Quantum Physics (UCSD Physics 130)

1 Quantum Physics (UCSD Physics 130)April 2, 20032 Contents1 Course Problems with Classical Physics .. Thought Experiments on Diffraction .. Probability Amplitudes .. Wave Packets and Uncertainty .. Operators .. Expectation Values .. Commutators .. The Schr odinger Equation .. Eigenfunctions, Eigenvalues and Vector Spaces .. A Particle in a Box .. Piecewise Constant Potentials in One Dimension .. The Harmonic Oscillator in One Dimension .. Delta Function Potentials in One Dimension .. Harmonic Oscillator Solution with Operators .. More Fun with Operators .. Two Particles in 3 Dimensions .. Identical Particles .. Some 3D Problems Separable in Cartesian Coordinates .. Angular Momentum .. Solutions to the Radial Equation for Constant Potentials .. Hydrogen .. Solution of the 3D HO Problem in Spherical Coordinates.

2 Matrix Representation of Operators and States .. A Study of = 1 Operators and Eigenfunctions .. Spin 1/2 and other 2 State Systems .. Quantum Mechanics in an Electromagnetic Field .. Local Phase Symmetry in Quantum Mechanics and the Gauge Symmetry .. Addition of Angular Momentum .. Time Independent Perturbation Theory .. The Fine Structure of Hydrogen .. Hyperfine Structure .. The Helium Atom .. Atomic Physics .. Molecules .. Time Dependent Perturbation Theory .. Radiation in Atoms .. Classical Field Theory .. The Classical Electromagnetic Field .. Quantization of the EM Field .. Scattering of Photons .. Electron Self Energy .. The Dirac Equation .. The Dirac Equation .. 602 The Problems with Classical Black Body Radiation*.. The Photoelectric Effect .. The Rutherford Atom*.. Atomic Spectra*.. The Bohr Atom*.

3 Derivations and Computations .. Black Body Radiation Formulas*.. The Fine Structure Constant and the Coulomb Potential .. Examples .. The Solar Temperature*.. Black Body Radiation from the Early Universe*.. Compton Scattering*.. Rutherford s Nuclear Size*.. Sample Test Problems .. 823 Diffraction from Two Slits .. Single Slit Diffraction .. Diffraction from Crystals .. The DeBroglie Wavelength .. Computing DeBroglie Wavelengths .. Wave Particle Duality (Thought Experiments) .. Examples .. Intensity Distribution for Two Slit Diffraction*.. Intensity Distribution for Single Slit Diffraction*.. Sample Test Problems .. 964 The Solution: Probability Derivations and Computations .. Review of Complex Numbers .. Review of Traveling Waves .. Sample Test Problems .. 995 Wave Building a Localized Single-Particle Wave Packet.

4 Two Examples of Localized Wave Packets .. The Heisenberg Uncertainty Principle .. Position Space and Momentum Space .. Time Development of a Gaussian Wave Packet*.. Derivations and Computations .. Fourier Series*.. Fourier Transform*.. Integral of Gaussian .. Fourier Transform of Gaussian*.. Time Dependence of a Gaussian Wave Packet*.. Numbers .. The Dirac Delta Function .. Examples .. The Square Wave Packet .. The Gaussian Wave Packet*.. The Dirac Delta Function Wave Packet*.. Can I See inside an Atom .. Can I See inside a Nucleus .. Estimate the Hydrogen Ground State Energy .. Sample Test Problems .. 1156 Operators in Position Space .. The Momentum Operator .. The Energy Operator .. The Position Operator .. The Hamiltonian Operator .. Operators in Momentum Space .. Expectation Values .. Dirac Bra-ket Notation.

5 Commutators .. Derivations and Computations .. Verify Momentum Operator .. Verify Energy Operator .. Examples .. Expectation Value of Momentum in a Given State .. Commutator ofEandt.. Commutator ofEandx.. Commutator ofpandxn.. Commutator ofLxandLy.. Sample Test Problems .. 12457 The Schr odinger Deriving the Equation from Operators .. The Flux of Probability*.. The Schr odinger Wave Equation .. The Time Independent Schr odinger Equation .. Derivations and Computations .. Linear Operators .. Probability Conservation Equation*.. Examples .. Solution to the Schr odinger Equation in a Constant Potential .. Sample Test Problems .. 1318 Eigenfunctions, Eigenvalues and Vector Eigenvalue Equations .. Hermitian Conjugate of an Operator .. Hermitian Operators .. Eigenfunctions and Vector Space .. The Particle in a 1D Box.

6 The Same Problem with Parity Symmetry .. Momentum Eigenfunctions .. Derivations and Computations .. Eigenfunctions of Hermitian Operators are Orthogonal .. Continuity of Wavefunctions and Derivatives .. Examples .. Hermitian Conjugate of a Constant Operator .. Hermitian Conjugate of x.. Sample Test Problems .. 1429 One Dimensional Piecewise Constant Potentials in 1D .. The General Solution for a Constant Potential .. The Potential Step .. The Potential Well withE >0*.. Bound States in a Potential Well*.. The Potential Barrier .. The 1D Harmonic Oscillator .. The Delta Function Potential*.. The Delta Function Model of a Molecule*.. The Delta Function Model of a Crystal*.. The Quantum Rotor .. Derivations and Computations .. Probability Flux for the Potential Step*.. Scattering from a 1D Potential Well*.. Bound States of a 1D Potential Well*.

7 Solving the HO Differential Equation*.. 1D Model of a Molecule Derivation*.. 1D Model of a Crystal Derivation*.. Examples .. Sample Test Problems .. 17010 Harmonic Oscillator Solution using IntroducingAandA .. Commutators ofA,A andH.. Use Commutators to Derive HO Energies .. Raising and Lowering Constants .. Expectation Values ofpandx.. The Wavefunction for the HO Ground State .. Examples .. The expectation value ofxin eigenstate .. The expectation value ofpin eigenstate .. The expectation value ofxin the state1 2(u0+u1).. The expectation value of12m 2x2in eigenstate .. The expectation value ofp22min eigenstate .. Time Development Example .. Sample Test Problems .. 17911 More Fun with Operators in a Vector Space .. Review of Operators .. Projection Operators|jihj|and Completeness .. Unitary Operators .. A Complete Set of Mutually Commuting Operators.

8 Uncertainty Principle for Non-Commuting Operators .. Time Derivative of Expectation Values*.. The Time Development Operator*.. The Heisenberg Picture*.. Examples .. Time Development Example .. Sample Test Problems .. 18912 Extending QM to Two Particles and Three Quantum Mechanics for Two Particles .. Quantum Mechanics in Three Dimensions .. Two Particles in Three Dimensions .. Identical Particles .. Sample Test Problems .. 194713 3D Problems Separable in Cartesian Particle in a 3D Box .. Filling the Box with Fermions .. Degeneracy Pressure in Stars .. The 3D Harmonic Oscillator .. Sample Test Problems .. 20114 Angular Rotational Symmetry .. Angular Momentum Algebra: Raising and Lowering Operators .. The Angular Momentum Eigenfunctions .. Parity of the Spherical Harmonics .. Derivations and Computations .. Rotational Symmetry Implies Angular Momentum Conservation.

9 The Commutators of the Angular Momentum Operators .. Rewritingp22 UsingL2.. Spherical Coordinates and the Angular Momentum Operators.. The OperatorsL .. Examples .. The Expectation Value ofLz.. The Expectation Value ofLx.. Sample Test Problems .. 21615 The Radial Equation and Constant Potentials* The Radial Equation*.. Behavior at the Origin*.. Spherical Bessel Functions*.. Particle in a Sphere*.. Bound States in a Spherical Potential Well*.. Partial Wave Analysis of Scattering*.. Scattering from a Spherical Well*.. The Radial Equation foru(r) =rR(r)*.. Sample Test Problems .. 22716 The Radial Wavefunction Solutions .. The Hydrogen Spectrum .. Derivations and Calculations .. Solution of Hydrogen Radial Equation*.. Computing the Radial Wavefunctions*.. Examples .. Expectation Values in Hydrogen States .. The Expectation of1rin the Ground State.

10 The Expectation Value ofrin the Ground State .. The Expectation Value ofvrin the Ground State .. Sample Test Problems .. 24017 3D Symmetric HO in Spherical Coordinates*24318 Operators Matrices and The Matrix Representation of Operators and Wavefunctions .. The Angular Momentum Matrices*.. Eigenvalue Problems with Matrices .. An = 1 System in a Magnetic Field*.. Splitting the Eigenstates with Stern-Gerlach .. Rotation operators for = 1*.. A Rotated Stern-Gerlach Apparatus*.. Spin12.. Other Two State Systems*.. The Ammonia Molecule (Maser) .. The Neutral Kaon System*.. Oscillator Hamiltonian Matrix .. Oscillator Raising Operator .. Oscillator Lowering Operator .. ofLx.. 90 degree rotation about the z axis.. Eigenstates of an = 1 System in a B-field .. series of Stern-Gerlachs .. Development of an = 1 System in a B-field: Version I .. ofSxin General Spin12 State.