Subatomic Physics

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Chapter 8 Angular Momentum and Isospin In this chapter we shall show that invariance under rotation in space leads to conservation of angular momentum. We shall then introduce isospin, a quantity that has many properties similar to ordinary spin, and discuss the “breaking” of isospin invariance. 8.1 Invariance Under Spatial Rotation Invariance under spatial rotation provides an important application of the general considerations presented in Section 7.1. Consider an idealized experimental arrangement, shown in Fig. 8.1. We assume for simplicity that the equipment is in the xy plane; its orientation is described by the angle ϕ. We further assume that the result of the experiment is described by a wave function ψ(x). Next, the equipment is rotated by an angle α about the z axis. This rotation is denoted by Rz(α), and it carries apointx into a point x R : x R = Rz(α)x. (8.1) Figure 8.1: Rotation around the z axis. The angle ϕ fixes the position of the original equipment axis; it does not denote a rotation. The equipment is rotated about the z axis by an angle α. Invariance under rotation means that the outcome of the experiment is not affected by the rotation. The rotation changes the wave function; the relation between the rotated and unrotated wave function at point x is given by Eq. (7.7) as ψ R (x) =Uz(α)ψ(x). (8.2) 221
222 Angular Momentum and Isospin The notation indicates that the rotation is by an angle α about the z axis. So far, no invariance properties have been used and Eqs. (8.1) and (8.2) are valid even if the system changes during rotation. Invariance arguments can now be used to find U. If the state of the system is unaffected by rotation, the wave function at point x in the original system is identical to the rotated wave function at the rotated point x R , ψ(x) =ψ R (x R ). (8.3) Note the difference between Eqs. (8.2) and (8.3). The first connects ψ(x) toψR at the same point, and the second to ψR at the rotated point xR . U can be found if ψR (xR ) can be expressed in terms of ψR (x). Because the rotation is continuous, any rotation by a finite angle can be built up from rotations by infinitely small angles. An infinitesimal rotation suffices to find U. If the system is rotated by an infinitesimal angle δα about the z axis, ψR (xR ) becomes ψ R (x R )=ψ R (x)+ ∂ψR � (x) δα = 1+δα ∂ϕ ∂ � ψ ∂ϕ R (x). This relation can be inverted by multiplication by [1−δα(∂/∂ϕ)]. Neglecting terms in δα2 and using Eq. (8.3) then yields ψ R � (x) = 1 − δα ∂ � ψ(x). (8.4) ∂ϕ Comparison with Eq. (8.2) shows that the operator in front of ψ(x) isUz(δα). The general expression for the operator for an infinitesimal unitary transformation is given by Eq. (7.14). Identifying ɛ with δα and comparing the two expressions for U yields the desired Hermitian operator F , (1) F = i ∂ . (8.5) ∂ϕ If U commutes with H, so will F , according to Eq. (7.15), and we have found the desired conserved observable. We could start exploring the physical consequences 1Some confusion can arise because formally F † = −i∂/∂ϕ looks different from F . However, Hermiticity is not a property of an operator alone but also of the wave functions and the region of integration. For a Hermitian operator, with F † = F , the equation in footnote 3 in Chapter 7 reads � d 3 x(Fψ) ∗ � φ = d 3 xψ ∗ Fφ. F = i∂/∂ϕ satisfies this relation: � d 3 � x i ∂ψ �∗ � φ = d ∂ϕ 3 � x −i ∂ � ψ ∂ϕ ∗ � φ = d 3 xψ ∗ i ∂φ ∂ϕ . In the last step, a partial integration has brought the operator to the right of ψ∗ . The explicit form of a Hermitian operator depends on its position with respect to the wave functions.
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SUBAT MIC PHYSICS Third Edition
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SUBAT MIC PHYSICS Ernest M Henley A
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To Elaine and Viviana
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Acknowledgments In writing the pres
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Preface to the First Edition Subato
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Preface to the Third Edition Subato
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General Bibliography The reader of
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Contents Dedication v Acknowledgmen
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Contents xvii 6 Structure of Subato
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Contents xix 12.5 GeneralReferences
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Chapter 1 Background and Language H
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1.2. Units 3 1.2 Units Table 1.1: B
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1.3. Special Relativity, Feynman Di
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1.3. Special Relativity, Feynman Di
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1.4. References 9 Problems 1.1. ∗
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Part I Tools One of the most frustr
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Chapter 2 Accelerators 2.1 Why Acce
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2.1. Why Accelerators? 15 particle
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2.2. Cross Sections and Luminosity
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2.3. Electrostatic Generators (Van
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2.4. Linear Accelerators (Linacs) 2
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2.5. Beam Optics 23 Figure 2.8: Rec
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2.6. Synchrotrons 25 Figure 2.10: E
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2.6. Synchrotrons 27 Photo 3: The p
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2.7. Laboratory and Center-of-Momen
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2.8. Colliding Beams 31 or with Eqs
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2.10. Beam Storage and Cooling 33 l
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2.11. References 35 and in E. Byckl
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2.11. References 37 (b) Extreme rel
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Chapter 3 Passage of Radiation Thro
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3.2. Heavy Charged Particles 41 Fig
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3.2. Heavy Charged Particles 43 −
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3.3. Photons 45 Equation (3.2) also
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3.4. Electrons 47 Figure 3.8: Passa
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3.5. Nuclear Interactions 49 Figure
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3.6. References 51 3.8. A beam of 1
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Chapter 4 Detectors What would a ph
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4.1. Scintillation Counters 55 The
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4.2. Statistical Aspects 57 Or, to
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4.3. Semiconductor Detectors 59 kno
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4.3. Semiconductor Detectors 61 Fig
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4.4. Bubble Chambers 63 Photo 5: Bu
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4.6. Wire Chambers 65 voltage suppl
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4.8. Time Projection Chambers 67 Fi
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4.10. Calorimeters 69 Figure 4.18:
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4.11. Counter Electronics 71 Figure
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4.13. References 73 Table 4.1: Func
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4.13. References 75 4.6. Sketch the
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Part II Particles and Nuclei The si
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Chapter 5 The Subatomic Zoo A conve
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5.1. Mass and Spin. Fermions and Bo
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5.1. Mass and Spin. Fermions and Bo
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5.2. Electric Charge and Magnetic D
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5.3. Mass Measurements 87 Figure 5.
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5.3. Mass Measurements 89 Figure 5.
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5.3. Mass Measurements 91 Earlier,
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5.4. A First Glance at the Subatomi
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5.5. Gauge Bosons 95 Figure 5.13: A
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5.6. Leptons 97 to its momentum, re
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5.7. Decays 99 Figure 5.15: Exponen
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5.7. Decays 101 The function g(ω)
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5.8. Mesons 103 5.8 Mesons Table 5.
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5.9. Baryon Ground States 105 Japan
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5.9. Baryon Ground States 107 neutr
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5.10. Particles and Antiparticles 1
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5.10. Particles and Antiparticles 1
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5.11. Quarks, Gluons, and Intermedi
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5.12. Excited States and Resonances
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5.12. Excited States and Resonances
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5.13. Excited States of Baryons 123
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5.14. References 129 in Section 5.5
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5.14. References 131 5.14. Use the
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5.14. References 133 5.32. ∗ What
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Chapter 6 Structure of Subatomic Pa
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6.2. Rutherford and Mott Scattering
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6.2. Rutherford and Mott Scattering
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6.3. Form Factors 141 The scatterin
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6.4. The Charge Distribution of Sph
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6.4. The Charge Distribution of Sph
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6.5. Leptons Are Point Particles 14
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6.6. Nucleon Elastic Form Factors 1
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6.8. Inelastic Electron and Muon Sc
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6.8. Inelastic Electron and Muon Sc
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6.9. Deep Inelastic Electron Scatte
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6.10. Quark-Parton Model for Deep I
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6.10. Quark-Parton Model for Deep I
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Page 210 and 211: 6.12. References 189 Some character
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Page 216 and 217: Part III Symmetries and Conservatio
Page 218 and 219: Chapter 7 Additive Conservation Law
Page 220 and 221: 7.1. Conserved Quantities and Symme
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Page 224 and 225: 7.2. The Electric Charge 203 7.2 Th
Page 226 and 227: 7.2. The Electric Charge 205 or [Q,
Page 228 and 229: 7.3. The Baryon Number 207 antineut
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Page 252 and 253: 8.6. Isospin of Particles 231 The a
Page 254 and 255: 8.7. Isospin in Nuclei 233 If the e
Page 256 and 257: 8.8. References 235 Isospin doublet
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Page 260 and 261: Chapter 9 P , C, CP, andT In the pr
Page 262 and 263: 9.1. The Parity Operation 241 P is
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Page 268 and 269: 9.3. Conservation and Breakdown of
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Page 276 and 277: 9.4. Charge Conjugation 255 The π
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9.9. References 271 • There are t
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9.9. References 273 9.8. * Find inf
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9.9. References 275 9.27. Assume th
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9.9. References 277 9.39. Compare t
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Part IV Interactions In the previou
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Chapter 10 The Electromagnetic Inte
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10.1. The Golden Rule 283 Schrödin
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10.1. The Golden Rule 285 where it
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10.2. Phase Space 287 Figure 10.4:
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10.3. The Classical Electromagnetic
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10.3. The Classical Electromagnetic
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10.4. Photon Emission 293 is a twof
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10.4. Photon Emission 295 the form
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10.4. Photon Emission 297 where V (
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10.5. Multipole Radiation 299 The t
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10.5. Multipole Radiation 301 Figur
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10.6. Electromagnetic Scattering of
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10.6. Electromagnetic Scattering of
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10.7. Vector Mesons as Mediators of
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10.7. Vector Mesons as Mediators of
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10.8. Colliding Beams 311 10.8 Coll
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10.8. Colliding Beams 313 Figure 10
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10.9. Electron-Positron Collisions
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10.10. The Photon-Hadron Interactio
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10.11. Magnetic Monopoles 323 The s
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10.12. References 325 Quantum Elect
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10.12. References 327 (b) Compare t
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10.12. References 329 (a) How would
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Chapter 11 The Weak Interaction Thi
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11.1. The Continuous Beta Spectrum
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11.2. Beta Decay Lifetimes 335 The
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11.3. The Current-Current Interacti
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11.4. A Variety of Weak Processes 3
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11.4. A Variety of Weak Processes 3
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11.5. The Muon Decay 347 Linac Pion
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11.6. The Weak Current of Leptons 3
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11.6. The Weak Current of Leptons 3
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11.7. Chirality versus Helicity 353
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11.9. Weak Decays of Quarks and the
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11.10. Weak Currents in Nuclear Phy
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11.10. Weak Currents in Nuclear Phy
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11.11. Inverse Beta Decay: Reines a
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11.12. Massive Neutrinos 363 11.12
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11.13. Majorana versus Dirac Neutri
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11.14. The Weak Current of Hadrons
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11.15. References 375 11.15 Referen
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11.15. References 377 11.5. Beta sp
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11.15. References 379 (b) * Discuss
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11.15. References 381 11.41. For nu
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Chapter 12 Introduction to Gauge Th
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12.1. Introduction 385 D0 ≡ 1 ∂
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12.2. Potentials in Quantum Mechani
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12.3. Gauge Invariance for Non-Abel
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12.3. Gauge Invariance for Non-Abel
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12.4. The Higgs Mechanism; Spontane
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12.5. General References 401 12.5 G
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Chapter 13 The Electroweak Theory o
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13.2. The Gauge Bosons and Weak Iso
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13.2. The Gauge Bosons and Weak Iso
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13.3. The Electroweak Interaction 4
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13.4. Tests of the Standard Model 4
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13.4. Tests of the Standard Model 4
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13.5. References 419 Physics. 112,
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Chapter 14 Strong Interactions All
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14.1. Range and Strength of the Low
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14.2. The Pion-Nucleon Interaction
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14.3. The Form of the Pion-Nucleon
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14.4. The Yukawa Theory of Nuclear
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14.5. Low-Energy Nucleon-Nucleon Fo
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14.6. Meson Theory of the Nucleon-N
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14.7. Strong Processes at High Ener
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14.8. The Standard Model, Quantum C
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14.9. QCD at Low Energies 457 An ex
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14.10. Grand Unified Theories, Supe
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14.11. References 461 and Partons,
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14.11. References 463 Fig. 14.28 14
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14.11. References 465 14.25. The lo
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14.11. References 467 (b) What comb
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Part V Models “A model is like an
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Chapter 15 Quark Models of Mesons a
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15.2. Quarks as Building Blocks of
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15.4. Mesons as Bound Quark States
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15.4. Mesons as Bound Quark States
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15.5. Baryons as Bound Quark States
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15.6. The Hadron Masses 481 Figure
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15.7. QCD and Quark Models of the H
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15.8. Heavy Mesons: Charmonium, Ups
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15.9. Outlook and Problems 493 wher
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15.10. References 495 On a more adv
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15.10. References 497 where J± = J
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15.10. References 499 (b) Apply the
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Chapter 16 Liquid Drop Model, Fermi
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16.1. The Liquid Drop Model 503 Fig
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16.1. The Liquid Drop Model 505 Bey
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16.2. The Fermi Gas Model 507 N = V
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E/A(MeV) 16.3. Heavy Ion Reactions
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16.4. Relativistic Heavy Ion Collis
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16.4. Relativistic Heavy Ion Collis
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16.5. References 515 medium (i.e. s
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16.5. References 517 16.12. Symmetr
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16.5. References 519 16.25. At RHIC
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Chapter 17 The Shell Model The liqu
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17.1. The Magic Numbers 523 The res
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17.2. The Closed Shells 525 solved
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17.2. The Closed Shells 527 Figure
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17.3. The Spin-Orbit Interaction 52
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17.4. The Single-Particle Shell Mod
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17.5. Generalization of the Single-
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17.6. Isobaric Analog Resonances 53
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17.7. Nuclei Far From the Valley of
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17.8. References 539 An elementary
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17.8. References 541 (b) N odd. (c)
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Chapter 18 Collective Model Althoug
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18.1. Nuclear Deformations 545 spin
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18.2. Rotational Spectra of Spinles
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18.2. Rotational Spectra of Spinles
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18.3. Rotational Families 551 A spi
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18.3. Rotational Families 553 2. Th
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18.4. One-Particle Motion in Deform
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18.4. One-Particle Motion in Deform
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18.5. Vibrational States in Spheric
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18.5. Vibrational States in Spheric
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18.6. The Interacting Boson Model 5
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18.7. Highly Excited States; Giant
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18.8. Nuclear Models—Concluding R
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18.8. Nuclear Models—Concluding R
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18.9. References 571 J. S. Vaagen,
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18.9. References 573 18.13. Verify
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18.9. References 575 18.33. Conside
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18.9. References 577 18.51. (a) In
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Chapter 19 Nuclear and Particle Ast
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19.1. The Beginning of the Universe
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19.1. The Beginning of the Universe
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19.2. Primordial Nucleosynthesis 58
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19.3. Stellar Energy and Nucleosynt
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19.4. Stellar Collapse and Neutron
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19.4. Stellar Collapse and Neutron
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19.5. Cosmic Rays 597 We are consta
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19.5. Cosmic Rays 599 Figure 19.11:
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19.6. Neutrino Astronomy and Cosmol
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19.8. References 603 baryon excess
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19.8. References 605 Neutron Stars
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19.8. References 607 (a) What is th
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Index Abundance, 585-586 Accelerato
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Index 611 drift chambers, 66-67 ele
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Index 613 diagrams, 279 electromagn
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Index 615 outlook, 567 Nuclear stru
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Index 617 poles, 494 recurrences, 4
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Index 619 TCP theorem, 269, 448 Tem
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