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QuantumPhysics.dvi

(Wang) #1

  • 1 Introduction

    • 1.1 Brief History

    • 1.2 Constants of Nature

    • 1.3 Scales

    • 1.4 Reductionism



  • 2 Two-state quantum systems

    • 2.1 Polarization of light

    • 2.2 Polarization of photons

    • 2.3 General parametrization of the polarization of light

    • 2.4 Mathematical formulation of the photon system

    • 2.5 The Stern-Gerlach experiment on electron spin



  • 3 Mathematical Formalism of Quantum Physics

    • 3.1 Hilbert spaces

      • 3.1.1 Triangle and Schwarz Inequalities

      • 3.1.2 The construction of an orthonormal basis

      • 3.1.3 Decomposition of an arbitrary vector

      • 3.1.4 Finite-dimensional Hilbert spaces

      • 3.1.5 Infinite-dimensional Hilbert spaces



    • 3.2 Linear operators on Hilbert space

      • 3.2.1 Operators in finite-dimensional Hilbert spaces

      • 3.2.2 Operators in infinite-dimensional Hilbert spaces



    • 3.3 Special types of operators

    • 3.4 Hermitian and unitary operators in finite-dimension

      • 3.4.1 Unitary operators

      • 3.4.2 The exponential map



    • 3.5 Self-adjoint operators in infinite-dimensional Hilbert spaces



  • 4 The Principles of Quantum Physics

    • 4.1 Conservation of probability

    • 4.2 Compatible versus incompatible observables

    • 4.3 Expectation values and quantum fluctuations

    • 4.4 Incompatible observables, Heisenberg uncertainty relations

    • 4.5 Complete sets of commuting observables



  • 5 Some Basic Examples of Quantum Systems

    • 5.1 Propagation in a finite 1-dimensional lattice

      • 5.1.1 Diagonalizing the translation operator

      • 5.1.2 Position and translation operator algebra

      • 5.1.3 The spectrum and generalized Hamiltonians

      • 5.1.4 Bilateral and reflection symmetric lattices



    • 5.2 Propagation in an infinite 1-dimensional lattice

    • 5.3 Propagation on a circle

    • 5.4 Propagation on the full line

      • 5.4.1 The Diracδ-function



    • 5.5 General position and momentum operators and eigenstates

    • 5.6 The harmonic oscillator

      • 5.6.1 Lowering and Raising operators

      • 5.6.2 Constructing the spectrum

      • 5.6.3 Harmonic oscillator wave functions



    • 5.7 The angular momentum algebra

      • 5.7.1 Complete set of commuting observables

      • 5.7.2 Lowering and raising operators

      • 5.7.3 Constructing the spectrum



    • 5.8 The Coulomb problem

      • 5.8.1 Bound state spectrum

      • 5.8.2 Scattering spectrum



    • 5.9 Self-adjoint operators and boundary conditions

      • 5.9.1 Example 1: One-dimensional Schr ̈odinger operator on half-line

      • 5.9.2 Example 2: One-dimensional momentum in a box

      • 5.9.3 Example 3: One-dimensional Dirac-like operator in a box





  • 6 Quantum Mechanics Systems

    • 6.1 Lagrangian mechanics

    • 6.2 Hamiltonian mechanics

    • 6.3 Constructing a quantum system from classical mechanics

    • 6.4 Schr ̈odinger equation with a scalar potential

    • 6.5 Uniqueness questions of the correspondence principle



  • 7 Charged particle in an electro-magnetic field

    • 7.1 Gauge transformations and gauge invariance

    • 7.2 Constant Magnetic fields

      • 7.2.1 Map onto harmonic oscillators



    • 7.3 Landau Levels

      • 7.3.1 Complex variables



    • 7.4 The Aharonov-Bohm Effect

      • 7.4.1 The scattering Aharonov-Bohm effect

      • 7.4.2 The bound state Aharonov-Bohm effect



    • 7.5 The Dirac magnetic monopole



  • 8 Theory of Angular Momentum

    • 8.1 Rotations

    • 8.2 The Lie algebra of rotations – angular momentum

    • 8.3 General Groups and their Representations

    • 8.4 General Lie Algebras and their Representations

    • 8.5 Direct sum and reducibility of representations

    • 8.6 The irreducible representations of angular momentum

    • 8.7 Addition of two spin 1/2 angular momenta

    • 8.8 Addition of a spin 1/2 with a general angular momentum

    • 8.9 Addition of two general angular momenta

    • 8.10 Systematics of Clebsch-Gordan coefficients

    • 8.11 Spin Models

    • 8.12 The Ising Model

    • 8.13 Solution of the 1-dimensional Ising Model

    • 8.14 Ordered versus disordered phases



  • 9 Symmetries in Quantum Physics

    • 9.1 Symmetries in classical mechanics

    • 9.2 Noether’s Theorem

    • 9.3 Group and Lie algebra structure of classical symmetries

    • 9.4 Symmetries in Quantum Physics

    • 9.5 Examples of quantum symmetries

    • 9.6 Symmetries of the multi-dimensional harmonic oscillator

      • 9.6.1 The orthogonal groupSO(N)

      • 9.6.2 The unitary groupsU(N) andSU(N)

      • 9.6.3 The groupSp(2N)



    • 9.7 Selection rules

    • 9.8 Vector Observables

    • 9.9 Selection rules for vector observables

    • 9.10 Tensor Observables

    • 9.11 P,C, andT



  • 10 Bound State Perturbation Theory

    • 10.1 The validity of perturbation theory

      • 10.1.1 Smallness of the coupling

      • 10.1.2 Convergence of the expansion for finite-dimensional systems

      • 10.1.3 The asymptotic nature of the expansion for infinite dimensional systems



    • 10.2 Non-degenerate perturbation theory

    • 10.3 Some linear algebra

    • 10.4 The Stark effect for the ground state of the Hydrogen atom

    • 10.5 Excited states and degenerate perturbation theory

    • 10.6 The Zeeman effect

    • 10.7 Spin orbit coupling

    • 10.8 General development of degenerate perturbation theory

      • 10.8.1 Solution to first order

      • 10.8.2 Solution to second order



    • 10.9 Periodic potentials and the formation of band structure

    • 10.10Level Crossing



  • 11 External Magnetic Field Problems

    • 11.1 Landau levels

    • 11.2 Complex variables

    • 11.3 Calculation of the density of states in each Landau level

    • 11.4 The classical Hall effect

    • 11.5 The quantum Hall effect



  • 12 Scattering Theory

    • 12.1 Potential Scattering

    • 12.2 Theiεprescription

    • 12.3 The free particle propagator

    • 12.4 The Lippmann-Schwinger equation in position space

    • 12.5 Short range versus long rangeV and massless particles

    • 12.6 The wave-function solution far from the target

    • 12.7 Calculation of the cross section

    • 12.8 The Born approximation

      • 12.8.1 The case of the Coulomb potential

      • 12.8.2 The case of the Yukawa potential



    • 12.9 The optical Theorem

    • 12.10Spherical potentials and partial wave expansion

      • 12.10.1Bessel Functions

      • 12.10.2Partial wave expansion of wave functions

      • 12.10.3Calculating the radial Green function



    • 12.11Phase shifts

    • 12.12The example of a hard sphere

    • 12.13The hard spherical shell

    • 12.14Resonance scattering



  • 13 Time-dependent Processes

    • 13.1 Magnetic spin resonance and driven two-state systems

    • 13.2 The interaction picture

    • 13.3 Time-dependent perturbation theory

    • 13.4 Switching on an interaction

    • 13.5 Sinusoidal perturbation



  • 14 Path Integral Formulation of Quantum Mechanics

    • 14.1 The time-evolution operator

    • 14.2 The evolution operator for quantum mechanical systems

    • 14.3 The evolution operator for a free massive particle

    • 14.4 Derivation of the path integral

    • 14.5 Integrating out the canonical momentump

    • 14.6 Dominant paths

    • 14.7 Stationary phase approximation

    • 14.8 Gaussian fluctuations

    • 14.9 Gaussian integrals

    • 14.10Evaluating the contribution of Gaussian fluctuations



  • 15 Applications and Examples of Path Integrals

    • 15.1 Path integral calculation for the harmonic oscillator

    • 15.2 The Aharonov-Bohm Effect

    • 15.3 Imaginary time path Integrals

    • 15.4 Quantum Statistical Mechanics

    • 15.5 Path integral formulation of quantum statistical mechanics

    • 15.6 Classical Statistical Mechanics as the high temperature limit



  • 16 Mixtures and Statistical Entropy

    • 16.1 Polarized versus unpolarized beams

    • 16.2 The Density Operator

      • 16.2.1 Ensemble averages of expectation values in mixtures

      • 16.2.2 Time evolution of the density operator



    • 16.3 Example of the two-state system

    • 16.4 Non-uniqueness of state preparation

    • 16.5 Quantum Statistical Mechanics

      • 16.5.1 Generalized equilibrium ensembles



    • 16.6 Classical information and Shannon entropy

    • 16.7 Quantum statistical entropy

      • 16.7.1 Density matrix for a subsystem

      • 16.7.2 Example of relations between density matrices of subsystems

      • 16.7.3 Lemma

      • 16.7.4 Completing the proof of subadditivity



    • 16.8 Examples of the use of statistical entropy

      • 16.8.1 Second law of thermodynamics

      • 16.8.2 Entropy resulting from coarse graining





  • 17 Entanglement, EPR, and Bell’s inequalities

    • 17.1 Entangled States for two spin 1/2

    • 17.2 Entangled states from non-entangled states

    • 17.3 The Schmidt purification theorem

    • 17.4 Generalized description of entangled states

    • 17.5 Entanglement entropy

    • 17.6 The two-state system once more

    • 17.7 Entanglement in the EPR paradox

    • 17.8 Einstein’s locality principle

    • 17.9 Bell’s inequalities

    • 17.10Quantum predictions for Bell’s inequalities

    • 17.11Three particle entangled states



  • 18 Introductory Remarks on Quantized Fields

    • 18.1 Relativity and quantum mechanics

    • 18.2 Why Quantum Field Theory?

    • 18.3 Further conceptual changes required by relativity

    • 18.4 Some History and present significance of QFT



  • 19 Quantization of the Free Electro-magnetic Field

    • 19.1 Classical Maxwell theory

    • 19.2 Fourrier modes and radiation oscillators

    • 19.3 The Hamiltonian in terms of radiation oscillators

    • 19.4 Momentum in terms of radiation oscillators

    • 19.5 Canonical quantization of electro-magnetic fields

    • 19.6 Photons – the Hilbert space of states

      • 19.6.1 The ground state orvacuum

      • 19.6.2 One-photon states

      • 19.6.3 Multi-photon states



    • 19.7 Bose-Einstein and Fermi-Dirac statistics

    • 19.8 The photon spin and helicity

    • 19.9 The Casimir Effect on parallel plates



  • 20 Photon Emission and Absorption

    • 20.1 Setting up the general problem of photon emission/absorption

    • 20.2 Single Photon Emission/Absorption

    • 20.3 Application to the decay rate of 2p state of atomic Hydrogen

    • 20.4 Absorption and emission of photons in a cavity

    • 20.5 Black-body radiation



  • 21 Relativistic Field Equations

    • 21.1 A brief review of special relativity

    • 21.2 Lorentz vector and tensor notation

    • 21.3 General Lorentz vectors and tensors

      • 21.3.1 Contravariant tensors

      • 21.3.2 Covariant tensors

      • 21.3.3 Contraction and trace



    • 21.4 Classical relativistic kinematics and dynamics

    • 21.5 Particle collider versus fixed target experiments

    • 21.6 A physical application of time dilation

    • 21.7 Relativistic invariance of the wave equation

    • 21.8 Relativistic invariance of Maxwell equations

      • 21.8.1 The gauge field and field strength

      • 21.8.2 Maxwell’s equations in Lorentz covariant form



    • 21.9 Structure of the Poincar ́e and Lorentz algebras

    • 21.10Representations of the Lorentz algebra



  • 22 The Dirac Field and the Dirac Equation

    • 22.1 The Dirac-Clifford algebra

    • 22.2 Explicit representation of the Dirac algebra

    • 22.3 Action of Lorentz transformations onγ-matrices

    • 22.4 The Dirac equation and its relativistic invariance

    • 22.5 Elementary solutions to the free Dirac equation

    • 22.6 The conserved current of fermion number

    • 22.7 The free Dirac action and Hamiltonian

    • 22.8 Coupling to the electro-magnetic field



  • 23 Quantization of the Dirac Field

    • 23.1 The basic free field solution

    • 23.2 Spinor Identities

    • 23.3 Evaluation of the electric charge operator and Hamiltonian

    • 23.4 Quantization of fermion oscillators

    • 23.5 Canonical anti-commutation relations for the Dirac field



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