Quantum Physics - UCLA

1 Quantum PhysicsEric D HokerDepartment of Physics and Astronomy,University of California, Los Angeles, CA 90095, USA15 September 20121 Contents1 Brief History .. Constants of Nature .. Scales .. Reductionism .. 152 Two-state Quantum Polarization of light .. Polarization of photons .. General parametrization of the polarization of light .. Mathematical formulation of the photon system .. The Stern-Gerlach experiment on electron spin .. 223 Mathematical Formalism of Quantum Hilbert spaces .. Triangle and Schwarz Inequalities .. The construction of an orthonormal basis .. Decomposition of an arbitrary vector .. Finite-dimensional Hilbert spaces .. Infinite-dimensional Hilbert spaces .. Linear operators on Hilbert space .. Operators in finite-dimensional Hilbert spaces .. Operators in infinite-dimensional Hilbert spaces.

2 Special types of operators .. Hermitian and unitary operators in finite-dimension .. Unitary operators .. The exponential map .. Self-adjoint operators in infinite-dimensional Hilbert spaces .. 374 The Principles of Quantum Conservation of probability .. Compatible versus incompatible observables .. Expectation values and Quantum fluctuations .. Incompatible observables, Heisenberg uncertainty relations .. Complete sets of commuting observables .. 4325 Some Basic Examples of Quantum Propagation in a finite 1-dimensional lattice .. Diagonalizing the translation operator .. Position and translation operator algebra .. The spectrum and generalized Hamiltonians .. Bilateral and reflection symmetric lattices .. Propagation in an infinite 1-dimensional lattice .. Propagation on a circle .. Propagation on the full line .. The Dirac -function.

3 General position and momentum operators and eigenstates .. The harmonic oscillator .. Lowering and Raising operators .. Constructing the spectrum .. Harmonic oscillator wave functions .. The angular momentum algebra .. Complete set of commuting observables .. Lowering and raising operators .. Constructing the spectrum .. The Coulomb problem .. Bound state spectrum .. Scattering spectrum .. Self-adjoint operators and boundary conditions .. Example 1: One-dimensional Schr odinger operator on half-line.. Example 2: One-dimensional momentum in a box .. Example 3: One-dimensional Dirac-like operator in a box .. 676 Quantum Mechanics Lagrangian mechanics .. Hamiltonian mechanics .. Constructing a Quantum system from classical mechanics .. Schr odinger equation with a scalar potential .. Uniqueness questions of the correspondence principle.

4 737 Charged particle in an electro-magnetic Gauge transformations and gauge invariance .. Constant Magnetic fields .. Map onto harmonic oscillators .. Landau Levels .. Complex variables .. The Aharonov-Bohm Effect .. The scattering Aharonov-Bohm effect .. The bound state Aharonov-Bohm effect .. The Dirac magnetic monopole .. 838 Theory of Angular Rotations .. The Lie algebra of rotations angular momentum .. General Groups and their Representations .. General Lie Algebras and their Representations .. Direct sum and reducibility of representations .. The irreducible representations of angular momentum .. Addition of two spin 1/2 angular momenta .. Addition of a spin 1/2 with a general angular momentum .. Addition of two general angular momenta .. Systematics of Clebsch-Gordan coefficients .. Spin Models .. The Ising Model.

5 Solution of the 1-dimensional Ising Model .. Ordered versus disordered phases .. 1029 Symmetries in Quantum Symmetries in classical mechanics .. Noether s Theorem .. Group and Lie algebra structure of classical symmetries .. Symmetries in Quantum Physics .. Examples of Quantum symmetries .. Symmetries of the multi-dimensional harmonic oscillator .. The orthogonal groupSO(N) .. The unitary groupsU(N) andSU(N) .. The groupSp(2N) .. Selection rules .. Vector Observables .. Selection rules for vector observables .. Tensor Observables .. ,C, andT.. 12110 Bound State Perturbation The validity of perturbation theory .. Smallness of the coupling .. Convergence of the expansion for finite-dimensional systems .. The asymptotic nature of the expansion for infinite dimensional Non-degenerate perturbation theory .. Some linear algebra .. The Stark effect for the ground state of the Hydrogen atom.

6 Excited states and degenerate perturbation theory .. The Zeeman effect .. Spin orbit coupling .. General development of degenerate perturbation theory .. Solution to first order .. Solution to second order .. Periodic potentials and the formation of band structure .. Crossing ..14011 External Magnetic Field Landau levels .. Complex variables .. Calculation of the density of states in each Landau level .. The classical Hall effect .. The Quantum Hall effect .. 14612 Scattering Potential Scattering .. Thei prescription .. The free particle propagator .. The Lippmann-Schwinger equation in position space .. Short range versus long rangeVand massless particles .. The wave-function solution far from the target .. Calculation of the cross section .. The Born approximation .. The case of the Coulomb potential .. The case of the Yukawa potential.

7 The optical Theorem .. potentials and partial wave expansion .. Functions .. wave expansion of wave functions .. the radial Green function .. shifts .. example of a hard sphere .. hard spherical shell .. scattering .. 17113 Time-dependent Magnetic spin resonance and driven two-state systems .. The interaction picture .. Time-dependent perturbation theory .. Switching on an interaction .. Sinusoidal perturbation .. 18014 Path Integral Formulation of Quantum The time-evolution operator .. The evolution operator for Quantum mechanical systems .. The evolution operator for a free massive particle .. Derivation of the path integral .. Integrating out the canonical momentump.. Dominant paths .. Stationary phase approximation .. Gaussian fluctuations .. Gaussian integrals .. the contribution of Gaussian fluctuations .. 19115 Applications and Examples of Path Path integral calculation for the harmonic oscillator.

8 The Aharonov-Bohm Effect .. Imaginary time path Integrals .. Quantum Statistical Mechanics .. Path integral formulation of Quantum statistical mechanics .. Classical Statistical Mechanics as the high temperature limit .. 20016 Mixtures and Statistical Polarized versus unpolarized beams .. The Density Operator .. Ensemble averages of expectation values in mixtures .. Time evolution of the density operator .. Example of the two-state system .. Non-uniqueness of state preparation .. Quantum Statistical Mechanics .. Generalized equilibrium ensembles .. Classical information and Shannon entropy .. Quantum statistical entropy .. Density matrix for a subsystem .. Example of relations between density matrices of subsystems.. Lemma 1 .. Completing the proof of subadditivity .. Examples of the use of statistical entropy .. Second law of thermodynamics.

9 Entropy resulting from coarse graining ..21817 Entanglement, EPR, and Bell s Entangled States for two spin 1/2 .. Entangled states from non-entangled states .. The Schmidt purification theorem .. Generalized description of entangled states .. Entanglement entropy .. The two-state system once more .. Entanglement in the EPR paradox .. Einstein s locality principle .. Bell s inequalities .. predictions for Bell s inequalities .. particle entangled states .. 23218 Introductory Remarks on Quantized Relativity and Quantum mechanics .. Why Quantum Field Theory ? .. Further conceptual changes required by relativity .. Some History and present significance of QFT .. 23719 Quantization of the Free Electro-magnetic Classical Maxwell theory .. Fourrier modes and radiation oscillators .. The Hamiltonian in terms of radiation oscillators .. Momentum in terms of radiation oscillators.

10 Canonical quantization of electro-magnetic fields .. Photons the Hilbert space of states .. The ground state orvacuum.. One-photon states .. Multi-photon states .. Bose-Einstein and Fermi-Dirac statistics .. The photon spin and helicity .. The Casimir Effect on parallel plates ..253720 Photon Emission and Setting up the general problem of photon emission/absorption.. Single Photon Emission/Absorption .. Application to the decay rate of 2p state of atomic Hydrogen .. Absorption and emission of photons in a cavity .. Black-body radiation .. 26321 Relativistic Field A brief review of special relativity .. Lorentz vector and tensor notation .. General Lorentz vectors and tensors .. Contravariant tensors .. Covariant tensors .. Contraction and trace .. Classical relativistic kinematics and dynamics .. Particle collider versus fixed target experiments.