| About the Authors |
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v | |
| Preface |
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vii | |
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Chapter 1 Wave-Particle Dualism, Complementarity Principle, Bell's Theorem and Related Experiments |
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1 | (124) |
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1.1 The Buildup of the Electron Interference Pattern |
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5 | (6) |
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1.2 Experimental Verification of Complementarity Principle by Atomic Interferometry |
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11 | (13) |
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1.3 Verification of Complementarity by Quantum Optical Experiments |
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24 | (5) |
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1.4 Single Photon Interference Experiment |
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29 | (8) |
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1.4.1 Interference between independent photon beams |
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34 | (3) |
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1.5 Multiparticle Interferometry |
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37 | (14) |
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1.5.1 Two-particle double-slit interferometry |
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37 | (3) |
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1.5.2 Interference experiment of a pair of down-converted photons |
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40 | (4) |
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1.5.3 Interference between different emission times |
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44 | (3) |
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1.5.4 Coherence and the distinguishability of paths |
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47 | (4) |
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1.6 Double Photon Interferometer as a Quantum Eraser |
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51 | (12) |
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1.7 Dispute between Einstein and Bohr on Quantum Mechanics, Bell's Theorem |
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63 | (12) |
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1.7.1 Criticism by Einstein in 1930: "quantum mechanics is not self-consistent" |
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63 | (2) |
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1.7.2 The Einstein-Podolsky-Rosen paradox: "quantum mechanical description is incomplete" |
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65 | (4) |
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69 | (3) |
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1.7.4 Bell's inequality generalized to real systems |
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72 | (3) |
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1.8 Experimental Verifications of Bell's Inequality |
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75 | (9) |
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1.9 Wheeler's Delayed Choice Experiment |
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84 | (5) |
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1.10 Bell's Theorem without Inequalities |
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89 | (27) |
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1.10.1 Complete correlation between three particles |
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89 | (5) |
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1.10.2 Bell's theorem without inequalities: Two-particle correlation |
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94 | (5) |
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1.10.3 Experimental verification of Bell's theorem without inequalities for two-particle systems |
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99 | (3) |
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1.10.4 Experimental realization of 3-photon entangled states |
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102 | (6) |
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1.10.5 EPR paradox in the context of entanglement and non-locality |
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108 | (5) |
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1.10.6 Non-locality of a single photon |
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113 | (3) |
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1.11 A Brief Introduction to Quantum Non-Demolition Experiment |
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116 | (9) |
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1.11.1 The standard quantum limit and the Back Action Evading (BAE) experiments |
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117 | (3) |
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1.11.2 The quantum non-demolition experiment |
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120 | (2) |
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122 | (3) |
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Chapter 2 Quantum Entanglement and Its Applications to Quantum Information and Quantum Computing |
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125 | (34) |
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2.1 A Brief Introduction to Quantum Computing |
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126 | (9) |
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2.1.1 Quantum data and data processing |
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126 | (2) |
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2.1.2 Quantum parallelism and efficient quantum algorithms |
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128 | (2) |
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2.1.3 Quantum information |
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130 | (5) |
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135 | (10) |
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2.2.1 Density matrix characterization of an entangled state |
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135 | (3) |
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2.2.2 Schmidt decomposition |
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138 | (3) |
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2.2.3 More about EPR-Bell states |
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141 | (4) |
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2.3 Entanglement in Applications to Quantum Information |
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145 | (7) |
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145 | (1) |
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2.3.2 Quantum cryptography: EPR quantum key distribution |
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146 | (2) |
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2.3.3 The quantum no-cloning theorem |
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148 | (2) |
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2.3.4 Quantum teleportation |
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150 | (2) |
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2.4 Six-photon Singlet --- An Example of Decoherence Free States |
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152 | (3) |
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155 | (4) |
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157 | (2) |
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Chapter 3 Geometrical Phases in Quantum Mechanics |
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159 | (50) |
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160 | (7) |
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3.2 Experimental Verifications of Aharonov-Bohm Effect |
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167 | (7) |
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3.3 Aharonov-Casher Effect |
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174 | (3) |
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3.4 Parallel Transport, Connexion, Curvature and Anholonomy |
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177 | (8) |
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185 | (10) |
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3.6 Aharonov-Anandan Phase |
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195 | (2) |
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3.7 Experimental Revelations of Berry's Phase |
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197 | (12) |
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3.7.1 Manifestation of Berry's phase in the rotation of polarization of a photon |
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199 | (2) |
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3.7.2 Manifestation of Berry's phase in neutron spin rotation |
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201 | (3) |
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3.7.3 Adiabatic rotational splitting and Berry's phase in nuclear quadrupole resonance |
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204 | (3) |
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207 | (2) |
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Chapter 4 Boundary between Quantum and Classical Mechanics, Entanglement and Decoherence |
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209 | (106) |
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4.1 Schrodinger's Harmonic Oscillator Wave Packets, Coherent States |
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212 | (11) |
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4.1.1 Basic properties of coherent states |
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215 | (2) |
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4.1.2 Canonical coherent states |
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217 | (6) |
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4.2 Circular Orbit Wave Packet and the Radial Wave Packet of the Hydrogen Atom |
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223 | (9) |
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4.3 50(4) Symmetry of H atom, the Runge-Lenz Vector and the Kepler Elliptic Orbit Wave Packet |
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232 | (26) |
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4.3.1 The Runge-Lenz Vector of the Classical Kepler motion |
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233 | (3) |
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4.3.2 The quantum mechanical Runge-Lenz vector, the dynamical symmetry and the energy spectrum of the hydrogen atom |
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236 | (4) |
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4.3.3 Construction of the Kepler Elliptic Orbit |
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240 | (7) |
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4.3.4 Rutherford atom in quantum mechanics |
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247 | (11) |
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4.4 Revivals and Fractional Revivals of the Wave Packets |
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258 | (4) |
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4.5 Principle of Superposition of States and the Quantum Decoherence |
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262 | (7) |
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4.6 Decoherence due to Interaction with Environment |
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269 | (6) |
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4.7 A Dynamical Model of Decoherence |
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275 | (4) |
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4.8 Classical Limit of Quantum Dynamics |
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279 | (4) |
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4.9 Schrodinger Cat Realized in the Laboratory |
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283 | (21) |
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4.9.1 A Schrodinger cat at single atom level |
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283 | (8) |
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4.9.2 A phase Schrodinger cat |
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291 | (6) |
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297 | (5) |
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4.9.4 Decoherence by the emission of thermal radiations |
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302 | (2) |
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4.10 The Collapse of Wave Function and the Quantum Zeno Effect |
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304 | (3) |
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4.11 The Squeeze Operator and the Squeezed Coherent State |
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307 | (8) |
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313 | (2) |
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Chapter 5 Path Integral Formulation of Quantum Mechanics |
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315 | (54) |
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5.1 The Path Integral in Quantum Mechanics |
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316 | (13) |
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5.2 The Instanton and the Coherence Splitting of Energy Levels in a Double Well |
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329 | (8) |
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5.3 The Density Matrix and the Path Integral |
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337 | (10) |
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5.4 Instanton Method for a Decaying State |
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347 | (22) |
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5.4.1 Penetration of the "quadratic plus cubic" potential |
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354 | (8) |
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5.4.2 Shifting method for evaluating the second order variation in path integrals |
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362 | (2) |
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5.4.3 Alternative Evaluation of the Prefactor for the Harmonic Oscillator |
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364 | (3) |
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367 | (2) |
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Chapter 6 Quantum Mechanics on the Macroscopic Level |
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369 | (104) |
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6.1 Wave Function with a Macroscopic Significance |
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370 | (8) |
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6.2 Coupled Superconductors, Josephson Effect |
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378 | (8) |
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6.3 Superconducting Loop with a Josephson Junction, the SQUID |
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386 | (7) |
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6.4 Macroscopic Quantum Tunneling and Macroscopic Quantum Coherence in Josephson Systems |
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393 | (7) |
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6.5 The Influence of Environment on the Macroscopic Quantum Phenomena |
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400 | (28) |
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6.5.1 On the Canonical Transformations and the Adiabatic Approximation |
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402 | (3) |
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6.5.2 The Hamiltonian of a Dissipative Electromagnetic System |
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405 | (3) |
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6.5.3 The Non-adiabatic Correction, The Lagrangian of a Dissipative System |
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408 | (6) |
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6.5.4 Relation between Macroscopic Dissipation Parameter and the Microscopic Parameters |
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414 | (1) |
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6.5.5 Dissipation and the Macroscopic Quantum Tunneling |
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415 | (8) |
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6.5.6 Dissipation and the Macroscopic Quantum Coherence |
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423 | (5) |
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6.6 Experiments on the Macroscopic Quantum Tunneling in Josephson Systems |
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428 | (4) |
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6.7 Macroscopic Quantum Tunneling of Magnetism, the Spin Coherent State |
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432 | (6) |
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6.8 Macroscopic Quantum Phenomena of Single Domain Ferromagnetic Particles |
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438 | (11) |
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6.8.1 Quantum Coherence: Tunneling Splitting of Energy States |
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439 | (2) |
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6.8.2 The Quantum Tunneling |
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441 | (3) |
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6.8.3 Quantum Interference Phenomena, the Topological Quenching |
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444 | (5) |
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6.9 Macroscopic Quantum Phenomena of Single Domain Antiferromagnetic Particles |
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449 | (5) |
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6.10 Experiments on the Macroscopic Quantum Tunneling of Magnetic Systems |
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454 | (5) |
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6.11 Macroscopic Quantum Phenomena of Macromolecules |
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459 | (6) |
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6.12 Quantum Mechanical Foundation of the Spin-parity Effect |
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465 | (8) |
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470 | (3) |
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Chapter 7 Topological Phase Factors in Quantum Systems |
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473 | (52) |
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7.1 Spin Wave Theory in the Heisenberg Model |
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475 | (13) |
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7.2 O(3) Non-Linear Sigma Model, the Spontaneous Symmetry Breakdown and the Goldstone Theorem |
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488 | (4) |
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7.3 One-dimensional Quantum Antiferromagnetic Chain, Topological Phase Factor, Large-s Mapping Onto O(3) Non-linear Sigma Model |
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492 | (10) |
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7.4 Lieb-Schultz-Mattis Theorem |
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502 | (4) |
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7.5 The Significance of the Topological Term |
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506 | (4) |
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7.6 The Vacuum of the non-Abelian Gauge Fields |
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510 | (8) |
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7.6.1 The non-Abelian gauge fields |
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510 | (2) |
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7.6.2 The equivalent classes of gauge transformations |
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512 | (3) |
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515 | (3) |
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7.7 The Topological Term and the Anomalies |
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518 | (7) |
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522 | (3) |
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Chapter 8 Cavity Quantum Electrodynamics, van der Waals Forces and Casimir Effect |
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525 | (142) |
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8.1 Interaction between the Radiation Field and an Atom |
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527 | (17) |
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8.1.1 Quantization of the field of electron in an atom |
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527 | (1) |
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8.1.2 Quantization of the radiation field |
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528 | (3) |
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8.1.3 Interacting electron field and radiation field, the rate of spontaneous emission |
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531 | (4) |
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8.1.4 Dark states, electromagnetically induced transparency |
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535 | (9) |
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8.2 Jaynes-Cummings Model |
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544 | (9) |
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8.2.1 The eigenstates of the coupled atom-cavity system |
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545 | (3) |
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8.2.2 The light shift of the atomic energy levels in the non-resonant case |
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548 | (4) |
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8.2.3 The evolution of state in time |
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552 | (1) |
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8.3 The Suppression and Enhancement of Spontaneous Emission |
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553 | (4) |
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557 | (4) |
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8.5 The Inverse Stern-Gerlach Effect |
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561 | (3) |
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8.6 The Atom-Cavity Dispersive Phase Shift Effects |
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564 | (11) |
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8.6.1 Quantum non-demolition detection (QND) of a single photon, QND of the birth and death of single photons |
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570 | (5) |
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8.7 Entangled States in Cavity QED |
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575 | (4) |
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8.8 Response of a System to External Disturbances, the Fluctuation-Dissipation Theorem |
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579 | (15) |
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8.8.1 The frequency dependence of the permittivity, the Kramers-Kronig dispersion relation |
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580 | (4) |
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8.8.2 The correlation between fluctuations and the generalized polarizability |
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584 | (4) |
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8.8.3 The fluctuation-dissipation theorem |
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588 | (6) |
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8.9 The van der Waals Interaction |
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594 | (4) |
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8.10 The Effect of Retardation on the van der Waals Interaction |
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598 | (8) |
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8.10.1 The field of an oscillating dipole |
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598 | (3) |
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8.10.2 The fluctuation of electromagnetic field in a homogeneous medium |
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601 | (4) |
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8.10.3 The Casimir-Polder interaction |
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605 | (1) |
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8.11 Zero-point Energy, Vacuum Fluctuation of the Electromagnetic Field and the van der Waals Interaction |
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606 | (9) |
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615 | (4) |
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8.13 Cavity QED in Strong Coupling Regime |
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619 | (6) |
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8.14 Radiative Corrections |
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625 | (6) |
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8.14.1 Spontaneous radiative correction: Free electron |
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626 | (2) |
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8.14.2 Spontaneous radiative correction: The bound electron |
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628 | (2) |
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8.14.3 Stimulated radiative corrections |
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630 | (1) |
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8.15 Interaction with Monochromatic Radiation, Bloch Sphere, and Light Shift |
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631 | (6) |
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8.15.1 Population change and Bloch sphere |
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631 | (3) |
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634 | (3) |
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8.16 Renormalization in Quantum Field Theory |
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637 | (6) |
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8.17 Renormalization Group Approach |
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643 | (24) |
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8.17.1 Renormalization group flow |
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643 | (3) |
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8.17.2 Physics on different energy scales |
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646 | (3) |
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8.17.3 Anderson localization |
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649 | (1) |
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8.17.4 Path to phase transition and critical phenomena |
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650 | (8) |
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8.17.5 0(n) Non-linear sigma model in d dimensions, weak coupling approximation |
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658 | (6) |
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664 | (3) |
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Chapter 9 The Quantum Hall Effect |
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667 | (134) |
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9.1 The Classical Hall Effect |
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668 | (1) |
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9.2 Electrons in Uniform Magnetic Field, the Landau Level |
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669 | (7) |
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9.3 Quantization of Magnetic Flux |
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676 | (2) |
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9.4 The Integral Quantum Hall Effect |
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678 | (16) |
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9.4.1 Magnetic translation, topological significance of the Hall conductance |
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689 | (5) |
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9.5 Fractional Quantum Hall Effect, Laughlin Wave Function |
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694 | (37) |
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9.5.1 Quantized motion of a small number of electrons |
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696 | (6) |
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9.5.2 Laughlin wave function for the v = 1/m states |
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702 | (6) |
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9.5.3 Quasiparticle excitations |
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708 | (3) |
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9.5.4 Collective modes in the incompressible quantum Hall liquid |
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711 | (10) |
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9.5.5 Gapless edge states of fractional quantum Hall liquids |
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721 | (7) |
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9.5.6 Hierarchy states of the fractional quantum Hall effect |
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728 | (1) |
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9.5.7 The composite Fermion, Jain's construction |
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729 | (2) |
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9.6 The Landau-Ginzburg Theory of the Fractional Quantum Hall Effect |
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731 | (15) |
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9.6.1 Zhang-Hansson-Kivelson mapping, the Chern-Simons-Landau-Ginzburg action |
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732 | (5) |
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9.6.2 The mean field solution, phenomenology of the fractional quantum Hall effect |
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737 | (3) |
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9.6.3 The algebraic off-diagonal long-range order |
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740 | (3) |
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9.6.4 Topological order of the fractional quantum Hall fluid |
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743 | (3) |
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9.7 The Global Phase Diagram of the Quantum Hall Effect |
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746 | (7) |
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9.8 Quantum Spin Hall Effect |
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753 | (8) |
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9.9 Topology in the Physics of 2 + 1 Dimensions, Anomalous Quantum Hall Effect |
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761 | (23) |
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9.9.1 The 3 + 1 dimensional chiral anomaly |
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762 | (4) |
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9.9.2 2 + 1 dimensional Dirac Field interacting with Gauge Field |
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766 | (2) |
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9.9.3 Topologically non-trivial vacuum current with abnormal parity |
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768 | (2) |
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9.9.4 Relation between fractional charge, vacuum current and zero mode |
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770 | (3) |
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9.9.5 A condensed matter simulation of the 2 + 1 D anomaly |
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773 | (7) |
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9.9.6 Anomalous quantum Hall effect |
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780 | (4) |
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9.10 Topology in Condensed Matter Physics, the Topological Insulators and Superconductors |
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784 | (17) |
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9.10.1 The quantum spin Hall insulator, topological classifications |
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787 | (4) |
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9.10.2 Topological Superconductors, Majorana fermions |
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791 | (6) |
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797 | (4) |
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Chapter 10 Bose-Einstein Condensation |
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801 | (130) |
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10.1 Some Fundamental Relations Pertaining to the Bose-Einstein Condensation |
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803 | (14) |
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10.1.1 The BEC as a quantum statistical phenomenon |
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804 | (1) |
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10.1.2 The Bose-Einstein temperature |
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805 | (2) |
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10.1.3 Thermodynamic properties of Bose gas |
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807 | (10) |
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10.2 Order Parameter of Bose-Einstein Condensation, the Off-Diagonal Long Range Order |
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817 | (3) |
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10.3 The Nature of Bose-Einstein Condensation: The Spontaneous Symmetry-Breaking and the Phase Coherence |
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820 | (14) |
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10.4 Weakly Interacting Bose Gas: Homogeneous Condensate |
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834 | (12) |
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10.4.1 Bogoliubov Theory of Weakly Interacting Bose Gas |
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835 | (8) |
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10.4.2 Non-ideal Bose gas |
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843 | (3) |
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10.5 Weakly Interacting Bose Gas: Inhomogeneous Condensate |
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846 | (18) |
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10.5.1 Dependence of the properties of the condensation on the external fields |
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846 | (4) |
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10.5.2 Bose-Einstein condensation of weakly interacting Bose gas in a trap |
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850 | (6) |
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10.5.3 Gross-Pitaevskii equation |
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856 | (3) |
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10.5.4 The quantum phase dynamics |
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859 | (5) |
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10.6 Bose-Einstein Condensation in Anisotropic Potential Traps |
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864 | (5) |
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10.7 Vortices and the Stability of Bose-Einstein Condensates against Attractive Interactions |
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869 | (7) |
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10.8 The Spinor Condensate |
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876 | (15) |
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10.8.1 Ground state structure |
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878 | (3) |
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10.8.2 Collective modes of trapped spinor condensate |
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881 | (2) |
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10.8.3 Intrinsic stability of vortices in the ferromagnetic state |
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883 | (2) |
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885 | (1) |
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10.8.5 Fragmented condensates |
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885 | (6) |
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10.9 Cold Bosonic Atoms in Optical Lattices |
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891 | (14) |
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10.10 Feshbach Resonance and Resonance Superfluidity |
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905 | (26) |
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10.10.1 The Feshbach resonance |
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906 | (5) |
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10.10.2 The degenerate Fermi gas |
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911 | (3) |
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10.10.3 Molecules formed by Fermi atoms and the molecular condensates |
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914 | (2) |
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10.10.4 Condensates of the pairs of Fermi atoms |
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916 | (10) |
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926 | (5) |
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Chapter 11 The Yangian Commutation Relations in Quantum Mechanics |
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931 | (40) |
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11.1 Tensor Operators for the Hydrogen Atom and the Yangian |
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932 | (5) |
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11.2 The Yangian Algebras |
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937 | (5) |
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11.3 Other Realizations of Y(SL(2)) in Quantum Mechanics |
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942 | (8) |
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11.4 The One-dimensional Chain Model of Long-range Interaction |
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950 | (7) |
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957 | (8) |
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11.6 Yangian Transition between the Singlet and Octet in SU{3) |
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965 | (6) |
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969 | (2) |
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Chapter 12 RTT Relation and the Yang-Baxter Equation, Physical Significance of the Quantum Algebras |
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971 | |
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12.1 Commutation Relations in Direct Product Form of Matrices |
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972 | (10) |
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982 | (4) |
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12.3 Yang-Baxter Equation |
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986 | (7) |
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12.4 Sets of Conserved Quantities, the Hamiltonian |
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993 | (8) |
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12.5 The Quantum Determinant, and the Co-product |
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1001 | (10) |
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12.6 Expansion of the RTT Relation and the Commutation Relations |
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1011 | (3) |
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12.7 The Hydrogen Atom and the RTT Relation |
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1014 | (5) |
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12.8 Representations of Yangian and the Hydrogen Atomic Spectrum |
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1019 | (9) |
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12.9 Yangian and the Bell Basis |
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1028 | (2) |
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12.10 Transition from S- To P-wave Superconductivity |
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|
1030 | (5) |
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|
|
1035 | (5) |
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12.12 Bi-frequency Harmonic Oscillators and the Commutation Relations of Quantum Algebras |
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|
1040 | (6) |
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12.13 The Translation Operators for the Coherent States and the Hofstadter model |
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|
1046 | (6) |
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12.14 The Cyclic Representation of the Quantum Algebras |
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1052 | |
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|
1061 | |
