You are previewing Quantum Mechanics with Basic Field Theory.
O'Reilly logo
Quantum Mechanics with Basic Field Theory

Book Description

Students and instructors alike will find this organized and detailed approach to quantum mechanics ideal for a two-semester graduate course on the subject. This textbook covers, step-by-step, important topics in quantum mechanics, from traditional subjects like bound states, perturbation theory and scattering, to more current topics such as coherent states, quantum Hall effect, spontaneous symmetry breaking, superconductivity, and basic quantum electrodynamics with radiative corrections. The large number of diverse topics are covered in concise, highly focused chapters, and are explained in simple but mathematically rigorous ways. Derivations of results and formulae are carried out from beginning to end, without leaving students to complete them. With over 200 exercises to aid understanding of the subject, this textbook provides a thorough grounding for students planning to enter research in physics. Several exercises are solved in the text, and password-protected solutions for remaining exercises are available to instructors at www.cambridge.org/9780521877602.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Physical constants
  7. 1 Basic formalism
    1. 1.1 State vectors
    2. 1.2 Operators and physical observables
    3. 1.3 Eigenstates
    4. 1.4 Hermitian conjugation and Hermitian operators
    5. 1.5 Hermitian operators: their eigenstates and eigenvalues
    6. 1.6 Superposition principle
    7. 1.7 Completeness relation
    8. 1.8 Unitary operators
    9. 1.9 Unitary operators as transformation operators
    10. 1.10 Matrix formalism
    11. 1.11 Eigenstates and diagonalization of matrices
    12. 1.12 Density operator
    13. 1.13 Measurement
    14. 1.14 Problems
  8. 2 Fundamental commutator and time evolution of state vectors and operators
    1. 2.1 Continuous variables: X and P operators
    2. 2.2 Canonical commutator [X, P]
    3. 2.3 P as a derivative operator: another way
    4. 2.4 X and P as Hermitian operators
    5. 2.5 Uncertainty principle
    6. 2.6 Some interesting applications of uncertainty relations
    7. 2.7 Space displacement operator
    8. 2.8 Time evolution operator
    9. 2.9 Appendix to Chapter 2
    10. 2.10 Problems
  9. 3 Dynamical equations
    1. 3.1 Schrödinger picture
    2. 3.2 Heisenberg picture
    3. 3.3 Interaction picture
    4. 3.4 Superposition of time-dependent states and energy–time uncertainty relation
    5. 3.5 Time dependence of the density operator
    6. 3.6 Probability conservation
    7. 3.7 Ehrenfest’s theorem
    8. 3.8 Problems
  10. 4 Free particles
    1. 4.1 Free particle in one dimension
    2. 4.2 Normalization
    3. 4.3 Momentum eigenfunctions and Fourier transforms
    4. 4.4 Minimum uncertainty wave packet
    5. 4.5 Group velocity of a superposition of plane waves
    6. 4.6 Three dimensions – Cartesian coordinates
    7. 4.7 Three dimensions – spherical coordinates
    8. 4.8 The radial wave equation
    9. 4.9 Properties of Ylm(θ, ϕ)
    10. 4.10 Angular momentum
    11. 4.11 Determining L2 from the angular variables
    12. 4.12 Commutator [Li, Lj] and [L2, Lj]
    13. 4.13 Ladder operators
    14. 4.14 Problems
  11. 5 Particles with spin ½
    1. 5.1 Spin ½ system
    2. 5.2 Pauli matrices
    3. 5.3 The spin ½ eigenstates
    4. 5.4 Matrix representation of σx and σy
    5. 5.5 Eigenstates of σx and σy
    6. 5.6 Eigenstates of spin in an arbitrary direction
    7. 5.7 Some important relations for σi
    8. 5.8 Arbitrary 2 × 2 matrices in terms of Pauli matrices
    9. 5.9 Projection operator for spin ½ systems
    10. 5.10 Density matrix for spin ½ states and the ensemble average
    11. 5.11 Complete wavefunction
    12. 5.12 Pauli exclusion principle and Fermi energy
    13. 5.13 Problems
  12. 6 Gauge invariance, angular momentum, and spin
    1. 6.1 Gauge invariance
    2. 6.2 Quantum mechanics
    3. 6.3 Canonical and kinematic momenta
    4. 6.4 Probability conservation
    5. 6.5 Interaction with the orbital angular momentum
    6. 6.6 Interaction with spin: intrinsic magnetic moment
    7. 6.7 Spin–orbit interaction
    8. 6.8 Aharonov–Bohm effect
    9. 6.9 Problems
  13. 7 Stern–Gerlach experiments
    1. 7.1 Experimental set-up and electron’s magnetic moment
    2. 7.2 Discussion of the results
    3. 7.3 Problems
  14. 8 Some exactly solvable bound-state problems
    1. 8.1 Simple one-dimensional systems
    2. 8.2 Delta-function potential
    3. 8.3 Properties of a symmetric potential
    4. 8.4 The ammonia molecule
    5. 8.5 Periodic potentials
    6. 8.6 Problems in three dimensions
    7. 8.7 Simple systems
    8. 8.8 Hydrogen-like atom
    9. 8.9 Problems
  15. 9 Harmonic oscillator
    1. 9.1 Harmonic oscillator in one dimension
    2. 9.2 Problems
  16. 10 Coherent states
    1. 10.1 Eigenstates of the lowering operator
    2. 10.2 Coherent states and semiclassical description
    3. 10.3 Interaction of a harmonic oscillator with an electric field
    4. 10.4 Appendix to Chapter 10
    5. 10.5 Problems
  17. 11 Two-dimensional isotropic harmonic oscillator
    1. 11.1 The two-dimensional Hamiltonian
    2. 11.2 Problems
  18. 12 Landau levels and quantum Hall effect
    1. 12.1 Landau levels in symmetric gauge
    2. 12.2 Wavefunctions for the LLL
    3. 12.3 Landau levels in Landau gauge
    4. 12.4 Quantum Hall effect
    5. 12.5 Wavefunction for filled LLLs in a Fermi system
    6. 12.6 Problems
  19. 13 Two-level problems
    1. 13.1 Time-independent problems
    2. 13.2 Time-dependent problems
    3. 13.3 Problems
  20. 14 Spin ½ systems in the presence of magnetic fields
    1. 14.1 Constant magnetic field
    2. 14.2 Spin precession
    3. 14.3 Time-dependent magnetic field: spin magnetic resonance
    4. 14.4 Problems
  21. 15 Oscillation and regeneration in neutrinos and neutral K-mesons as two-level systems
    1. 15.1 Neutrinos
    2. 15.2 The solar neutrino puzzle
    3. 15.3 Neutrino oscillations
    4. 15.4 Decay and regeneration
    5. 15.5 Oscillation and regeneration of stable and unstable systems
    6. 15.6 Neutral K-mesons
    7. 15.7 Problems
  22. 16 Time-independent perturbation for bound states
    1. 16.1 Basic formalism
    2. 16.2 Harmonic oscillator: perturbative vs. exact results
    3. 16.3 Second-order Stark effect
    4. 16.4 Degenerate states
    5. 16.5 Linear Stark effect
    6. 16.6 Problems
  23. 17 Time-dependent perturbation
    1. 17.1 Basic formalism
    2. 17.2 Harmonic perturbation and Fermi’s golden rule
    3. 17.3 Transitions into a group of states and scattering cross-section
    4. 17.4 Resonance and decay
    5. 17.5 Appendix to Chapter 17
    6. 17.6 Problems
  24. 18 Interaction of charged particles and radiation in perturbation theory
    1. 18.1 Electron in an electromagnetic field: the absorption cross-section
    2. 18.2 Photoelectric effect
    3. 18.3 Coulomb excitations of an atom
    4. 18.4 Ionization
    5. 18.5 Thomson, Rayleigh, and Raman scattering in second-order perturbation
    6. 18.6 Problems
  25. 19 Scattering in one dimension
    1. 19.1 Reflection and transmission coefficients
    2. 19.2 Infinite barrier
    3. 19.3 Finite barrier with infinite range
    4. 19.4 Rigid wall preceded by a potential well
    5. 19.5 Square-well potential and resonances
    6. 19.6 Tunneling
    7. 19.7 Problems
  26. 20 Scattering in three dimensions – a formal theory
    1. 20.1 Formal solutions in terms of Green’s function
    2. 20.2 Lippmann–Schwinger equation
    3. 20.3 Born approximation
    4. 20.4 Scattering from a Yukawa potential
    5. 20.5 Rutherford scattering
    6. 20.6 Charge distribution
    7. 20.7 Probability conservation and the optical theorem
    8. 20.8 Absorption
    9. 20.9 Relation between the T-matrix and the scattering amplitude
    10. 20.10 The S-matrix
    11. 20.11 Unitarity of the S-matrix and the relation between S and T
    12. 20.12 Properties of the T-matrix and the optical theorem (again)
    13. 20.13 Appendix to Chapter 20
    14. 20.14 Problems
  27. 21 Partial wave amplitudes and phase shifts
    1. 21.1 Scattering amplitude in terms of phase shifts
    2. 21.2 χl, Kl, and Tl
    3. 21.3 Integral relations for χl, Kl, and Tl
    4. 21.4 Wronskian
    5. 21.5 Calculation of phase shifts: some examples
    6. 21.6 Problems
  28. 22 Analytic structure of the S-matrix
    1. 22.1 S-matrix poles
    2. 22.2 Jost function formalism
    3. 22.3 Levinson’s theorem
    4. 22.4 Explicit calculation of the Jost function for l = 0
    5. 22.5 Integral representation of F0(k)
    6. 22.6 Problems
  29. 23 Poles of the Green’s function and composite systems
    1. 23.1 Relation between the time-evolution operator and the Green’s function
    2. 23.2 Stable and unstable states
    3. 23.3 Scattering amplitude and resonance
    4. 23.4 Complex poles
    5. 23.5 Two types of resonances
    6. 23.6 The reaction matrix
    7. 23.7 Composite systems
    8. 23.8 Appendix to Chapter 23
  30. 24 Approximation methods for bound states and scattering
    1. 24.1 WKB approximation
    2. 24.2 Variational method
    3. 24.3 Eikonal approximation
    4. 24.4 Problems
  31. 25 Lagrangian method and Feynman path integrals
    1. 25.1 Euler–Lagrange equations
    2. 25.2 N oscillators and the continuum limit
    3. 25.3 Feynman path integrals
    4. 25.4 Problems
  32. 26 Rotations and angular momentum
    1. 26.1 Rotation of coordinate axes
    2. 26.2 Scalar functions and orbital angular momentum
    3. 26.3 State vectors
    4. 26.4 Transformation of matrix elements and representations of the rotation operator
    5. 26.5 Generators of infinitesimal rotations: their eigenstates and eigenvalues
    6. 26.6 Representations of J2 and Ji for j = 1/2 and j = 1
    7. 26.7 Spherical harmonics
    8. 26.8 Problems
  33. 27 Symmetry in quantum mechanics and symmetry groups
    1. 27.1 Rotational symmetry
    2. 27.2 Parity transformation
    3. 27.3 Time reversal
    4. 27.4 Symmetry groups
    5. 27.5 Dj (R) for j = 1/2 and j = 1: examples of SO(3) and SU (2) groups
    6. 27.6 Problems
  34. 28 Addition of angular momenta
    1. 28.1 Combining eigenstates: simple examples
    2. 28.2 Clebsch–Gordan coefficients and their recursion relations
    3. 28.3 Combining spin ½ and orbital angular momentum l
    4. 28.4 Appendix to Chapter 28
    5. 28.5 Problems
  35. 29 Irreducible tensors and Wigner–Eckart theorem
    1. 29.1 Irreducible spherical tensors and their properties
    2. 29.2 The irreducible tensors: Ylm(θ, ϕ) and Dj(χ)
    3. 29.3 Wigner–Eckart theorem
    4. 29.4 Applications of the Wigner–Eckart theorem
    5. 29.5 Appendix to Chapter 29: SO(3), SU(2) groups and Young’s tableau
    6. 29.6 Problems
  36. 30 Entangled states
    1. 30.1 Definition of an entangled state
    2. 30.2 The singlet state
    3. 30.3 Differentiating the two approaches
    4. 30.4 Bell’s inequality
    5. 30.5 Problems
  37. 31 Special theory of relativity: Klein–Gordon and Maxwell’s equations
    1. 31.1 Lorentz transformation
    2. 31.2 Contravariant and covariant vectors
    3. 31.3 An example of a covariant vector
    4. 31.4 Generalization to arbitrary tensors
    5. 31.5 Relativistically invariant equations
    6. 31.6 Appendix to Chapter 31
    7. 31.7 Problems
  38. 32 Klein–Gordon and Maxwell’s equations
    1. 32.1 Covariant equations in quantum mechanics
    2. 32.2 Klein–Gordon equations: free particles
    3. 32.3 Normalization of matrix elements
    4. 32.4 Maxwell’s equations
    5. 32.5 Propagators
    6. 32.6 Virtual particles
    7. 32.7 Static approximation
    8. 32.8 Interaction potential in nonrelativistic processes
    9. 32.9 Scattering interpreted as an exchange of virtual particles
    10. 32.10 Appendix to Chapter 32
  39. 33 The Dirac equation
    1. 33.1 Basic formalism
    2. 33.2 Standard representation and spinor solutions
    3. 33.3 Large and small components of u(p)
    4. 33.4 Probability conservation
    5. 33.5 Spin ½ for the Dirac particle
  40. 34 Dirac equation in the presence of spherically symmetric potentials
    1. 34.1 Spin–orbit coupling
    2. 34.2 K-operator for the spherically symmetric potentials
    3. 34.3 Hydrogen atom
    4. 34.4 Radial Dirac equation
    5. 34.5 Hydrogen atom states
    6. 34.6 Hydrogen atom wavefunction
    7. 34.7 Appendix to Chapter 34
  41. 35 Dirac equation in a relativistically invariant form
    1. 35.1 Covariant Dirac equation
    2. 35.2 Properties of the γ -matrices
    3. 35.3 Charge–current conservation in a covariant form
    4. 35.4 Spinor solutions: ur(p) and υr(p)
    5. 35.5 Normalization and completeness condition for ur(p) and υr(p)
    6. 35.6 Gordon decomposition
    7. 35.7 Lorentz transformation of the Dirac equation
    8. 35.8 Appendix to Chapter 35
  42. 36 Interaction of a Dirac particle with an electromagnetic field
    1. 36.1 Charged particle Hamiltonian
    2. 36.2 Deriving the equation another way
    3. 36.3 Gordon decomposition and electromagnetic current
    4. 36.4 Dirac equation with EM field and comparison with the Klein–Gordon equation
    5. 36.5 Propagators: the Dirac propagator
    6. 36.6 Scattering
    7. 36.7 Appendix to Chapter 36
  43. 37 Multiparticle systems and second quantization
    1. 37.1 Wavefunctions for identical particles
    2. 37.2 Occupation number space and ladder operators
    3. 37.3 Creation and destruction operators
    4. 37.4 Writing single-particle relations in multiparticle language: the operators, N, H, and P
    5. 37.5 Matrix elements of a potential
    6. 37.6 Free fields and continuous variables
    7. 37.7 Klein–Gordon/scalar field
    8. 37.8 Complex scalar field
    9. 37.9 Dirac field
    10. 37.10 Maxwell field
    11. 37.11 Lorentz covariance for Maxwell field
    12. 37.12 Propagators and time-ordered products
    13. 37.13 Canonical quantization
    14. 37.14 Casimir effect
    15. 37.15 Problems
  44. 38 Interactions of electrons and phonons in condensed matter
    1. 38.1 Fermi energy
    2. 38.2 Interacting electron gas
    3. 38.3 Phonons
    4. 38.4 Electron–phonon interaction
  45. 39 Superconductivity
    1. 39.1 Many-body system of half-integer spins
    2. 39.2 Normal states (∆ = 0, G ≠ 0)
    3. 39.3 BCS states (∆ ≠ 0)
    4. 39.4 BCS condensate in Green’s function formalism
    5. 39.5 Meissner effect
    6. 39.6 Problems
  46. 40 Bose–Einstein condensation and superfluidity
    1. 40.1 Many-body system of integer spins
    2. 40.2 Superfluidity
    3. 40.3 Problems
  47. 41 Lagrangian formulation of classical fields
    1. 41.1 Basic structure
    2. 41.2 Noether’s theorem
    3. 41.3 Examples
    4. 41.4 Maxwell’s equations and consequences of gauge invariance
  48. 42 Spontaneous symmetry breaking
    1. 42.1 BCS mechanism
    2. 42.2 Ferromagnetism
    3. 42.3 SSB for discrete symmetry in classical field theory
    4. 42.4 SSB for continuous symmetry
    5. 42.5 Nambu–Goldstone bosons
    6. 42.6 Higgs mechanism
  49. 43 Basic quantum electrodynamics and Feynman diagrams
    1. 43.1 Perturbation theory
    2. 43.2 Feynman diagrams
    3. 43.3 T(HI (x1) HI (x2)) and Wick’s theorem
    4. 43.4 Feynman rules
    5. 43.5 Cross-section for 1 + 2 → 3 + 4
    6. 43.6 Basic two-body scattering in QED
    7. 43.7 QED vs. nonrelativistic limit: electron–electron system
    8. 43.8 QED vs. nonrelativistic limit: electron–photon system
  50. 44 Radiative corrections
    1. 44.1 Radiative corrections and renormalization
    2. 44.2 Electron self-energy
    3. 44.3 Appendix to Chapter 44
  51. 45 Anomalous magnetic moment and Lamb shift
    1. 45.1 Calculating the divergent integrals
    2. 45.2 Vertex function and the magnetic moment
    3. 45.3 Calculation of the vertex function diagram
    4. 45.4 Divergent part of the vertex function
    5. 45.5 Radiative corrections to the photon propagator
    6. 45.6 Divergent part of the photon propagator
    7. 45.7 Modification of the photon propagator and photon wavefunction
    8. 45.8 Combination of all the divergent terms: basic renormalization
    9. 45.9 Convergent parts of the radiative corrections
    10. 45.10 Appendix to Chapter 45
  52. Bibliography
  53. Index