You are previewing Advanced Quantum Mechanics.
O'Reilly logo
Advanced Quantum Mechanics

Book Description

An accessible introduction to advanced quantum theory, this graduate-level textbook focuses on its practical applications rather than mathematical technicalities. It treats real-life examples, from topics ranging from quantum transport to nanotechnology, to equip students with a toolbox of theoretical techniques. Beginning with second quantization, the authors illustrate its use with different condensed matter physics examples. They then explain how to quantize classical fields, with a focus on the electromagnetic field, taking students from Maxwell's equations to photons, coherent states and absorption and emission of photons. Following this is a unique master-level presentation on dissipative quantum mechanics, before the textbook concludes with a short introduction to relativistic quantum mechanics, covering the Dirac equation and a relativistic second quantization formalism. The textbook includes 70 end-of-chapter problems. Solutions to some problems are given at the end of the chapter and full solutions to all problems are available for instructors at www.cambridge.org/9780521761505.

Table of Contents

  1. Coverpage
  2. Advanced Quantum Mechanics
  3. Title page
  4. Copyright page
  5. Contents
  6. Figure Credits
  7. Preface
  8. Part I Second Quantization
    1. 1 Elementary quantum mechanics
      1. 1.1 Classical mechanics
      2. 1.2 Schrödinger equation
      3. 1.3 Dirac formulation
      4. 1.4 Schrödinger and Heisenberg pictures
      5. 1.5 Perturbation theory
      6. 1.6 Time-dependent perturbation theory
        1. 1.6.1 Fermi’s golden rule
      7. 1.7 Spin and angular momentum
        1. 1.7.1 Spin in a magnetic field
        2. 1.7.2 Two spins
      8. 1.8 Two-level system: The qubit
      9. 1.9 Harmonic oscillator
      10. 1.10 The density matrix
      11. 1.11 Entanglement
      12. Exercises
      13. Solutions
    2. 2 Identical particles
      1. 2.1 Schrödinger equation for identical particles
      2. 2.2 The symmetry postulate
        1. 2.2.1 Quantum fields
      3. 2.3 Solutions of the N-particle Schrödinger equation
        1. 2.3.1 Symmetric wave function: Bosons
        2. 2.3.2 Antisymmetric wave function: Fermions
        3. 2.3.3 Fock space
      4. Exercises
      5. Solutions
    3. 3 Second quantization
      1. 3.1 Second quantization for bosons
        1. 3.1.1 Commutation relations
        2. 3.1.2 The structure of Fock space
      2. 3.2 Field operators for bosons
        1. 3.2.1 Operators in terms of field operators
        2. 3.2.2 Hamiltonian in terms of field operators
        3. 3.2.3 Field operators in the Heisenberg picture
      3. 3.3 Why second quantization?
      4. 3.4 Second quantization for fermions
        1. 3.4.1 Creation and annihilation operators for fermions
        2. 3.4.2 Field operators
      5. 3.5 Summary of second quantization
      6. Exercises
      7. Solutions
  9. Part II Examples
    1. 4 Magnetism
      1. 4.1 Non-interacting Fermi gas
      2. 4.2 Magnetic ground state
        1. 4.2.1 Trial wave function
      3. 4.3 Energy
        1. 4.3.1 Kinetic energy
        2. 4.3.2 Potential energy
        3. 4.3.3 Energy balance and phases
      4. 4.4 Broken symmetry
      5. 4.5 Excitations in ferromagnetic metals
        1. 4.5.1 Single-particle excitations
        2. 4.5.2 Electron-hole pairs
        3. 4.5.3 Magnons
        4. 4.5.4 Magnon spectrum
      6. Exercises
      7. Solutions
    2. 5 Superconductivity
      1. 5.1 Attractive interaction and Cooper pairs
        1. 5.1.1 Trial wave function
        2. 5.1.2 Nambu boxes
      2. 5.2 Energy
        1. 5.2.1 Energy minimization
      3. 5.3 Particles and quasiparticles
      4. 5.4 Broken symmetry
      5. Exercises
      6. Solutions
    3. 6 Superfluidity
      1. 6.1 Non-interacting Bose gas
      2. 6.2 Field theory for interacting Bose gas
        1. 6.2.1 Hamiltonian and Heisenberg equation
      3. 6.3 The condensate
        1. 6.3.1 Broken symmetry
      4. 6.4 Excitations as oscillations
        1. 6.4.1 Particles and quasiparticles
      5. 6.5 Topological excitations
        1. 6.5.1 Vortices
        2. 6.5.2 Vortices as quantum states
        3. 6.5.3 Vortex lines
      6. Exercises
      7. Solutions
  10. Part III Fields and Radiation
    1. 7 Classical fields
      1. 7.1 Chain of coupled oscillators
      2. 7.2 Continuous elastic string
        1. 7.2.1 Hamiltonian and equation of motion
        2. 7.2.2 Solution of the equation of motion
        3. 7.2.3 The elastic string as a set of oscillators
      3. 7.3 Classical electromagnetic field
        1. 7.3.1 Maxwell equations
        2. 7.3.2 Useful relations
        3. 7.3.3 Vector and scalar potentials
        4. 7.3.4 Gauges
        5. 7.3.5 Electromagnetic field as a set of oscillators
        6. 7.3.6 The LC-oscillator
      4. Exercises
      5. Solutions
    2. 8 Quantization of fields
      1. 8.1 Quantization of the mechanical oscillator
        1. 8.1.1 Oscillator and oscillators
      2. 8.2 The elastic string: phonons
      3. 8.3 Fluctuations of magnetization: magnons
      4. 8.4 Quantization of the electromagnetic field
        1. 8.4.1 Photons
        2. 8.4.2 Field operators
        3. 8.4.3 Zero-point energy, uncertainty relations,and vacuum fluctuations
        4. 8.4.4 The simple oscillator
      5. Exercises
      6. Solutions
    3. 9 Radiation andmatter
      1. 9.1 Transition rates
      2. 9.2 Emission and absorption: General considerations
        1. 9.2.1 Master equations
        2. 9.2.2 Equilibrium and black-body radiation
      3. 9.3 Interaction of matter and radiation
      4. 9.4 Spontaneous emission by atoms
        1. 9.4.1 Dipole approximation
        2. 9.4.2 Transition rates
        3. 9.4.3 Selection rules
      5. 9.5 Blue glow: Cherenkov radiation
        1. 9.5.1 Emission rate and spectrum of Cherenkov radiation
      6. 9.6 Bremsstrahlung
      7. 9.7 Processes in lasers
        1. 9.7.1 Master equation for lasers
        2. 9.7.2 Photon number distribution
      8. Exercises
      9. Solutions
    4. 10 Coherent states
      1. 10.1 Superpositions
      2. 10.2 Excitation of an oscillator
      3. 10.3 Properties of the coherent state
      4. 10.4 Back to the laser
        1. 10.4.1 Optical coherence time
        2. 10.4.2 Maxwell-Bloch equations
      5. 10.5 Coherent states of matter
        1. 10.5.1 Cooper pair box
      6. Exercises
      7. Solutions
  11. Part IV Dissipative Quantum Mechanics
    1. 11 Dissipative quantummechanics
      1. 11.1 Classical damped oscillator
        1. 11.1.1 Dynamical susceptibility
        2. 11.1.2 Damped electric oscillator
      2. 11.2 Quantum description
        1. 11.2.1 Difficulties with the quantum description
        2. 11.2.2 Solution: Many degrees of freedom
        3. 11.2.3 Boson bath
        4. 11.2.4 Quantum equations of motion
        5. 11.2.5 Diagonalization
      3. 11.3 Time-dependent fluctuations
        1. 11.3.1 Fluctuation-dissipation theorem
        2. 11.3.2 Kubo formula
      4. 11.4 Heisenberg uncertainty relation
        1. 11.4.1 Density matrix of a damped oscillator
      5. Exercises
      6. Solutions
    2. 12 Transitions and dissipation
      1. 12.1 Complicating the damped oscillator: Towards a qubit
        1. 12.1.1 Delocalization criterion
      2. 12.2 Spin-boson model
      3. 12.3 Shifted oscillators
      4. 12.4 Shake-up and P(E)
      5. 12.5 Orthogonality catastrophe
      6. 12.6 Workout of P(E)
      7. 12.7 Transition rates and delocalization
      8. 12.8 Classification of environments
        1. 12.8.1 Subohmic
        2. 12.8.2 Ohmic
        3. 12.8.3 Superohmic
      9. 12.9 Vacuum as an environment
      10. Exercises
      11. Solutions
  12. Part V Relativistic Quantum Mechanics
    1. 13 Relativistic quantummechanics
      1. 13.1 Principles of the theory of relativity
        1. 13.1.1 Lorentz transformation
        2. 13.1.2 Minkowski spacetime
        3. 13.1.3 The Minkowski metric
        4. 13.1.4 Four-vectors
      2. 13.2 Dirac equation
        1. 13.2.1 Solutions of the Dirac equation
        2. 13.2.2 Second quantization
        3. 13.2.3 Interaction with the electromagnetic field
      3. 13.3 Quantum electrodynamics
        1. 13.3.1 Hamiltonian
        2. 13.3.2 Perturbation theory and divergences
      4. 13.4 Renormalization
      5. Exercises
      6. Solutions
  13. Index