasa Rao and Rajeswari 1993), |j1 , j2 ; j, m = j1 , m1 ; j2 , m2 |j, m |j1 , m1 ; j2 , m2 , (11.4.8) m1 ,m2 where the coefficients j1 ,m1 ,j2 ,m2 j1 , m1 ; j2 , m2 |j, m ≡ Cj,m are the Clebsch–Gordan coefficients (after the work of Clebsch (1872) and Gordan (1875) on the invariant theory of algebraic forms) By construction, they are non-vanishing if and only if m1 + m2 = m, j ∈ {j1 + j2 , , |j1 − j2 |} 414 Angular momentum operators A set of recurrence relations for the evaluation of all Clebsch–Gordan coefficients is obtained on applying the raising and lowering operators to Eq (11.4.8) This yields J± |j1 , j2 ; j, m = = (j ∓ m)(j ± m + 1)|j1 , j2 ; j, m ± 1 (j1 ∓ m1 )(j1 ± m1 + 1) |j1 , m1 ± 1; j2 , m2 m1 ,m2 + j1 ,m1 ,j2 ,m2 (j2 ∓ m2 )(j2 ± m2 + 1) |j1 , m1 ; j2 , m2 ± 1 Cj,m (11.4.9) At this stage, it is convenient to set m1 ≡ m1 ± 1, m2 ≡ m2 ± (11.4.10) On the right-hand side of Eq (11.4.9) one can then express, for k = 1, 2, jk ∓ mk = jk ∓ (mk ∓ 1) = jk ∓ mk + 1, (11.4.11) jk ± mk = jk ± (mk ∓ 1) = jk ± mk − (11.4.12) Thus, bearing in mind that summation over all values of m1 and m2 makes it possible to re-label them, one finds from (11.4.8) and (11.4.9) the very useful formula j1 ,m1 ,j2 ,m2 (j ∓ m)(j ± m + 1)Cj,m±1 |j1 , m1 ; j2 , m2 m1 ,m2 = j1 ,m1 ∓1,j2 ,m2 (j1 ∓ m1 + 1)(j1 ± m1 )Cj,m |j1 , m1 ; j2 , m2 m1 ,m2 + j1 ,m1 ,j2 ,m2 ∓1 (j2 ∓ m2 + 1)(j2 ± m2 )Cj,m |j1 , m1 ; j2 , m2 m1 ,m2 (11.4.13) This leads in turn to the desired recursive algorithm: j1 ,m1 ,j2 ,m2 (j ∓ m)(j ± m + 1)Cj,m±1 = j1 ,m1 ∓1,j2 ,m2 (j1 ∓ m1 + 1)(j1 ± m1 )Cj,m + j1 ,m1 ,j2 ,m2 ∓1 (j2 ∓ m2 + 1)(j2 ± m2 )Cj,m (11.4.14) Interestingly, the orthogonal transformation (11.4.8) can be inverted, in the form |j1 , m1 ; j2 , m2 = j,m j1 ,m1 ,j2 ,m2 Cj,m |j1 , j2 ; j, m, (11.4.15) 11.4 Clebsch–Gordan coefficients and the Regge map 415 by virtue of the orthogonality properties satisfied by the Clebsch–Gordan coefficients (Srinivasa Rao and Rajeswari 1993): j1 ,m1 ,j2 ,m2 ,m1 ,j2 ,m2 Cj,m Cjj1,m = δj,j δm,m (11.4.16) m1 ,m2 and j ,m1 ,j2 ,m2 j1 ,m1 ,j2 ,m2 Cj,m Cj,m = δm1 ,m1 δm2 ,m2 (11.4.17) j,m In 1940, Wigner defined the 3-j symbol as j1 m1 j2 m2 j3 m3 ≡ (−1)j1 −j2 −m3 j1 ,m1 ,j2 ,m2 , Cj3 ,−m3 [j3 ] where [j3 ] ≡ (11.4.18) 2j3 + 1, (11.4.19) and the mk quantum numbers in the 3-j coefficients satisfy the condition m1 + m2 + m3 = (11.4.20) In the current literature, the 3-j coefficient is defined as j1 m1 j2 m2 j3 m3 ≡ δm1 +m2 +m3 ,0 (−1)j1 −j2 −m3 f (j1 , j2 , j3 ) × × (−1)t t! t (t − αk )! k=1 −1 (βl − t)! l=1 (ji + mi )!(ji − mi )!, (11.4.21) tmin ≡ max(0, α1 , α2 ), tmax ≡ min(β1 , β2 , β3 ), (11.4.22) α1 ≡ j1 − j3 + m2 = (j1 − m1 ) − (j3 + m3 ), (11.4.23) α2 ≡ j2 − j3 − m1 = (j2 + m2 ) − (j3 − m3 ), (11.4.24) β1 ≡ j1 − m1 , β2 ≡ j2 + m2 , β3 ≡ j1 + j2 − j3 , (11.4.25) i=1 where t ∈ tmin , tmax , and f (x, y, z) ≡ (−x + y + z)!(x − y + z)!(x + y − z)! (x + y + z + 1)! The function f vanishes unless |j1 − j2 | ≤ j ≤ j1 + j2 (11.4.26) 416 Angular momentum operators Further details can be found in Biedenharn and Louck (1981) Major progress in the understanding of the symmetries of the 3-j coefficients began when Regge (1958) arranged the nine non-negative integer parameters: −j1 + j2 + j3 , j1 − j2 + j3 , j1 + j2 − j3 , j1 − m1 , j2 − m2 , j3 − m3 , j1 + m1 , j2 + m2 , j3 + m3 , into a × square symbol and represented the 3-j coefficient as where j1 m1 j2 m2 j3 m3 −j1 + j2 + j3 (aik ) ≡ j1 − m1 j1 + m1 → (aik ) i, k = 1, 2, 3, j1 − j2 + j3 j2 − m2 j2 + m2 (11.4.27a) j1 + j2 − j3 j3 − m3 , j3 + m3 (11.4.27b) and noted that aik = j1 + j2 + j3 ∀i = 1, 2, 3, (11.4.28) ail = j1 + j2 + j3 ∀l = 1, 2, (11.4.29) k=1 i=1 Regge concluded that the 3-j coefficient has 72 symmetries, being invariant under 3! column permutations, 3! row permutations and a reflection about the diagonal of the × square symbol 11.5 Postulates of quantum mechanics with spin We can now give a thorough formulation of quantum mechanics when the effects of spin are included To begin, note that the Pauli matrices (5.7.10)–(5.7.12) provide a particular example of a Clifford algebra Their basic properties are as follows (hereafter we replace x, y, z by 1, 2, respectively) (i) They form an algebra (ii) They obey the property σj σk = δjk 1I + iεjkl σl , (11.5.1) σj σk − σk σj = 2iεjkl σl , (11.5.2) σj σk + σk σj = 2δjk 1I (11.5.3) which implies 11.5 Postulates of quantum mechanics with spin 417 (iii) The triple σ1 , σ2 , σ3 ≡ σ is an operator with vector-valued expectation values (iv) One finds by direct calculation a · σ b · σ = a · b 1I + i a ∧ b · σ (11.5.4) Thus, if one considers the rotation through an angle ω about a generic axis n, which is image of Un (ω/2) ≡ e−i(σ·n)ω/2 , one finds Un (ω/2) = cos (11.5.5) ω ω σ · n 1I − i sin 2 (11.5.6) Since double-valued representations of the rotation group exist, the next step is to consider the class of all projective transformations (i.e up to a phase): D(R)D(S) = ω(R, S)D(RS), (11.5.7) |ω(R, S)| = (11.5.8) where Denoting by S1 , S2 and S3 the generators of R → D(R), one can write D(n, α) ≡ D(R(n, α)) = e−iαn·S (11.5.9) By virtue of (11.5.7), one recovers from (11.5.9) the commutation rules for angular momentum operators, i.e Sj , Sk = i¯ hεjkl Sl (11.5.10) We are thus led to say that the spin of a (quantum) particle behaves like an angular momentum Note that spin does not result from the translational motion of the particle, and its magnitude can only take a fixed value The basic postulates of a formalism which incorporates spin are thus as follows a spin angular momentum (a) Besides the orbital angular momentum L, exists with components S1 , S2 , S3 Such operators commute with position and momentum (b) The operator +S J ≡ L (11.5.11) 418 Angular momentum operators plays the role of total angular momentum, and hence, by definition, commutes with all the operators invariant under rotations Within this framework, wave functions are complex-valued maps ψ : R3 × C2s+1 → C which are linear in the second argument Therefore we may consider maps ψ : R3 → C2s+1 , where C2s+1 is identified with its dual vector space Moreover, the operators become matrix-valued: xk ψ1 x ˆk ψ = , xk ψ2s+1 h ¯ ∂ i ∂xk pˆk ψ = (11.5.12) ψ1 , h ¯ ∂ ψ2s+1 i ∂xk (11.5.13) ψ1 (Sk )1,2s+1 ψ2s+1 (Sk )2s+1,2s+1 (11.5.14) In other words, the wave function ψ is a column vector of functions belonging to L2 (R3 ), since the Hilbert space of quantum mechanics incorporating spin is isomorphic to the direct product of L2 (R3 ) with a finite-dimensional Hilbert space If ψ undergoes a linear transformation under a rotation R, i.e (Sk )1,1 Sˆk ψ = (Sk )2s+1,1 (Sk )1,2 (Sk )2s+1,2 ψ (x ) = D(R)ψ(x) x = Rx, (11.5.15) then, on setting ψ = U (R)ψ, one obtains the overall transformation law (U (R)ψ)(x) = D(R)ψ(R−1 x), (11.5.16) and the composition of two rotations yields (U (R1 )U (R2 )ψ)(x) = (D(R1 )D(R2 )ψ)((R1 R2 )−1 x) (11.5.17) Nothing prevents us from fulfilling Eq (11.5.17) with the help of projective transformations defined by (11.5.7) and (11.5.8) For the group SO(3), the analysis of section 5.7 shows that the phase factor σ(R1 , R2 ) = ±1 The overall representation R → U (R) obeys the composition law U (R1 R2 ) = σ(R1 , R2 )U (R1 )U (R2 ), (11.5.18) 11.6 Spin and Weyl systems 419 with exponentiated form U (n, α) = e−iαn·J , (11.5.19) where J is given by (11.5.11) Once again, we stress that S1 , S2 , S3 are the generators of the projective representation R → D(R), which maps the rotation R into the matrix D(n, α ) ≡ D(R(n, α)) = e−iαn·S (11.5.20) 11.6 Spin and Weyl systems Recall from section 9.5 that a Weyl system is a continuous unitary map of a symplectic space into the set of unitary operators on a Hilbert space such that Eq (9.5.11) holds, with W a strongly continuous function of z It is clear that there we had a projective unitary representation of the Abelian group S ≡ R2n with multiplier σ associated with the symplectic structure on S: i σ(x, y) = e h¯ ω(x,y) (11.6.1) For the case of quantum mechanics with spin we consider the semidirect group structure SU (2) ⊗ρ S where ρ is the action of SU (2) on S preserving the symplectic structure Moreover, the multiplier is provided by the one of S plus the σ(R1 , R2 ) of the previous section The Hilbert space H carries a projective representation of the group SU (2) ⊗ρ S A specific realization is obtained by selecting any Lagrangian sub-space left invariant by the SU (2)-action and building on it square-integrable functions with values in C2s+1 A particular realization of H is therefore provided by H ≡ L2 (R3 ) ⊗ C2s+1 (11.6.2) For one-parameter groups one can always write σ(R1 , R2 ) = eiΩ(R1 (s),R2 (t)) , (11.6.3) where R1 (s) and R2 (t) are such that U (R1 (s)) = e−isJ1 , U (R2 (t)) = e−itJ2 (11.6.4) On writing the generators in the form J1 = λ1k jk , J2 = λ2l jl , (11.6.5) one obtains k jk l jl U (R1 (s)) = e−isλ1 U (R2 (t)) = e−itλ2 , (11.6.6) (11.6.7) 420 Angular momentum operators For the rotation group, however, the phase factor reduces to ±1 as we mentioned after Eq (11.5.17) 11.7 Monopole harmonics Recall from appendix 6.B that, in his attempt to understand the quantization of the electric charge, Dirac was led to assume that particles have both magnetic and electric charge, with corresponding magnetic and electric currents (Dirac 1931) The resulting macroscopic Maxwell equations would therefore read as + curl E 4π ∂B = − Jm , c ∂t c (11.7.1) = 4πρm , div B (11.7.2) − ∂ D = 4π Je , curl H c ∂t c (11.7.3) = 4πρe div D (11.7.4) Note now that under the duality transformations, defined by (with ξ ∈ R) E H D B ρe ρm Je Jm = = = = cos ξ − sin ξ sin ξ cos ξ cos ξ − sin ξ sin ξ cos ξ cos ξ − sin ξ sin ξ cos ξ cos ξ − sin ξ sin ξ cos ξ E H D B , (11.7.5) , (11.7.6) ρ e ρm Je Jm , (11.7.7) , (11.7.8) , B , H , with , D the Maxwell equations for the transformed fields E charge densities ρe , ρm and current densities Je , Jm , remain formally analogous to Eqs (11.7.1)–(11.7.4) Thus, what is really crucial is that for all particles the ratio of magnetic to electric charge is the same If this is the case, one can perform a duality transformation to obtain vanishing values for the magnetic charge density ρm and magnetic current density Jm (Jackson 1975) Note also that, by virtue of Eq (11.7.2), one can no longer describe on the whole of R3 The correct B as the curl of a vector potential A 11.7 Monopole harmonics 421 mathematical description is instead obtained by considering a vector B such that = in R3 − {0} , (11.7.9) div B i.e in a space where not all closed surfaces can be shrunk continuously to a point without passing outside The integer that classifies solutions of the Maxwell equations in R3 − {0} is called the magnetic charge An example of monopole solution with charge n is given below, with the notation of differential geometry: n F = dA = x dy ∧ dz + y dz ∧ dx + z dx ∧ dy , (11.7.10) 2r = n r, B (11.7.11) 2r3 where r ≡ x2 + y + z Dirac pointed out that if magnetic monopoles exist, their charge is inversely proportional to the electric charge It follows that, if monopoles exist, we can understand charge quantization! The occurrence of monopoles, well defined mathematically, but for which the experimental evidence is still lacking, makes it possible to introduce the corresponding monopole harmonics They are the next logical step after the spherical harmonics of section 5.6, and hence we focus on them hereafter For this purpose, we study the wave function of a charged particle in the field of a magnetic monopole Our presentation follows the work of Wu and Yang (1976), whereas, for further developments, the reader is referred to the work by Kazama et al (1977), Dray (1985) and Weinberg (1994) First it should be remarked that, since the space around a monopole is spherically symmetric and singularity-free, the wave function of a charged particle should also be singularity-free On the other hand, cusps and discontinuities are found to occur because any choice of the vector poten around the monopole must have singularities One can overcome tial A this difficulty after a careful consideration of a simpler problem, i.e the choice of a coordinate system on the surface of a sphere It is indeed well known that all possible choices have some singularities, whereas the geometry of the sphere is, clearly, singularity-free To avoid introducing (fictitious) singularities in the coordinate system one can divide the sphere into more than one overlapping region, defining a singularity-free coordinate system in each region Moreover, in the overlap one has a singularity-free coordinate transformation between the different coordinate systems Similarly, one can divide the space outside a magnetic monopole into (1) two regions R1 and R2 , and define a vector potential Ak in R1 and a