FUNDAMENTALS OF QUANTUM MECHANICS
CONTENTS BIRTH OF QUANTUM MECHANICS 1 1 DIF?CULTIES OF CLASSICAL PHYSICS 1 1 1 INTRODUCTION 1 1 2 RADIOACTIVE DECAY 1 1 3 BLACK BODY RADIATION 1 1 4 PHOTOELECTRIC EFFECT 1 1 5 COMPTON EFFECT 1 1 6 RULE OF ATOMIC LINE SPECTRA 1 1 7 STABILITY OF ATOM 1 1 8 SPECI?C HEAT OF SOLID 1 2 PLANCK-EINSTEIN’S QUANTUM POSTULATE OF LIGHT 1 3 BOHR POSTULATES OF ATOMIC STATES 1 4 DE BLOGLIE MATTER WAVE HYPOTHESES 1 5 DAVIDSON AND GERMER EXPERIMENT 1 6 YOUNG’S DOUBLE SLIT EXPERIMENT 1 7 ESTABLISHMENT OF QUANTUM MECHANICS 1 8 HEISENBERG UNCERTAINTY RELATION 1 9 STATISTICAL INTERPRETATION OF QUANTUM MECHANICS 1 10 EXERCISES AND SOLUTIONS CLASSICAL MECHANICS AND MATHEMATICS 2 1 INTRODUCTION 2 2 FOUNDATION OF CLASSICAL MECHANICS 2 2 1 LAGRANGIAN FORMULATION OF CLASSICAL MECHANICS 2 2 2 HAMILTONIAN FORMULATION OF CLASSICAL MECHANICS 2 2 3 POISON BRACKET 2 2 4 CANONICAL TRANSFORMATIONS 2 2 5 JACOBI-HAMILTON EQUATION 2 3 HILBERT SPACE AND OPERATORS 2 3 1 STATE VECTOR AND DIRAC NOTATION 2 3 2 HILBERT SPACE 2 4 FOURIER TRANSFORMATION 2 4 1 INTEGRATION OF FUNCTIONS ON REAL LINE 2 4 2 DE?NITION OF FOURIER TRANSFORM 2 5 MATHEMATICAL UNCERTAINTY RELATIONS 2 5 1 DEMONSTRATION 2 5 2 APPLICATIONS OF MATHEMATICAL UNCERTAINTY RELATIONS WAVEPACKET-ONLY THEORY 3 1 INTRODUCTION 3 2 EXISTENCE OF UNIVERSAL CONSTANT 3 3 UNCERTAINTY RELATIONS 3 3 1 GENERAL THEORY OF FOURIER TRANSFORMATION 3 3 2 UNCERTAINTY RELATIONS 3 4 WAVEPACKET-ONLY THEORY 3 4 1 FREE WAVEPACKET COMPOSITIONS 3 4 2 NON-SPREADING OF FREE WAVEPACKET 3 4 3 AVERAGE WAVENUMBER OF WAVEPACKET 3 4 4 EXPRESSIONS OF FREE WAVEPACKET 3 4 5 SCHR¨ODINGER EQUATION DETERMINING WAVEFUNCTION 3 4 6 OTHER METHODS TO OBTAIN SCHR¨ODINGER EQUATION 3 4 7 WAVEPACKET MOTION LAW AND EXPLANATIONS FOR EXPERIMENTS 3 4 8 PROPERTIES OF WAVEFUNCTION 3 5 SCHR¨ODINGER EQUATION 3 5 1 ONE DIMENSIONAL TIME-INDEPENDENT SCHR¨ODINGER EQUATION 3 5 2 STATIONARY STATE 3 5 3 SUPERPOSITION OF STATES 3 5 4 REQUIREMENTS FOR WAVEFUNCTION 3 5 5 OBSERVABLE AS OPERATOR 3 5 6 PROBABILITY CURRENT DENSITY OF SINGLE PARTICLE 3 5 7 MORE ABOUT AVERAGE VALUES 3 6 QUANTUM OR CLASSICAL MECHANICS 3 6 1 CRITERION 3 6 2 EHRENFEST THEOREM 3 6 3 STILL MORE MICROSCOPIC PROCESS 3 7 EXERCISES AND SOLUTIONS STATIONARY STATES OF ONE DIMENSION SYSTEM 4 1 GENERAL PROPERTIES 4 2 IN?NITE SQUARE WELL AND DISCRETE SPECTRUM 4 3 ONE DIMENSIONAL POTENTIAL BARRIER AND WELL 4 3 1 BASIC RULES 4 3 2 POTENTIAL STEP 4 3 3 RECTANGULAR POTENTIAL BARRIER AND WELL 4 4 ONE DIMENSIONAL HARMONIC OSCILLATOR 4 4 1 HARMONIC OSCILLATOR POTENTIAL 4 4 2 ONE DIMENSIONAL HARMONIC OSCILLATOR 4 4 3 OPERATOR METHOD 4 5 PARTICLE IN GRAVITATIONAL FIELD 4 5 1 UNBOUND (DISSOCIATED) PARTICLE 4 5 2 BOUND PARTICLE 4 6 ENERGY QUANTIZATION OF BOUND STATE 4 7 EXERCISES AND SOLUTIONS GENERAL FORMALISM 5 1 EXPANSION IN EIGENFUNCTIONS 5 2 HERMITIAN OPERATORS 5 3 COMMUTATION RELATIONS OF OPERATORS 5 4 NORMALIZATION AND PERIODIC BOUNDARY CONDITION 5 5 MOTION EQUATION OF OPERATOR 5 6 SYMMETRIES AND CONSERVATIONS OF AVERAGE VALUES 5 6 1 COORDINATE TRANSFORMATIONS 5 6 2 UNIFORMITY OF TIME 5 6 3 UNIFORMITY OF POSITION SPACE 5 6 4 ISOTROPY OF POSITION SPACE 5 6 5 INVERSION INVARIANCE OF POSITION SPACE 5 7 PICTURES OF QUANTUM MECHANICS 5 7 1 UNITARY TRANSFORMATION 5 7 2 SCHR¨ODINGER PICTURE 5 7 3 HEISENBERG PICTURE 5 7 4 INTERACTION PICTURE 5 8 THREE FORMULATIONS OF QUANTUM MECHANICS 5 8 1 HEISENBERG MATRIX MECHANISM FORMULATION 5 8 2 SCHR¨ODINGER WAVE MECHANICS FORMULATION 5 8 3 FEYNMAN PATH INTEGRAL FORMULATION 5 9 EXERCISES AND SOLUTIONS ANGULAR MOMENTUM AND HYDROGEN-LIKE ATOMS 6 1 ORBITAL ANGULAR MOMENTUM 6 1 1 DE?NITION 6 1 2 ANGULAR MOMENTUM OPERATOR IN SPHERICAL POLAR COORDINATES 6 1 3 RISING AND LOWERING OPERATORS 6 1 4 EIGENVALUES AND EIGENFUNCTIONS OF L2 AND LZ 6 2 HYDROGEN ATOM 6 2 1 TWO BODY PROBLEM 6 2 2 ASYMPTOTIC BEHAVIOR 6 2 3 EIGENVALUES 6 2 4 EIGENFUNCTION 6 3 HELLMANN-FEYNMAN THEOREMS 6 3 1 HF THEOREM OF TIME-INDEPENDENT WAVEFUNCTIONS 6 3 2 HF THEOREM FOR TIME-DEPENDENT WAVEFUNCTIONS 6 4 GENERAL PROPERTIES OF CENTRAL FIELD 6 4 1 CONSERVATION OF ANGULAR MOMENTUM AND RADIAL EQUATION 6 4 2 BEHAVIOR OF RADIAL WAVEFUNCTION NEAR R = 0 6 4 3 EXPANSION OF PLANE WAVE IN SPHERICAL HARMONICS 6 5 EXERCISES AND SOLUTIONS 6 5 1 ANGULAR MOMENTUM 6 5 2 HYDROGEN-LIKE IONS 6 5 3 HF THEOREM 6 5 4 GENERAL PROPERTIES OF CENTRAL FIELD OF FORCE CHARGED PARTICLE IN ELECTROMAGNETIC FIELDS 7 1 ELECTROMAGNETIC FIELD IN CLASSICAL MECHANICS 7 2 ELECTROMAGNETIC FIELD IN QUANTUM MECHANICS 7 3 AHARONOV-BOHM EFFECT 7 4 EXERCISES AND SOLUTIONS 8 1 GENERAL CONCEPT 8 2 STATES OF SPIN 1/2 8 3 SPIN OPERATOR 8 4 MORE ABOUT EIGENSTATES OF SPIN 1/2 8 5 EXERCISES AND SOLUTIONS IDENTICAL PARTICLES 9 1 CHARACTERISTICS OF IDENTICAL PARTICLES 9 2 SYSTEM WITH TWO IDENTICAL PARTICLES 9 3 WAVEFUNCTIONS OF N IDENTICAL PARTICLE SYSTEM 9 3 1 THREE IDENTICAL PARTICLES 9 3 2 N IDENTICAL PARTICLE SYSTEM 9 4 SPIN STATES OF TWO ELECTRON SYSTEM 9 5 BELL BASIS AND ENTANGLED STATES 9 6 EXERCISES AND SOLUTIONS 9 6 1 CHARACTERISTICS OF IDENTICAL PARTICLES 9 6 2 TWO IDENTICAL PARTICLES 9 6 3 BELL BASIS AND ENTANGLED STATES 9 6 4 APPLICATIONS OF PAULI EXCLUSION THEOREM 10 TIME DEPENDENT PERTURBATION THEORY 10 1 THE PROBLEM 10 2 BASIC FORMULAS 10 3 PERIODIC PERTURBATION THEORY 10 3 1 SIMPLE CASE 10 3 2 GENERAL CASE 10 4 APPLICATIONS OF TRANSITION THEORY 10 4 1 ABSORPTION AND STIMULATING EMISSION OF LIGHT 10 4 2 SPONTANEOUS EMISSION 10 4 3 PERTURBATION IN FINITE TIME 10 4 4 SEMICONDUCTOR OPTICAL ABSORPTION 10 4 5 PHOTOEFFECT 10 4 6 SCATTERING 10 4 7 INTERNAL CONVERSION 10 5 GENERAL THEORY OF TRANSITION PROCESSES 10 5 1 OTHER EXAMPLES OF ENERGY NONCONSERVATION IN TRANSITION PROCESS 10 5 2 MORE GENERAL THEORY OF TRANSITION PROCESSES 10 5 3 QUANTUM ZENO EFFECT 10 6 EXERCISES AND SOLUTIONS REFERENCES CHAPTER BIRTH OF QUANTUM MECHANICS 1 1 DIF?CULTIES OF CLASSICAL PHYSICS 1 1 1 INTRODUCTION UNTIL THE END OF NINETEENTH CENTURY, CLASSICAL PHYSICS, INCLUDING CLASSICAL MECHANICS, CLASSICAL ELECTRODYNAMICS, THERMODYNAMICS, CLASSICAL ACOUSTICS, OPTICS, HYDROMECHANICS, AND CLASSICAL STATISTICAL PHYSICS, APPEARED TO BE SUF?CIENT TO EXPLAIN ALL PHYSICAL PHENOMENA DURING THE LATTER PART OF NINETEENTH CENTURY AND THE EARLY YEARS OF TWENTIETH CENTURY, THE EXPERIMENTAL FACTS MENTIONED IN SECTIONS 1 1 2 - 1 1 8 REVEALED FOUR DIF?CULTIES: (1) THE OBSERVED RADIOACTIVITY OF MATTER; (2) THE ENERGY OF HYDROGEN ATOM IS DISCRETE; (3) THE ELECTRON AND OTHER MASSY PARTICLES ARE OF WAVE PROPERTY, AND THE ELECTROMAGNETIC ?ELD HAS PARTICLE PROPERTY; (4) THE UNCERTAINTY RELATIONS BETWEEN COORDINATE AND MOMENTUM AND BETWEEN ENERGY AND TIME ALL THESE DIF?CULTIES REQUIRED NEW CONCEPTS AND IDEAS RADICALLY DIFFERENT FROM THOSE OF CLASSICAL PHYSICS OUTLINED BELOW ARE THE DISCOVERIES AND EVENTS THAT OCCURRED NEAR THE TURN OF THE CENTURY THAT REMOVED SOME ENIGMAS AND LED NATURALLY TO THE DEVELOPMENT OF THE QUANTUM MECHANICS AND QUANTUM ?ELD THEORY 1986 BACQUEREL RADIOACTIVE URANIUM 1901 MME CURIE RADIOACTIVE POLONIUM AND RADIUM (1903 NOBEL PRIZE) 1901 PLANCK BLACKBODY RADIATION (1918 NOBEL PRIZE) 1905 EINSTEIN PHOTOEFFECT (1921 NOBEL PRIZE) 1911 RUTHERFORD MODEL OF ATOM 1913 BOHR QUANTUM THEORY OF SPECTRA (1922 NOBEL PRIZE) 1913 STARK SPLITTING OF SPECTRAL LINE IN ELECTRIC ?ELD (1919 NOBEL PRIZE)