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 1 Kx, t, zo, to = 4 27Tiqt - t o / m We conclude by givi

662 GREEN'S FUNCTIONS AND PATH INTEGRALS

We can write the free particle propagator in terms of the phase space path integral as

1 K(x, t, zo, to) = 4 27Tiqt - t o ) / m

We conclude by giving the following useful rules for path integrals with

For the pinned Wiener measure:

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

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 d m 2 }

C [ l O , 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

Physicists, Academic Press, 2003

Arfken, G B., and H J Weber, Mathematical Methods of Physics, Aca- demic Press, sixth edition, 2005

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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

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

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Goldstein, H., C Poole, and J Safko, Classical Mechanics, Addison- Wesley, third edition, 2002

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668 REFERENCES

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Roach, G F., Green's Functions, Cambridge University Press, second edition, 1982

Ross, S.L , Differential Equations, Wiley, New York, third edition, 1984 Samko, S.G., A.A Kilbas, and 0.1 Marichev, Fkactional Integrals and Derivatives, Gordon and Breach Science Publishers, 1993

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Szekerez, P., A Course in Modern Mathematical Physics: Group, Halbert Space and Differential Geometry, Cambridge University Press, 2004 Titchmarsh, E.C., The Theory of finctions, Oxford University Press, New York, 1939

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boundary conditions, 91

channel waves factorization method, 155

tsunamis, 93

669

modified Bessel functions, 88

orthogonality and completeness,

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

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

symmetric top problem, 154

technique and categories, 132

theory, 124

momentum space, 659 quadratic momentum depen- Schrodinger equation, 655

derivation, 641

Feynman path integral

dence, 661 Feynman-Kac formula, 639 Fick’s equation, 380 Field strength tensor, 212 First canonical form

tor self-adjoint differential opera-

S t urm- Liou ville operat or, 108 Flexible chain

Bessel’s equation, 84 Flow around an obstacle conformal mappings, 319 Fokker-Planck equation fractional derivatives, 427 Four-momentum

conservation, 205 Four-scalars, 204 Four-tensors, 202 Four-vector space, 274 Four-vectors, 204 Four-velocity, 204 Fourier integral, 479 Fourier transforms, 481 convolution theorem, 485 cosine

sine, 482 diffusion equation, 488 existence, 486

in three dimensions, 486 Parceval theorems, 487 partial differential equations, transform of a derivative, 484 Caputo definition, 429 Cauchy integral formula, 390 Griinwald definition

differintegrals, 385 Laplace transforms, 396

484

Fractional derivatives

defining equation, 572 differential equations, 572 integral equations, 568 Dirac-delta function, 583 eigenfunction expansions, 579 first-order time dependence, 606 general boundary conditions, harmonic oscillator, 591 Helmholtz equation, 582 all space, 584

point source, 609 Poisson equation, 597 propagators, 609 wave equation, 618 Schrodinger's equation, 597 second-order time dependence, three-dimensional

603

6 16 continuum limit, 594 Group

definition, 224 terminology, 224 Group invariants, 231 Group representations, 246 R(3), 248

SU(2), 269 Group spaces, 272 Group theory group character, 248 invariants, 231 Lorentz group, 232, 241 Poincare group, 241

double root, 45 roots, 16

cosine function, 471 gamma function, 471 sine function, 470 Infinite series

convergence, 431 Infinitesimal ring Lie algebra, 226 Infinitesimal transformations orthogonal transformations, 175 Inhomogeneous boundary conditions Green’s functions, 575

Inhomogeneous Lorentz group, 241 Inner product

Inner product space, 273 Integral

Neumann series, 554 separable kernels, 556 successive iterations, 554 via integral transforms, 559 nonhermitian kernels, 564 Volterra equation, 548

vs differential equations, 548

Fourier transforms, 478 Integral transforms, 10

Mellin transforms, 51 1 fractional derivatives, 396

in n dimensions, 511 inverses

partial fractions, 501 theorems, 494 Laplacian

covariant, 194 Laurent series, 341 short cut, 346 Legendre equation, 13 Legendre polynomials, 18 Bromwich integral, 492

generating function, 19 normalization constant, 26 orthogonality and completeness, recursion relations, 21

Rodriguez formula, 19 Schlofli formula, 370 special integrals, 23 special values, 22 convergence

24

Legendre series, 15 Gauss test, 436 Leibniz formula, 25 Letnikov, 385 Levi-Civita symbol, 180 Lie algebra

generators of SU(2) differential, 240 group

differential operators, 228 infinitesimal ring, 226 rotation group R(3), 227 SU(2), 237

continuous groups, 224 Lie groups

Line element, 184, 199 Linear independence Wronskian, 41

Newton’s equations Normal form

error calculation, 556 covariant, 215 generators, 280 Orthogonal transformations, 167, 170 Orthogonality and completeness associated Laguerre polynomi- associated Legendre polynomi- Bessel functions, 90

Chebyshev polynomials, 78 Gegenbauer polynomials, 75 Hermite polynomials, 62 Hermitian operators Laguerre polynomials, 48 Legendre polynomials, 24

als, 53

als, 31

Sturm-Liouville operators, 111

Outer product, 179, 189 Parceval theorems, 487 Partial fractions Partial sum, 431 Path integrals Laplace transforms, 501

Bloch formula, 640 interpretation, 643 ESKC relation, 635, 649 Feynman path integral, 655 Feynman phase space path in- Feynman-Kac formula, 639 finite elements method, 650 methods of calculation, 646 Schrodinger equation, 658 semiclassical method, 650 time slice method, 647 Wiener path integral, 635 tegral, 659

Pauli spin matrices, 236 Permutation symbol, 190 Pinned Wiener measure, 637

Riemann sheets

Riemann theorem, 440 Riemann zeta function, 434 Riemann-Liouville

derivative, 387 Rodriguez formula associated Laguerre polynomi-

Hermite polynomials, 61 Laguerre polynomials, 47 Legendre polynomials, 19

representation, 248

branch cuts, 308

als, 53

Rotation group spherical harmonics, 249 Rotation matrix

differential equation, 262 evaluation, 260

inverse, 261 orthogonal transformations, 170 spherical harmonics, 258

Euler angles, 251 Rotation operator

Schlofli formula, 370 Schlofli integral formula

Schrodinger equation, 10, 43 Legendre polynomials, 370 bound states, 601

factorization method single electron atom, 151 Feynman path integral, 658 Green’s function, 615 propagator

free particle, 615 Schur’s lemma, 247 Schwartz inequality, 118 Schwaris-Cauchy inequality, 442 Schwarz-Christoffel transformations,

324 fringe effects, 325

678 INDEX

Second canonical form

Self-adjoint differential operator, 107

Semi-infinite parallel plate

Special unitary group

Spherical Bessel functions, 88

Spherical Hankel functions, 377

SU(2), 272 Step up/down operators ladder operators, 125 Stirling’s approximation, 377 Structure constants, 226 Sturm-Liouville operator completeness, 113 Sturm-Liouville equation, 108

expansion theorem first canonical form, 108 Green’s functions, 567 hermitian operators, 110 second canonical form, 122

boundary conditions, 109 variational integral, 535 generators, 237, 238 commutation relations, 238 differential, 240

irreducible representation, 269 relation to R(3), 269

spinor space, 272 Summation convention Einstein, 188 Summation of series, 452 Euler-Maclaurin sum formula, using differintegrals, 423, 462 using the residue theorem, 458 factorization method, 154

Cartesian, 178 covariant divergence, 194 Tensor density, 179, 189

Tensors

Trigonometric Fourier series, 479

generalized Fourier series, 114

presence of higher-order deriva-

several dependent and indepen-

several dependent variables, 523

several independent variables,

linear independence, 41

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