WORDS TO THE READER ABOUT HOW TO USE THIS TEXTBOOK
CONTENTS 1 QUANTIZATION OF A FREE PARTICLE IN N DIMENSIONAL SPACE 1 1 1 THE OPERATORS CORRESPONDING TO THE MEASUREMENT OF THE COMPONENTS OF POSITION 2 1 2 PARTIAL INTEGRATION AND AN AUXILIARY SEQUENCE 3 1 3 THE OPERATORS CORRESPONDING TO THE MEASUREMENT OF THE COMPONENTS OF MOMENTUM 7 1 4 THE HAMILTON OPERATOR GOVERNING FREE MOTION IN RN 9 1 5 TIME EVOLUTION GENERATED BY THE FREE HAMILTON OPERATOR 13 1 5 1 LARGE TIME ASYMPTOTICS OF THE EVOLUTION 14 1 5 2 TIME EVOLUTION AND CAUSALITY 16 1 5 3 FREE PROPAGATION OF GAUSSIANS 17 1 6 PERTURBATIONS OF THE FREE HAMILTON OPERATOR IN 3 SPACE DIMENSIONS 25 2 COMMUTATORS SYMMETRIES AND INVARIANCES 33 2 1 COMMUTING OPERATORS 33 2 2 CLOSED INVARIANT SUBSPACES OF OBSERVABLES 36 2 3 INSERT DECOMPOSITION OF SPECTRA OF DSLO S 40 2 4 QUANTIZATION OF ANGULAR MOMENTUM 43 2 4 1 A CHANGE OF REPRESENTATION INDUCED BY INTRODUCTION OF SPHERICAL COORDINATES 44 2 4 2 SPHERICAL HARMONICS 46 2 4 3 ANALYSIS OF THE REDUCED OPERATORS 49 2 4 4 THE OPERATOR CORRESPONDING TO THE SQUARE OF ANGULAR MOMENTUM 50 2 4 5 THE COMMUTING OF THE OPERATORS L3 AND L2 53 2 5 SYMMETRY AND INVARIANCE 53 SUBGROUPS 59 2 6 1 GENERATORS CORRESPONDING TO CONTINUOUS ONE PARAMETER 2 7 AN UNITARY REPRESENTATION OF ORTHOGONAL TRANSFORMATIONS IN EUCLIDEAN SPACE 63 2 7 1 AN UNITARY REPRESENTATION OF O N ON L2 C RN 65 2 7 2 GENERATORS CORRESPONDING TO ROTATIONS ABOUT THE COORDINATE AXES IN 3 SPACE DIMENSIONS 68 2 7 3 SYMMETRIES OF PERTURBATIONS OF THE FREE HAMILTON OPERATOR 70 2 8 AN ONE PARAMETER GROUP OF SYMMETRIES 74 2 9 GALILEAN INVARIANCE 75 2 10 GAUGE INVARIANCE 80 3 SIMPLE QUANTUM SYSTEMS IN 1 SPACE DIMENSION 85 3 1 AUXILIARY RESULTS ABOUT PERTURBATIONS OF THE FREE HAMILTON OPERATOR IN 1D 85 3 2 QUANTUM TUNNELING 91 3 3 POTENTIAL WELLS AND POTENTIAL BARRIERS 95 3 3 1 EIGENVALUES OF H FOR V0 0 100 3 3 2 MOTION IN A FUNCTION POTENTIAL 103 3 4 WEYL S LIMIT POINT LIMIT CIRCLE CRITERION 114 3 4 1 THE FREE HAMILTONIAN IN 1 SPACE DIMENSION A SIMPLE APPLICATION 116 3 5 FURTHER RESULTS ABOUT THE HAMILTON OPERATOR OF THE HARMONIC OSCILLATOR 117 3 6 MOTION IN A REPULSIVE P SCHL TELLER POTENTIAL AND RESONANCES 119 3 6 1 CALCULATION OF THE RESOLVENT AND THE RESONANCES OF A 120 3 6 2 CALCULATION OF THE FUNCTIONAL CALCULUS FOR A 128 4 MOTION IN A CENTRAL FORCE FIELD 133 4 1 A CHANGE OF REPRESENTATION INDUCED BY INTRODUCTION OF SPHERICAL COORDINATES 134 4 2 REDUCTION OF U H0U 1 135 4 3 ANALYSIS OF THE REDUCED OPERATORS 136 4 4 MOTION IN A COULOMB FIELD 137 4 4 1 AN APPLICATION OF WEYL S CRITERION 137 4 4 2 EXTENSIONS OF THE PURELY RADIAL OPERATOR H00 142 4 4 3 EIGENVALUES OF H 00E AND H LM FOR L M I 0 0 146 4 4 4 CORRESPONDING NORMALIZED EIGENFUNCTIONS 151 4 4 5 CONTINUOUS SPECTRUM OF H 00E AND H LM FOR L M I 0 0 153 4 4 6 THE SPECTRUM OF H 00E AND H LM FOR L M I 0 0 157 5 MOTION IN AN AXIALLY SYMMETRIC FORCE FIELD 161 5 1 A CHANGE OF REPRESENTATION INDUCED BY INTRODUCTION OF CYLINDRICAL COORDINATES 162 5 2 REDUCTION OF H0 164 5 3 MOTION OF A CHARGED PARTICLE IN A HOMOGENEOUS MAGNETIC FIELD 165 5 3 1 CONSTRUCTION OF A HILBERT BASIS OF L2 C 0 168 5 3 2 SYNTHESIS AND PROPERTIES OF THE HAMILTON OPERATOR H OF THE SYSTEM 182 5 3 3 CALCULATION OF THE TIME EVOLUTION GENERATED BY H 183 6 THE PATH INTEGRAL APPROACH TO QUANTUM MECHANICS 187 6 1 A CONNECTION BETWEEN THE SOLUTIONS OF THE SCHR DINGER AND THE HEAT EQUATION 191 7 CONCLUSION 195 REFERENCES 197 1 QUANTIZATION OF A FREE PARTICLE IN N DIMENSIONAL SPACE A BASIC CLASSICAL MECHANICAL SYSTEM IN N SPACE DIMENSIONS N N IS GIVEN BY A POINT PARTICLE OF MASS M 0 SOLELY INTERACTING WITH AN EXTERNAL POTENTIAL V THE CASE OF A VAN ISHING POTENTIAL CORRESPONDS TO A FREE PARTICLE IT IS TEMPTING TO QUALIFY THE CORRESPONDING SYSTEM AS SIMPLE IN THE SENSE THAT NOT MUCH CAN BE LEARNED FROM THIS SYSTEM ACCORDING TO CLASSICAL MECHANICS SINCE THERE IS NO EXTERNAL FORCE SUCH PARTICLES MOVE UNIFORMLY IN STRAIGHT LINES I E PERFORM GEODESIC MOTION IN EUCLIDEAN SPACE SO PARTICLES ARE SOMEHOW AWARE OF THE GEOMETRY OF THE SURROUNDING SPACE THIS IS NOT REALLY SIMPLE SIMILAR IS TRUE FOR QUANTUM MECHANICS THE HAMILTONIAN OF THE CORRESPONDING QUANTUM SYSTEM IS A MULTIPLE OF THE LAPLACE OPERATOR THAT IS ALSO TIED TO EUCLIDEAN GEOMETRY SIGNALING THE AWARENESS OF THE QUANTUM SYSTEM OF EUCLIDEAN GEOMETRY MORE CONCRETELY THE QUANTUM SYSTEM IS PHYSICALLY RELEVANT SINCE IT EXPLAINS EXPERIMENTS LIKE THE DAVISSON GERMER EXPERIMENT FROM 1927 20 BY CLINTON DAVISSON AND LESTER GERMER AT WESTERN ELECTRIC IN WHICH ELECTRONS SCAT TERED BY THE SURFACE OF A CRYSTAL OF NICKEL METAL DISPLAYED A DIFFRACTION PATTERN CON RMING THE HYPOTHESIS ADVANCED BY LOUIS DE BROGLIE IN 1924 OF WAVE PARTICLE DUALITY AN EXPER IMENTAL MILESTONE IN THE CREATION OF QUANTUM MECHANICS IN PHYSICS SUCH EXPERIMENTS ARE REFERRED AS DOUBLE SLIT EXPERIMENTS REFERRING TO THOMAS YOUNG S DOUBLE SLIT EXPERIMENT FROM 1801 78 THAT SHOWED THE WAVE CHARACTER OF LIGHT IN CERTAIN SITUATIONS MATHEMATICALLY IT MIGHT SURPRISE AS WE SHALL SEE LATER THAT THE HAMILTONIAN CORRESPONDING TO THE MOTION OF A CHARGED PARTICLE IN A COULOMB ELD CAN CONSIDERED A SMALL PERTURBATION OF THE FREE HAMILTONIAN I E IS A RELATIVELY BOUNDED PERTURBATION OF THE FREE HAMILTONIAN SEE SECT 1 6 1 ALTHOUGH NOT DIF CULT WE ARE NOT GOING TO GO THROUGH THE QUANTIZATION PROCESS FOR THIS PARTICULAR SYSTEM AGAIN BUT GIVE ONLY THE RESULTS FOR MORE DETAILS ABOUT THE PROCESS WE REFER THE READER TO 7 1 1 THE OPERATORS CORRESPONDING TO THE MEASUREMENT OF THE COMPONENTS OF POSITION IN A POSITION REPRESENTATION THE STATE SPACE OF THE CORRESPONDING QUANTUM SYSTEM IS GIVEN BY L2 C RN AND THE OPERATOR QK CORRESPONDING TO THE MEASUREMENT OF THE KTH K 1 N COMPONENT OF POSITION IS GIVEN BY THE MAXIMAL MULTIPLICATION OPERATOR TUK IN L2 C RN WITH THE KTH COORDINATE PROJECTION UK RN R DE NED BY UK U UK FOR ALL U U1 UN RN SEE THEOREMS 12 6 7 12 6 8 AND COROLLARY 12 6 9 IN THE APPENDIX THE QUANTITY 0 IS A CONSTANT WITH DIMENSION 1 LENGTH THAT IS GOING TO BE LEFT UNSPECI ED IN THE FOLLOWING BUT IS NOT GOING TO AFFECT THE PHYSICAL RESULTS 1 ITS SPECTRUM CONSISTS OF ALL REAL NUMBERS AND IS PURELY ABSOLUTELY CONTINUOUS FURTHER FOR ANY F L2 C RN WITH NORM 1 THE CORRESPONDING SPECTRAL MEASURE QK F IS GIVEN BY QK F I U RN UK I F 2 DVN FOR EVERY BOUNDED INTERVAL I OF R THE QUANTITY QK F I GIVES THE PROBABILITY IN A POSITION MEASUREMENT OF NDING THE K TH COORDINATE TO BE IN THE RANGE I IF THE PARTICLE IS IN THE STATE C F FURTHER FOR EVERY BOUNDED AND UNIVERSALLY MEASURABLE FUNCTION F R C F QK T F UK 1 1 WHERE T F UK IS THE MAXIMAL MULTIPLICATION OPERATOR WITH THE FUNCTION F UK DE NED BY T F UK G F UK G FOR EVERY G L2 C RN WE EXPECT THAT MEASUREMENT OF THE K TH COORDINATE IS INDEPENDENT OF THE MEASUREMENT OF THE LTH COORDINATE IF K L INDEED THIS IS TRUE SINCE THE CORRESPONDING OPERATORS COM MUTE SEE SECT 2 1 FOLLOWING THE LAWS OF PROBABILITY FOR INDEPENDENT RANDOM VARIABLES THE PROBABILITY OF NDING THE POSITION OF THE PARTICLE TO BELONG TO A BOX I1 IN WHERE I1 IN ARE INTERVALS IN R IS GIVEN BY THE PRODUCT OF THE PROBABILITIES OF NDING 1 IT IS REASONABLE TO USE THE INVERSE OF THE COMPTON WAVE LENGTH MC H WHERE C DENOTES THE SPEED OF LIGHT IN VACUUM ON THE OTHER HAND C IS A FOREIGN OBJECT IN A NON RELATIVISTIC THEORY LIKE A NATURAL DE BROGLIE WAVE LENGTH H MV DOES NOT EXIST FOR A FREE PARTICLE BECAUSE THERE IS NO NATURAL SPEED V FOR THE SYSTEM THIS WOULD BE DIFFERENT IF THE PARTICLE WOULD BE CON NED TO A NITE SPACE QUANTUM MECHANICS THEORY 2