Quantum Mechanics: Fundamental Principles and …

1 Quantum mechanics : Fundamental Principles andApplicationsJohn F. DawsonDepartment of Physics, University of New Hampshire, Durham, NH 03824 October 14, 2009, 9:08am ESTc 2007 John F. Dawson, all rights 2009 John F. Dawson, all rights Fundamental Principles11 Linear Linear vector spaces .. Linear independence .. Inner product .. dual space .. basis sets .. Operators .. and eigenvectors: .. basis vectors .. operators: .. representations: .. transformations .. operators .. sets of commuting operators .. Infinite dimensional spaces .. of the coordinate system .. Measurement .. uncertainty relation .. Time in non-relativistic Quantum mechanics .

2 272 Canonical Classical mechanics review .. of the action .. transformations .. Canonical quantization postulates .. Heisenberg picture .. Schr odinger picture .. Canonical transformations .. Schwinger s transformation theory ..423 Path Space-time paths .. Some path integrals .. Matrix elements of coordinate operators .. Generating functionals .. Closed time path integrals .. Initial value conditions .. Connected Green functions .. Classical expansion .. Some useful integrals ..544 In and Out The interaction representation .. The time development operator .. Forced oscillator.

3 585 Density matrix Classical theory .. time development operator .. averages .. correlation and Green functions .. generating functional .. Quantum theory ..686 Thermal The canonical ensemble .. Ensemble averages .. Imaginary time formalism .. Thermal Green functions .. Path integral representation .. Thermovariable methods ..767 Green functions778 Identical Coordinate representation .. Occupation number representation .. Particle fields ..819 Galilean transformations .. Galilean group .. structure .. Galilean transformations .. factors for the Galilean group.. transformations of the generators.

4 Relations of the generators .. of mass operator .. invariants .. of the Galilean group .. dimensional representations .. massless case .. Time translations .. Space translations and boosts .. Rotations .. rotation operator .. 105c 2009 John F. Dawson, all rights of the basis sets .. General Galilean transformations .. Improper transformations .. reversal .. conjugation .. Scale and conformal transformations .. transformations .. transformations .. The Schr odinger group .. 11310 Wave Scalars .. Spinors .. Spinor particles .. Spinor antiparticles .. Vectors .. Massless wave equations.

5 Massless scalers .. Massless vectors .. 12111 Grassmann variables .. Superspace and the 1D-N supersymmetry group .. 1D-N supersymmetry transformations in Quantum mechanics .. Supersymmetric generators .. R-symmetry .. Extension of the supersymmetry group .. Differential forms .. 133II Applications13512 Finite Quantum Diatomic molecules .. Periodic chains .. Linear chains .. Impurities .. Bound state .. Scattering .. 14513 One and two dimensional wave Introduction .. Schr odinger s equation in one dimension .. Transmission of a barrier .. Wave packet propagation .. Time delays for reflection by a potential step.

6 Schr odinger s equation in two dimensions .. 156c 2009 John F. Dawson, all rights The WKB Introduction .. Theory .. Connection formulas .. Positive slope .. Negative slope .. Examples .. Bound states .. Tunneling .. 16515 Spin Magnetic moments .. Pauli matrices .. The eigenvalue problem .. Spin precession in a magnetic field .. Driven spin system .. Spin decay:T1andT2.. The Ising model .. Heisenberg models .. 17616 The harmonic The Lagrangian .. Energy eigenvalue and eigenvectors .. Other forms of the Lagrangian .. Coherent states .. Completeness relations .. Generating function.

7 Squeezed states .. The forced oscillator .. The three-dimensional oscillator .. The Fermi oscillator .. Action for a Fermi oscillator .. 19617 Electrons and Electron-phonon action .. Equations of motion .. Numerical classical results .. Electron modes .. Vibrational modes .. Electron-phonon interaction .. The action revisited .. Quantization .. Block wave functions .. A one-dimensional periodic potential .. A lattice of delta-functions .. Numerical methods .. 21618 Schr odinger perturbation Time-independent perturbation theory .. Time-dependent perturbation theory .. 225c 2009 John F.

8 Dawson, all rights Variational Introduction .. Time dependent variations .. The initial value problem .. The eigenvalue problem .. Examples .. The harmonic oscillator .. The anharmonic oscillator .. Time-dependent Hartree-Fock .. 23620 Exactly solvable potential Supersymmetric Quantum mechanics .. The hierarchy of Hamiltonians .. Shape invariance .. 23721 Angular Eigenvectors of angular momentum .. Spin .. Orbital angular momentum .. Kinetic energy operator .. Parity and Time reversal .. Rotation of coordinate frames .. Rotation matrices .. Axis and angle parameterization .. Euler angles.

9 Cayley-Klein parameters .. Rotations in Quantum mechanics .. Rotations using Euler angles .. Properties ofD-functions .. Rotation of orbital angular momentum .. Sequential rotations .. Addition of angular momentum .. Coupling of two angular momenta .. Coupling of three and four angular momenta .. Rotation of coupled vectors .. Tensor operators .. Tensor operators and the Wigner-Eckart theorem .. Reduced matrix elements .. Angular momentum matrix elements of tensor operators .. Selected problems .. Spin-orbit force in hydrogen .. Transition rates for photon emission in Hydrogen .. Hyperfine splitting in Hydrogen.

10 The Zeeman effect in hydrogen .. The Stark effect in hydrogen .. Matrix elements of two-body nucleon-nucleon potentials .. Density matrix for the Deuteron .. 294c 2009 John F. Dawson, all rights The Lagrangian .. Probability conservation .. Gauge transformations .. Constant electric field .. Hydrogen atom .. Eigenvalues and eigenvectors .. Matrix elements of the Runge-Lenz vector .. Symmetry group .. Operator factorization .. Operators for the principle Quantum number .. (4,2) algebra .. The fine structure of hydrogen .. The hyperfine structure of hydrogen .. The Zeeman effect .. The Stark effect.