ESSENTIAL QUANTUM MECHANICS
CONTENTS INTRODUCTION: THREE WORLDS WORLDS 1 AND WORLD PROBLEMS THE QUANTUM POSTULATES POSTULATE 1: THE QUANTUM STATE POSTULATE 2: OBSERVABLES, OPERATORS, AND EIGENSTATES POSTULATE 3: QUANTUM SUPERPOSITIONS DISCRETE EIGENVALUES CONTINUOUS EIGENVALUES CLOSING COMMENTS WHAT IS A QUANTUM STATE? PROBABILITIES, AVERAGES, AND UNCERTAINTIES PROBABILITIES AVERAGES UNCERTAINTIES THE STATISTICAL INTERPRETATION BOHR, EINSTEIN, AND HIDDEN VARIABLES BACKGROUND FUNDAMENTAL ISSUES EINSTEIN REVISITED THE STRUCTURE OF QUANTUM STATES MATHEMATICAL PRELIMINARIES VECTOR SPACES FUNCTION SPACES DIRAC’S BRA-KET NOTATION BRAS AND KETS LABELING STATES THE SCALAR PRODUCT QUANTUM SCALAR PRODUCTS DISCUSSION REPRESENTATIONS BASICS SUPERPOSITIONS AND REPRESENTATIONS REPRESENTATIONAL FREEDOM OPERATORS INTRODUCTORY COMMENTS HERMITIAN OPERATORS ADJOINT OPERATORS HERMITIAN OPERATORS: DE?NITION AND PROPERTIES WAVEFUNCTIONS AND HERMITIAN OPERATORS PROJECTION AND IDENTITY OPERATORS PROJECTION OPERATORS THE IDENTITY OPERATOR UNITARY OPERATORS MATRIX MECHANICS ELEMENTARY MATRIX OPERATIONS VECTORS AND SCALAR PRODUCTS MATRICES AND MATRIX MULTIPLICATION VECTOR TRANSFORMATIONS STATES AS VECTORS OPERATORS AS MATRICES AN OPERATOR IN ITS EIGENBASIS MATRIX ELEMENTS AND ALTERNATIVE BASES CHANGE OF BASIS ADJOINT, HERMITIAN, AND UNITARY OPERATORS EIGENVALUE EQUATIONS COMMUTATORS AND UNCERTAINTY RELATIONS THE COMMUTATOR DE?NITION AND CHARACTERISTICS COMMUTATORS IN MATRIX MECHANICS THE UNCERTAINTY RELATIONS UNCERTAINTY PRODUCTS GENERAL FORM OF THE UNCERTAINTY RELATIONS INTERPRETATIONS RE?ECTIONS ANGULAR MOMENTUM ANGULAR MOMENTUM IN CLASSICAL MECHANICS BASICS OF QUANTUM ANGULAR MOMENTUM OPERATORS AND COMMUTATION RELATIONS EIGENSTATES AND EIGENVALUES RAISING AND LOWERING OPERATORS PHYSICAL INTERPRETATION MEASUREMENTS RELATING L2 AND LZ ORBITAL AND SPIN ANGULAR MOMENTUM ORBITAL ANGULAR MOMENTUM SPIN ANGULAR MOMENTUM REVIEW THE TIME-INDEPENDENT SCHR¨ODINGER EQUATION AN EIGENVALUE EQUATION FOR ENERGY USING THE SCHR¨ODINGER EQUATION CONDITIONS ON WAVEFUNCTIONS AN EXAMPLE: THE IN?NITE POTENTIAL WELL INTERPRETATION ENERGY EIGENSTATES IN POSITION SPACE OVERALL AND RELATIVE PHASES POTENTIAL BARRIERS AND TUNNELING THE STEP POTENTIAL THE STEP POTENTIAL AND SCATTERING TUNNELING WHAT’S WRONG WITH THIS PICTURE? WHY IS THE STATE COMPLEX? COMPLEX NUMBERS POLAR FORM ARGAND DIAGRAMS AND THE ROLE OF THE PHASE THE PHASE IN QUANTUM MECHANICS PHASES AND THE DESCRIPTION OF STATES PHASE CHANGES AND PROBABILITIES UNITARY OPERATORS REVISITED UNITARY OPERATORS, PHASES, AND PROBABILITIES EXAMPLE: A SPIN 2 SYSTEM WAVEFUNCTIONS TIME EVOLUTION THE TIME-DEPENDENT SCHR¨ODINGER EQUATION HOW TIME EVOLUTION WORKS TIME EVOLVING A QUANTUM STATE UNITARITY AND PHASES REVISITED EXPECTATION VALUES TIME DERIVATIVES CONSTANTS OF THE MOTION ENERGY-TIME UNCERTAINTY RELATIONS CONCEPTUAL BASIS SPIN 2: AN EXAMPLE WHAT IS A WAVEFUNCTION? EIGENSTATES AND COE?CIENTS REPRESENTATIONS AND OPERATORS CHANGING REPRESENTATIONS CHANGE OF BASIS REVISITED FROM X TO P AND BACK AGAIN GAUSSIANS AND BEYOND PHASES AND TIME EVOLUTION FREE PARTICLE EVOLUTION WAVEPACKETS BRA-KET NOTATION QUANTUM STATES EIGENSTATES AND TRANSFORMATIONS EPILOGUE