GREEN'S FUNCTIONS IN PHYSICS
CONTENTS THE VIBRATING STRING THE STRING FORCES ON THE STRING EQUATIONS OF MOTION FOR A MASSLESS STRING EQUATIONS OF MOTION FOR A MASSIVE STRING THE LINEAR OPERATOR FORM BOUNDARY CONDITIONS CASE 1: A CLOSED STRING CASE 2: AN OPEN STRING LIMITING CASES INITIAL CONDITIONS SPECIAL CASES NO TENSION AT BOUNDARY SEMI-IN?NITE STRING OSCILLATORY EXTERNAL FORCE SUMMARY REFERENCES GREEN’S IDENTITIES GREEN’S 1ST AND 2ND IDENTITIES USING GI #2 TO SATISFY RBC THE CLOSED STRING THE OPEN STRING A NOTE ON HERMITIAN OPERATORS ANOTHER BOUNDARY CONDITION PHYSICAL INTERPRETATIONS OF THE GIS THE PHYSICS OF GREEN’S 2ND IDENTITY A NOTE ON POTENTIAL ENERGY THE PHYSICS OF GREEN’S 1ST IDENTITY GREEN’S FUNCTIONS THE PRINCIPLE OF SUPERPOSITION THE DIRAC DELTA FUNCTION TWO CONDITIONS CONDITION 1 CONDITION 2 APPLICATION OPEN STRING THE FORCED OSCILLATION PROBLEM FREE OSCILLATION REFERENCE PROPERTIES OF EIGEN STATES EIGEN FUNCTIONS AND NATURAL MODES A CLOSED STRING PROBLEM THE CONTINUUM LIMIT SCHR¨ODINGER’S EQUATION NATURAL FREQUENCIES AND THE GREEN’S FUNCTION GF BEHAVIOR NEAR ? = ?N RELATION BETWEEN GF & EIG FN CASE 1: ? NONDEGENERATE CASE 2: ?N DOUBLE DEGENERATE SOLUTION FOR A FIXED STRING A NON-ANALYTIC SOLUTION THE BRANCH CUT ANALYTIC FUNDAMENTAL SOLUTIONS AND GF ANALYTIC GF FOR FIXED STRING GF PROPERTIES THE GF NEAR AN EIGENVALUE DERIVATION OF GF FORM NEAR EVAL RECONSIDER THE GEN SELF-ADJOINT PROBLEM SUMMARY, INTERP & ASYMPTOTICS GENERAL SOLUTION FORM OF GF ?-FN REPRESENTATIONS & COMPLETENESS EXTENSION TO CONTINUOUS EIGENVALUES ORTHOGONALITY FOR CONTINUUM 410 EXAMPLE: IN?NITE STRING 4101 THE GREEN’S FUNCTION 4102 UNIQUENESS 4103 LOOK AT THE WRONSKIAN 4104 SOLUTION 4105 MOTIVATION, ORIGIN OF PROBLEM 411 SUMMARY OF THE IN?NITE STRING 412 THE EIGEN FUNCTION PROBLEM REVISITED 413 SUMMARY 414 REFERENCES STEADY STATE PROBLEMS OSCILLATING POINT SOURCE THE KLEIN-GORDON EQUATION CONTINUOUS COMPLETENESS THE SEMI-IN?NITE PROBLEM A CHECK ON THE SOLUTION STEADY STATE SEMI-IN?NITE PROBLEM THE FOURIER-BESSEL TRANSFORM DYNAMIC PROBLEMS ADVANCED AND RETARDED GF’S PHYSICS OF A BLOW SOLUTION USING FOURIER TRANSFORM INVERTING THE FOURIER TRANSFORM SUMMARY OF THE GENERAL IVP ANALYTICITY AND CAUSALITY THE IN?NITE STRING PROBLEM DERIVATION OF GREEN’S FUNCTION PHYSICAL DERIVATION SEMI-IN?NITE STRING WITH FIXED END SEMI-IN?NITE STRING WITH FREE END ELASTICALLY BOUND SEMI-IN?NITE STRING 610 RELATION TO THE EIGEN FN PROBLEM 6101 ALTERNATIVE FORM OF THE GR PROBLEM 611 COMMENTS ON GREEN’S FUNCTION 6111 CONTINUOUS SPECTRA 6112 NEUMANN BC 6113 ZERO NET FORCE 612 SUMMARY 613 REFERENCES SURFACE WAVES AND MEMBRANES INTRODUCTION ONE DIMENSIONAL SURFACE WAVES ON FLUIDS THE PHYSICAL SITUATION SHALLOW WATER CASE TWO DIMENSIONAL PROBLEMS EXAMPLE: 2D SURFACE WAVES EXTENSION TO N-DIMENSIONS REGIONS OF INTEREST EXAMPLES OF N-DIMENSIONAL PROBLEMS GENERAL RESPONSE NORMAL MODE PROBLEM FORCED OSCILLATION PROBLEM GREEN’S FIRST IDENTITY GREEN’S SECOND IDENTITY CRITERION FOR HERMITIAN L0 THE RETARDED PROBLEM GENERAL SOLUTION OF RETARDED PROBLEM THE RETARDED GREEN’S FUNCTION IN N-DIM REDUCTION TO EIGENVALUE PROBLEM REGION R INTERIOR EXTERIOR THE METHOD OF IMAGES EIGENFUNCTION METHOD METHOD OF IMAGES CYLINDRICAL PROBLEMS COORDINATES DELTA FUNCTION GF PROBLEM FOR CYLINDRICAL SYM EXPANSION IN TERMS OF EIGENFUNCTIONS PARTIAL EXPANSION SUMMARY OF GF FOR CYL SYM USES OF THE GF GM(R, R?; ?) EIGENFUNCTION PROBLEM NORMAL MODES/NORMAL FREQUENCIES THE STEADY STATE PROBLEM FULL TIME DEPENDENCE THE WEDGE PROBLEM GENERAL CASE SPECIAL CASE: FIXED SIDES THE HOMOGENEOUS MEMBRANE THE RADIAL EIGENVALUES THE PHYSICS 10 HEAT CONDUCTION 101 INTRODUCTION 1011 CONSERVATION OF ENERGY 1012 BOUNDARY CONDITIONS CONTENTS 102 THE STANDARD FORM OF THE HEAT EQ 1021 CORRESPONDENCE WITH THE WAVE EQUATION 1022 GREEN’S FUNCTION PROBLEM 1023 LAPLACE TRANSFORM 1024 EIGEN FUNCTION EXPANSIONS 103 EXPLICIT ONE DIMENSIONAL CALCULATION 1031 APPLICATION OF TRANSFORM METHOD 1032 SOLUTION OF THE TRANSFORM INTEGRAL 1033 THE PHYSICS OF THE FUNDAMENTAL SOLUTION 1034 SOLUTION OF THE GENERAL IVP 1035 SPECIAL CASES 104 SUMMARY 105 REFERENCES 11 SPHERICAL SYMMETRY 111 SPHERICAL COORDINATES 112 DISCUSSION OF L?? 113 SPHERICAL EIGENFUNCTIONS 1131 REDUCED EIGENVALUE EQUATION 1132 DETERMINATION OF UM L (X) 1133 ORTHOGONALITY AND COMPLETENESS OF UM L (X) 114 SPHERICAL HARMONICS 1141 OTHONORMALITY AND COMPLETENESS OF Y M L 115 GF’S FOR SPHERICAL SYMMETRY 1151 GF DI?ERENTIAL EQUATION 1152 BOUNDARY CONDITIONS 1153 GF FOR THE EXTERIOR PROBLEM 116 EXAMPLE: CONSTANT PARAMETERS 1161 EXTERIOR PROBLEM 1162 FREE SPACE PROBLEM 117 SUMMARY 118 REFERENCES 12 STEADY STATE SCATTERING 121 SPHERICAL WAVES 122 PLANE WAVES 123 RELATION TO POTENTIAL THEORY CONTENTS 124 SCATTERING FROM A CYLINDER 125 SUMMARY 126 REFERENCES 13 KIRCHHO?’S FORMULA 131 REFERENCES 14 QUANTUM MECHANICS 141 QUANTUM MECHANICAL SCATTERING 142 PLANE WAVE APPROXIMATION 143 QUANTUM MECHANICS 144 REVIEW 145 SPHERICAL SYMMETRY DEGENERACY 146 COMPARISON OF CLASSICAL AND QUANTUM 147 SUMMARY 148 REFERENCES 15 SCATTERING IN 3-DIM 151 ANGULAR MOMENTUM 152 FAR-FIELD LIMIT 153 RELATION TO THE GENERAL PROPAGATION PROBLEM 154 SIMPLI?CATION OF SCATTERING PROBLEM 155 SCATTERING AMPLITUDE 156 KINEMATICS OF SCATTERED WAVES 157 PLANE WAVE SCATTERING 158 SPECIAL CASES 1581 HOMOGENEOUS SOURCE; INHOMOGENEOUS OBSERVER 1582 HOMOGENEOUS OBSERVER; INHOMOGENEOUS SOURCE 1583 HOMOGENEOUS SOURCE; HOMOGENEOUS OBSERVER 1584 BOTH POINTS IN INTERIOR REGION 1585 SUMMARY 1586 FAR FIELD OBSERVATION 1587 DISTANT SOURCE: R? ?? 159 THE PHYSICAL SIGNI?CANCE OF XL 1591 CALCULATING ?L(K) 1510SCATTERING FROM A SPHERE 15101A RELATED PROBLEM CONTENTS 1511CALCULATION OF PHASE FOR A HARD SPHERE 1512EXPERIMENTAL MEASUREMENT 15121CROSS SECTION 15122NOTES ON CROSS SECTION 15123GEOMETRICAL LIMIT 1513OPTICAL THEOREM 1514CONSERVATION OF PROBABILITY INTERPRETATION: 15141HARD SPHERE 1515RADIATION OF SOUND WAVES 15151STEADY STATE SOLUTION 15152FAR FIELD BEHAVIOR 15153SPECIAL CASE 15154ENERGY FLUX 15155SCATTERING FROM PLANE WAVES 15156SPHERICAL SYMMETRY 1516SUMMARY 1517REFERENCES 16 HEAT CONDUCTION IN 3D 161 GENERAL BOUNDARY VALUE PROBLEM 162 TIME DEPENDENT PROBLEM 163 EVALUATION OF THE INTEGRALS 164 PHYSICS OF THE HEAT PROBLEM 1641 THE PARAMETER ? 165 EXAMPLE: SPHERE 1651 LONG TIMES 1652 INTERIOR CASE 166 SUMMARY 167 REFERENCES 17 THE WAVE EQUATION 171 INTRODUCTION 172 DIMENSIONALITY 1721 ODD DIMENSIONS 1722 EVEN DIMENSIONS 173 PHYSICS 1731 ODD DIMENSIONS CONTENTS 1732 EVEN DIMENSIONS 1733 CONNECTION BETWEEN GF’S IN 2 & 3-DIM 175 SUMMARY 176 REFERENCES 18 THE METHOD OF STEEPEST DESCENT 181 REVIEW OF COMPLEX VARIABLES 182 SPECI?CATION OF STEEPEST DESCENT 183 INVERTING A SERIES 184 EXAMPLE 1: EXPANSION OF ?–FUNCTION 1841 TRANSFORMING THE INTEGRAL 1842 THE CURVE OF STEEPEST DESCENT 185 EXAMPLE 2: ASYMPTOTIC HANKEL FUNCTION 186 SUMMARY 187 REFERENCES 19 HIGH ENERGY SCATTERING 191 FUNDAMENTAL INTEGRAL EQUATION OF SCATTERING 192 FORMAL SCATTERING THEORY 1921 A SHORT DIGRESSION ON OPERATORS 193 SUMMARY OF OPERATOR METHOD 1932 BORN APPROXIMATION 194 PHYSICAL INTEREST 1941 SATISFYING THE SCATTERING CONDITION 195 PHYSICAL INTERPRETATION 196 PROBABILITY AMPLITUDE 197 REVIEW 198 THE BORN APPROXIMATION 1981 GEOMETRY 1982 SPHERICALLY SYMMETRIC CASE 1983 COULOMB CASE 199 SCATTERING APPROXIMATION 1910PERTURBATION EXPANSION 19101PERTURBATION EXPANSION 19102USE OF THE T-MATRIX CONTENTS 1911SUMMARY 1912REFERENCES CHAPTER MOST OF THE PHYSICAL INTERPRETATIONS OF SMALL VIBRATIONS IN THIS CHAPTER WE WILL INVESTIGATE THE MATHEMATICS OF THIS PROBLEM BY DETERMINING THE EQUATIONS OF MOTION 11 THE STRING WRITE (UI ?UI?1) A = ?U ?X I (12) THE EQUATIONS OF MOTION BECOME (AFTER DIVIDING BOTH SIDES BY ?X) ??U(X, T) ?X (17) 12 THE LINEAR OPERATOR FORM SO THAT ?U ?X I+1 ??I ?U ?X I # = TO OBTAIN A UNIQUE SOLUTION FOR THE DI?ERENTIAL EQUATION, WE MUST PLACE RESTRICTIVE CONDITIONS ON IT IN THIS CASE WE PLACE CONDITIONS ON THE ENDS OF THE STRING EITHER THE STRING IS TIED TOGETHER (IE CLOSED), OR ITS ENDS ARE LEFT APART (OPEN) HA(T) ?FA ?A AND ?A ?KA ?A 13 BOUNDARY CONDITIONS [ˆNS · ?+ ?S]U(X, T) = HS(T) = 1 U(A, T) = U(B, T), D DXU(A, T) = D DXU(B, T) (PERIODIC), OR 2 [ˆNS · ?+ ?S]U(X, T) = SPECIAL CASES THIS MATERIAL WAS ORIGINALLY IN CHAPTER CONDITION LEADING TO THE HELMHOLTZ EQUATION THE CONDITIONS ?RST TWO CASES WILL ENSURE THAT THE RIGHT-HAND SIDE OF GREEN’S SECOND IDENTITY (INTRODUCED IN CHAPTER 2) VANISHES THIS IS NECESSARY FOR A PHYSICAL SYSTEM 14 SPECIAL CASES EQUATION IN CHAPTER 3 # U(X, T) = ?(X)F(X, T) ON WHERE L0U = " ?? ?X ?(X) ? ?X ! + V (X) # U 2 REGULAR BOUNDARY CONDITIONS REFER TO THE BOUNDARY CONDITIONS FOR EITHER (A) A CLOSED STRING: U(A, T) = U(B, T) (CONTINUOUS) ?U(A, T) ?X X=A = ?U(B, T) ?X X=B (NO BENDS) OR (B) AN OPEN STRING: [ˆNS · ?+ ?S]U(X, T) = HS(T) = X ON S, ALL T 3 THE INITIAL CONDITIONS ARE GIVEN BY THE EQUATIONS U(X, T)|T=0 = U0(X) FOR A < X < B (127) 16 REFERENCES