MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

662 GREEN'S FUNCTIONS AND PATH INTEGRALS We can write the free particle propagator in terms of the phase space path integral as BpDxexp { d~ [p2 - f] } . (20.167) s K(x, t, zo, to) = fl C[zo,to;z,t] After we take the momentum integral and after putting all the new constants coming into D, Equation (20.167) becomes Dxexp { l: d~ ( imX2)}. (20.168) s K(z, t, 50, to) = fl c [zo $0; =, tl We can convert this into a Wiener path integral by the t + -it rotation, and after evaluating it, we return to real time to obtain the propagator as (20.169) i m(z - xo)2 2(t -to) . exp - 1 K(x, t, zo, to) = 4 27Tiqt -to)/m We conclude by giving the following useful rules for path integrals with For the pinned Wiener measure: N + 1 segments [Eq (20.15)]: For the unpinned Wiener measure: Also, N+l (20.172) PROBLEMS 663 Problems 20.1 Show that satisfies the normalization condition 00 &W(z, t, xo, to) = 1. L 20.2 equation is true: By differentiating both sides with respect to t show that the following 20.3 Show that V(z) in Equation (20.64): is defined as 1 1 dF(x) 4q2 D 27 dx . V(x) = -F2(x) + 20.4 Show that the propagator satisfies the ESKC relation [Eq. (20.14)]. 20.5 Derive equation 664 GREEN’S FUNCTlONS AND PATH 1NTEGRALS given in Section 20.3. 20.6 integral Using the semiclassical method show that the result of the Wiener W(z,t, 20, to) = 1 d,z(T) exp { -k2 lot dm2 } C[lO,O;W] is given as (22 + 2;) cosh(2kJiS(t - to)) - 2202 2v%sinh(Zkfi(t -to)) 20.7 By diagonalizing the real synimetric matrix, A, show that 20.8 Use the formula dqexp { -a(q - v’)~ - b(q - v”)~} I-, to evaluate the integral 20.9 propagator Equation (20.164): By taking the momentum integral in Equation (20.159) derive the where S is given as N S=C. 1 =o Ref e re n ces . . Akhiezer, N.I., The Calculus of Variations, Blaisdell, New York, 1962 Arfken, G. B., and H. J. Weber, Essential Mathematical Methods for Physicists, Academic Press, 2003. Arfken, G. B., and H. J. Weber, Mathematical Methods of Physics, Aca- demic Press, sixth edition, 2005. Artin, E., The Gamma Function, Holt, Rinehart and Winston, New York, 1964. . . . . Beiser, A., Concepts of Modern Physics, McGraw-Hill, sixth edition, 2002. Bell, W.W., Special Functions for Scientists and Engineers, Dover Publi- cations, 2004. Bluman, W. B., and Kumei, S., Symmetries and Differential Equations, Springer Verlag, New York, 1989. Boas, M.L., Mathematical Methods in the Physical Sciences, Wiley, third edition, 2006. . Bradbury, T.C., Theoretical Mechanics, Wiley, international edition, 1968. . Bromwich, T. J.I., Infinite Series, Chelsea Publishing Company, 1991. 665 666 REFERENCES . Brown, J.W., and R.V. Churchill, Complex Variables and Applications, McGraw-Hill, New York, 1995. Butkov, E., Mathematical Physics, Addison-Wesley, New York, 1968. Byron, W. Jr., and R.W. Fuller, Mathematics of Classical and Quantum Physics, Dover Publications, New York, 1970. Chaichian, M., and A. Dernichev, Path Integrals in Physics, Volume I and 11, Institute of Physics Publishing, 2001. Churchill, R.V., Fourier Series and Boundary Value Problems, McGraw- Hill, New York, 1963. Courant, E., and D. Hilbert, Methods of Mathematical Physics, Volume I and 11, Wiley, New York, 1991. Dennery, P., and A. Krzywicki, Mathematics for Physics, Dover Publica- tions, New York, 1995. Doniach, S., and E.H. Sondheimer, Green’s hnctions for Solid State Physics, World Scientific, 1998. Dwight, H.B., Tables of Integrals and Other Mathematical Data, Prentice Hall, fourth edition, 1961. Erdelyi, A., Asymptotic Expansions, Dover Publications, New York, 1956. Erdelyi, A., Oberhettinger, M.W., and Tricomi. F.G., Higher Tmnscen- dental Functions, Krieger, vol. I, New York,1981. Feynman, R., R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Addison-Wesley, 1966. Feynman, R., and Hibbs, A.R., Quantum Mechanics and Path Integrals, McGraw-Hill, 1965. Gantmacher, F.R., The Theory of Matrices, Chelsea Publishing Company, New York, 1960. Gluzman, S., and D. Sornette, Log Periodic Route to Fractal Functions, Physical Review, E65, 036142, (2002). Goldstein, H., C. Poole, and J. Safko, Classical Mechanics, Addison- Wesley, third edition, 2002. Hamermesh, M., Group Theory and its Application to Physical Problems, Addison-Wesley, 1962. Hartle, J.B., An Introduction to Einstein’s General Relativity, Addison- Wesley, 2003. . . . . . . . . . . . . . . . REFERENCES 667 Hassani, S., Mathematical Methods: for Students of Physics and Related Fields, Springer Verlag, 2000. Hassani, S., Mathematical Physics, Springer Verlag, second edition, 2002. Hildebrand, F.B., Methods of Applied, Mathematics, Dover Publications, second reprint edition, 1992. Hilfer, R., Applications of Fkactional Calculus, World Scientific, 2000. Hydon, P. E., Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge, 2000. Ince, E.L., Ordinary Digerential Equations, Dover Publications, New York, 1958. Infeld, L., and T.E. Hull, The Factorization Method, Reviews of Modern Physics, 23, 21-68 (1951). Jackson, J.D., Classical Electrodynamics, Wiley, third edition, 1999. Jacobson, T.A., and R. Parentani, An Echo of Black Holes, Scientific American, December, 48-55 (2005). Kaplan, W., Advanced Calculus, Addison- Wesley, New York, 1973. Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets, World Scientific, third edition, 2003. Lamhrecht, A., The Casimir Effect: World, September, 29-32 (2002). Lebedev, N.N., Special Functions and their Applications, Prentice-Hall, 1965. Lebedev, N.N., I.P. Skolskaya, and Uflyand, Problems of Mathematical Physics, Prentice-Hall, Englewood Cliffs, 1965. Marion, J. B., Classical Dynamics of Particles and Systems, Academic Press, second edition, 1970. Mathews, J., and R.W. Walker, Mathematical Methods of Physics, Addison-Wesley, Marlo Park, second edition, 1970. McCollum, P.A., and B.F. Brown, Laplace Il-ansform Tables and Theo- rems, Holt, Rinehart and Winston, New York, 1965. Milton, K.A., The Casimir Effect, World Scientific, 2001. Morse, P. M., and H. Feshbach, Methods of Theoretical Physics, McGraw- Hill, 1953. A Force from Nothing, Physics 668 REFERENCES Oldham, B.K., and J. Spanier, The Fractional Calculus, Academic Press, 1974. Osler, T.J., Leibniz Rule for Fractional Derivatives and an Application to Infinite Series, SIAM, Journal of Applied Mathematics, 18, 658-674 (1970). Osler, T.J., The Integral Analogue of the Leibniz Rule, Mathematics of Computation, 26, 903-915 (1972). Pathria, R. K., Statistical Mechanics, Pergamon Press, 1984. Podlubny, I., Fractional Differential Equations, Academic Press, 1999. 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Index: Abel test, 444 Abel’s formula, 569 Absolute convergence, 432 Active and passive views, 174 Addition of velocities, 201 Addition theorem spherical harmonics, 264 Advanced Green’s functions, 624 Algebra of vectors, 164 Alternating series Leibniz rule, 439 Analytic continuation, 350 A naIytic functions Angular momentum, 116 Cauchy-Riemann conditions, 297 factorization method, 143 quantum mechanics, 249 Angular momentum operators eigenvalue equations matrix elements quantum mechanics, 255 quantum mechanics, 257 Argument, 294 Associated Laguerre polynomials, 45, 51 generating function, 52 orthogonality and completeness, recursion relations, 53 Rodriguez formula, 53 53 Associated Legendre equation, 13, factorization method, 137 Associated Legendre polynomials, 28 30 31 orthogonality and completeness, Asymptotic series, 462 Bernoulli numbers, 453 Bernoulli periodic function, 454 Bernoulli polynomials Bessel functions generating function, 453 boundary conditions, 91 channel waves factorization method, 155 tsunamis, 93 669 670 lNDEX first kind, 86 flexible chain problem, 92 generating functions, 89 integral definitions, 90 modified Bessel functions, 88 orthogonality and completeness, recursion relations, 90 second kind, 87 spherical, 88 third kind, 87 Wronskians, 95 Laplace transforms, 507 90 Bessel's equation, 86 Beta function, 362 Binomial coefficient, 447 Binomial formula Binomial theorem, 447 Bloch equation, 640 Bohr energy levels, 45 Boosts Boundary conditions relativistic energy, 447 Lorentz transformation, 244 Dirichlet, 109 Green's functions, 572, 594 Hermitian operators, 110 inhomogeneous Green's functions, 575 Neumann, 109 single point Green's functions, 572 Sturm-Liouville system, 108 unmixed mixed, 109 Branch cut Branch line, 306 Branch point, 306 Bromwich integral Riemann sheet, 306 inverse Laplace transform Laplace transform, 492 Caputo derivative, 429 Cartesian coordinates, 163 Cartesian tensors, 178 contraction, 179 pseudotensor rank, 178 trace, 179 Casimir effect, 466 MEMS, 468 Cauchy formula, 388 Cauchy integral formula fractional derivative, 390 Cauchy integral theorem, 336 Cauchy principal value, 365 Cauchy theorem, 339 convergence tests, 435 Cauchy-Goursat theorem, 335 Cauchy-Riemann conditions, 297 Chebyshev equation, 75 second kind, 76 Chebyshev polynomials first kind, 75, 76 Gegenbauer polynomials, 76 generating function, 78 orthogonality and completeness, second kind, 76 tensor density, 180 78 another definition, 78 Chebyshev series Raabe test, 437 Christoffel symbols first kind, 192 second kind, 192 Commutation relations angular momentum, 249 Completeness of eigenfunctions, 276 Complex algebra, 293 Complex conjugate, 295 Complex derivative, 296 Complex functions, 295 Complex numbers argument, 294 conjugate, 295 modulus, 294 Complex plane, 294 Complex techniques INDEX 671 definite integrals, 352 Conditional convergence, 432 Abel test, 444 Condon-Shortley phase, 140 spherical harmonics, 34 Confluent Gauss equation, 104 Conformal mappings, 313 electrostatics, 3 14 fluid mechanics, 318 Conjugate harmonic functions, 299 Continuous groups generators, 278 Lie groups, 224, 278 Continuous random walk fractional derivatives, 424 Contour integral complex, 335 Contour integral techniques, 352 Contour integrals special functions, 369 Contraction of indices, 188 Contravariant/covariant components, 182 Convergence absolu te Convergence tests conditional, 432 Cauchy root test, 433 comparison, 433 Gauss test, 436 integral test, 434 Raabe test, 435 ratio test, 433 Fourier transforms, 485 Laplace transforms, 498 Convolution theorem Covariance, 197 Covariant divergence, 194 Covariant/contravariant components, 182 nents, 186 contravariant/covariant compe Curl, 193 Cut line, 306 d’Alembert operator, 72, 209, 215, 473,619 De Moivre’s formula, 295 Derivative Derivative and integral 385 Differential equations 550 n-fold, 382 unification for integer orders, conversion to integral equations, Differentiation of vectors, 166 Differintegrals composition, 400 CTRW dependence on the lower limit, evaluation of definite integrals, extraordinary differential equa- Fokker-Planck equation, 427 heat transfer equation, 415 homogeneity, 399 Leibniz rule, 407 linearity, 399 properties, 399 right and left handed, 407 scale transformation, 400 semidifferential equations, 419 series, 400 some examples, 409 special functions, 424 techniques, 413 Diffusion equation, 379 Brownian motion path integrals, 633 Feynman-Kac formula, 639 Fourier transforms, 488 propagator, 610 Dipoles, 23 Dirac-Delta function, 481 Direction cosines, 167 Divergence, 194 Brownian motion, 424 408 421 tions, 417 [...]... Problems 10 7 10 7 I 08 11 0 11 0 11 1 11 1 11 2 11 3 11 4 11 5 11 8 SYSTEMSand the FACTORIZATION 9 STURM-LIOUVILLE METHOD 12 1 9 .1 Another Form for the Sturm-Liouville Equation 12 2 9.2 Method of Factorization 12 3 9.3 Theory of Factorization and the Ladder Operators 12 4 9.4 Solutions via the Factorization Method 13 0 9.4 .1 Case I ( m > 0 and p ( m ) is a n increasing function) 13 0 1 9.4.2 Case 1 m > 0 and p ( m... ) 269 11 .13 Relation of S U ( 2 ) and R(3) 11 .14 Group Spaces 272 11 .14 .1 Real Vector Space 272 11 .14 .2 Inner Product Space 273 11 .14 .3 Four- Vector Space 274 11 .14 .4 Complex Vector Space 274 11 .14 .5 Function Space and Hilbert Space 274 11 .14 .6 Completeness of the Set of Eigenfunctions 275 (urn (XI} 11 .15 Hilbert Space and Quantum Mechanics 276 11 .16 Continuous Groups and Symmetries 277 11 .16 .1 One-Parameter... Spherical Harmonics 257 11 .11 .7 Matrix Elements ofL,,Lv, and L, 11 .11 .8 Rotation Matrices for the Spherical 258 Harmonics 260 11 .11 .9 Evaluation of the di,m(,f?) Matrices 2 61 11. 19 .10 Inverse of the d i r m ( p ) Matrices 262 11 .11 .11 Differential Equation f o r dk,m(/3) 11 .11 .12 Addition Theorem f o r Spherical Harmonics 264 11 .11 .13 Determination of I, in the Addition Theorem 266 268 11 .12 Irreducible Representations... Fields 10 .10 .14 Maxwell’s Equations in Terms of Potentials 10 .10 .15 Covariance of Newton’s Dynamical Theory Problems 11 CONTINUOUS G R O U P S and R E P R E S E N T A T I O N S 11 .1 Definition of a Group 11 .1. 1 Terminology 11 .2 Infinitesimal Ring or Lie Algebra 11 .3 Lie Algebra of the Rotation Group R(3) 11 .3 .1 Another Approach to T R ( 3 ) 11 .4 Group Invariants 11 .4 .1 Lorentz Transformation 11 .5 Unitary... 208 209 21 1 213 214 215 216 223 224 224 226 227 228 2 31 232 234 236 237 239 24 1 24 6 24 7 24 7 248 248 249 249 CONTENTS xi 250 11 .11 .2 Rotation of the Physical System 11 .11 .3 Rotation Operator in Terms of the Euler 2 51 Angles 11 .11 .4 Rotation Operator in Terms of the 2 51 Original Coordinates 255 1 1 .1 1.5 Eigenvalue Equations f o r L,, L k , and L2 11 .11 .6 Generalized Fourier Expansion in 255 Spherical... Tensor Densities 19 0 10 .9.7 Diflerentiation of Tensors 19 3 10 .9.8 Some Covariant Derivatives 19 5 10 .9.9 Riemann Curvature Tensor 19 6 10 .9 .10 Geodesics 19 7 10 .9 .11 Invariance and Covariance 19 7 10 .10 Spacetime and Four- Tensors 19 7 10 .10 .1 Minkowski Spacetime 10 .10 .2 Lorentz Transformation and the Theory 19 9 of Special Relativity 8 01 10.20.3 T i m e Dilation and Length Contraction 2 01 10 .10 .4 Addition... Velocities 10 C O O R D I N A T E Sand T E N S O R S x CONTENTS 10 .10 .5 Four- Tensors in Minkowski Spacetime 10 .10 .6 Four- Velocity 10 .10 .7 Four-Momentum and Conservation Laws 10 .10 .8 Mass of a Moving Particle 10 .10 .9 Wave Four- Vector 10 .10 .10 Derivative Operators in Spacetime 10 .10 .11 Relative Orientation of Axes in R and K Frames 10 .10 .12 Maxwell’s Equations in Minkowski Space time 10 .10 .13 Transformation... Asymptotic Series 15 .10 Divergent Series in Physics 15 .10 .1 Casimir Eflect and Renormalization 15 .10 .2 Casimir Egect and MEMS 15 .11 Infinite Products 15 .11 .1 Sine, Cosine, and the Gamma Functions Problems 16 I N T E G R A L T R A NSFORMS 16 1 Some Commonly Encountered Integral Transforms 478 16 .2 Derivation of the Fourier Integral 16 .2 .1 Fourier Series 16 .2.2 Dirac-Delta Function 16 .3 Fourier and Inverse Fourier... Constant 409 14 .5.2 Differintegral of [x- u] 410 14 .5.3 Differintegral of [x- u ] p ( p > -1) 411 14 .5.4 Differintegral of [I -XI* 412 14 .5.5 Diflerintegral of exp(fx) 412 14 .5.6 Differintegral of In( x) 412 14 .5.7 Some Semiderivatives and Semi-integrals 413 Mathematical Techniques with Differintegrals 413 14 .6 .1 Laplace Transform of Differintegrals 413 14 .6.2 Extraordinary Diflerential Equations 427 14 .6.3... Tensors 10 .8 .1 Contravariant and Covariant Components 18 1 18 3 10 .8.2 Metric Tensor and the Line Element 10 .8.3 Geometric Interpretation of Covariant 18 6 and Contravariant Components 18 8 10 .9 Operations with General Tensors 18 8 10 .9 .1 Einstein Summation Convention 18 8 10 .9.2 Contraction of Indices 18 9 10 .9.3 Multiplication of Tensors 18 9 10 .9.4 The Quotient Theorem 18 9 10 .9.5 Equality of Tensors 18 9 10 .9.6 . equation, 14 double root, 45 roots, 16 cosine function, 4 71 gamma function, 4 71 sine function, 470 Infinite series convergence, 4 31 Infinitesimal ring Lie algebra, 226 Infinitesimal. Cartesian, 17 8 covariant divergence, 19 4 Tensor density, 17 9, 18 9 Tensors INDEX 679 covariant gradiant, 19 3 curl, 19 3 differentiation, 19 1 equality, 18 9 general, 18 1 Laplacian, 19 4. states, 6 01 factorization method single electron atom, 15 1 Feynman path integral, 658 Green’s function, 615 propagator free particle, 615 Schur’s lemma, 247 Schwartz inequality, 11 8 Schwaris-Cauchy