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
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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
672
INDEX
Divergent series, 465
Casimir effect, 466
quantum vacuum energyy, 467
Doppler shift, 208
Dot product, 165
Double factorial, 377
Dual field strength tensor, 212
Eigenvalue problems
Einstein summation convention,
188
Elastic beam
deformation, 527
Electrostatics
Green’s functions, 604

Entire function, 297, 347
Equivalent representations, 246
ESKC relation, 635
Essential singular point, 347
Euler angles, 172
Euler equation, 518
Euler’s theorem, 228
Euler-Maclaurin
sum
formula, 454
Euler-Masheroni constant, 471
Expansion theorem,
113
eigenfunctions, 276
Extension
prolongat ion
Green’s functions, 579
another form, 520
generators, 282
Extraordinary differential equations,
417
Factorization method
137
associated Legendre equation,
Bessel functions,
155
Gegenbauer polynomials, 153
harmonic wcillator, 156
single electron atom,
151

solutions,
130
spherical harmonics, 141
Sturm-Liouville equation, 123
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
INDEX
673
notation,
381

Riemann formula,
395
Riemann-Liouville definition, 387
Fredholm equation, 548
Frobenius method, 13,
16
Function spaces
Fundamental tensor, 184
Hilbert space, 274
Galilean transformation, 2 15
Gamma function, 360, 462
infinite product, 471
Gauss equation
special functions,
104
Gegenbauer equation, 75
factorization method, 153
Gegenbauer polynomials, 75
Chebyshev polynomials, 76
cosmology, 72
generating function, 75
orthogonality and completeness,
75
Generalized Fourier series, 114
Generating function
associated Laguerre polynomi-
Bessel functions, 89
Chebyshev polynomials, 78
Gegenbauer polynomials, 75
Hermite polynomials, 60

Laguerre polynomials, 46
Legendre polynomials,
19
continuous groups
Lie groups, 278
ex tension
prolongation, 282
normal form, 280
R(3), 227
als,
52
Generators
commutation relations, 227
differential, 228
transformations, 279
Geodesics, 197
Griinwald, 385
Gradient, 193
Green's functions,
10
advanced and retarded, 621
boundary conditions, 568
compounding propagators, 609
construction, 569
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
three-dimensional, 593
604
inhomogeneous boundary con-
ditions, 575
Laplace operator, 597
Lippmann-Schwinger equation,
one-dimensional, 567
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
674
lNDEX
Holder inequality, 442
Hamilton’s principle, 533
Hankel function, 87
Harmonic functions, 299
Harmonic oscillator
damped
Laplace transforms, 505
factorization method, 156
Green’s functions,
591
quantum mechanical
three dimensional, 56
Hermite polynomials, 57
Harmonic series, 432
Heat transfer equation
differintegrals, 415
Helmholtz equation, 9
continuum limit, 584
Green’s functions, 582
three dimensions, 593
Hermite equation, 58, 60

Hermite polynomials, 59
contour integral, 373
dipole calculations, 64
Gaussian,
63
generating function, 60
harmonic oscillator, 57
orthogonality and completeness,
recursion relations, 62
Rodriguez formula, 61
Hermitian operators
boundary conditions,
110
eigenvalues
eigenfunctions,
110
quantum mechanics,
116
Sturm-Liouville operator,
110
function spaces, 274
inner product, 117
quantum mechanics, 277
Hilbert-Schmidt theory,
560
completeness of eigenfunctions,
nonhermitian operators, 564
Homogeneous Lorentz group, 241
62
Hilbert space

563
Hypergeometric equation, 99
Hypergeometric functions, 99
Improper transformations, 170
Incomplete beta function, 364
Incomplete gamma function, 364
Indicia1 equation, 14
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
Integral equations
Infinite products, 468
Hilhert
space,
117

n-fold, 384
Cauchy formula, 549
classification, 548
eigenvalue problems
Fredholm equation, 548
Green’s functions, 568
homogeneous, 548
methods of solution
Hilbert-Schmidt theory, 560
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
INDEX
675
general, 477
Hankel transform
integral equations, 559
Laplace transforms, 478
Mellin transform, 479
relations, 51
1

Fourier-Bessel transform, 479
Invariance,
197
Inverse Laplace transforms
Bromwich integral, 492
Lerch theorem, 491
Inversion
of
power series, 451
Irreducible representation, 247
Isolated singular point, 297, 347
Isomorphism, 239
Isoperimetric problems, 529
Jacobi polynomials, 41
Jacobian
of
transformation,
190
Jordan’s lemma, 357
contour integral, 375
Kronecker delta, 179
Kummer formula, 106
Ladder operators
step up/down operators, 124,
125
Laguerre equation, 45
Laguerre polynomials, 46
contour integral, 371, 372
generating function, 46
orthogonality and completeness,

recursion relations,
50
Rodriguez formula, 47
special values, 50
48
Laguerre series, 46
Laplace equation,
9
Laplace transforms, 490
variational analysis, 525
basic, 492
Bessel’s equation, 507
damped oscillator, 505
definite integrals, 502
derivatives, 503
differintegrals, 413
electromagnetic waves, 506
Fourier transforms
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
676
INDEX
Lorentz contraction
Lorentz group
length contraction, 201
commutation relations, 244
generators, 244
homogeneous
inhomogeneous, 241
Lorentz transformation, 199
boost, 244
group invariants, 232
orientation of axis, 209
M-test
Weierstrass M-test, 444
Maclaurin series, 446
Mappings,
300

conformal, 313
inversion,
301,
302
many-to-one,
306
one-to-one, 304
one-to-two, 306
rotation,
301
Schwarz-Christoffel transforma-
translation,
300
two-to-one, 304
Maxwell’s equations, 21
1
potentials, 214
transformations, 213
tions, 322
Mean square displacement, 380
Mellin transforms, 512
MEMS
Casimir effect, 468
Metric tensor, 184
covariant derivative, 194
Minkowski metric, 202
Minkowski spacetime, 198
Minkowski’s inequality, 442
Mittag-Leffler functions, 418
Mittag-Leffler theorem

infinite products, 470
Modified Bessel functions,
88
Modulus, 294
Multipole expansion, 267
Neumann function, 87
Neumann series
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
INDEX
677
Poincare group, 241

Point groups, 278
Point source initial condition
Poisson equation
Power
series,
449
Prolongation
extension
generators, 282
Propagators,
609
Proper time, 204, 205
Proper transformations, 170
Pseudo-Euclidean
,
199
Pseudotensor, 180
Green’s functions, 609
Green’s functions, 597
Quantum mechanics
Quotient theorem, 189
Hermitian operators, 116
relation
to
SU(
2), 269
R(3)
R(3)
and SU(2), 269
Rank, 178

Rayleigh-Ritz met hod
Recursion relation
variational integrals, 539
associated Laguerre polynomi-
Bessel functions, 90
Hermite polynomials, 62
Laguerre polynomials, 50
Legendre polynomials, 21
Reducible representation, 247
Regular singular point
Frobenius method, 16
Regularization
Renormalization, 465
Relativistic energy
binomial formula, 447
Relativistic
mass,
207
Renormalization, 465
Representation space, 246
Residue theorem, 347
Rest mass, 205
als, 53
Retarded Green’s functions, 624
Riemann curvature scalar, 195
Riemann curvature tensor, 195
Riemann formula, 395
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
Sturm-Liouville operator, 122
mappings
Semi-integrals, 413
Semiderivatives, 413
Semidifferential equations, 419
Separation of variables, 10
Series
fringe fields, 325
algebra, 439
inequalities, 442
rearrangement,

440
Similarity transformations, 175
Simple pole, 347
Singleelectron atom models,
44
Singular points
classification, 347
essential, 347
isolated, 347
simple pole, 347
Soap film, 521
Spacetime
derivatives, 208
Minkowski, 198
confluent hypergeometric func-
contour integrals, 369
differintegral representations, 424
hypergeometric functions, 104
postulates, 199
SU(2), 236
Special functions
tions,
104
Special relativity
Special unitary group
Spherical Bessel functions, 88
Spherical Hankel functions, 377
contour integral, 376
Spherical harmonics,
33,

249
addition theorem, 264
Condon-Shortley phase, 34
expansions, 255
factorization method, 141
Gegenbauer polynomials, 73
ladder operators, 141, 150
Spinor space
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
differential equations, 285
Sturm-Liouville system
SUP)
454
Symmetric top
Symmetries
Taylor series, 339
with multiple variables, 448
with the remainder, 445
pseudotensor, 180
Cartesian, 178
covariant divergence, 194
Tensor density, 179, 189
Tensors
INDEX
679
covariant gradiant,
193
curl,
193
differentiation,

19
1
equality,
189
general,
181
Laplacian,
194
some covariant derivatives,
193
Time dilation,
201
Trace,
179
Triangle inequality,
117
Trigonometric Fourier series,
479
generalized Fourier series,
114
Uniform convergence,
443
properties,
445
Unitary group
U(2),
234
Unitary representations,
248
Unpinned Wiener measure,

638
Variational integrals
eigenvalue problems,
535
elastic beam,
527
Euler equation,
518
geodesics,
520
Hamilton’s principle,
533
Lagrangian,
533
loaded
cable,
540
presence of constraints,
529
presence
of
higher-order deriva-
several dependent and indepen-
several dependent variables,
523
several independent variables,
524
soap
film,
521

upper bound to eigenvalues,
537
tives,
527
dent variables,
526
Vector product,
165
Vector spaces
complex,
274
inner product,
273
Minkowski,
274
real,
272
Volterra equation,
548
Wallis’s formula,
471
Wave four-vector,
208
Weierstrass function,
479
Weierstrass M-test,
444
Weight
of
a

tensor,
189
Wiener measure
pinned
unpinned,
638
Brownian motion,
635
unpinned,
637
Wiener path integral
Worldline,
204
Wronskian
Bessel functions,
95
linear independence,
41
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
This Page Intentionally Left Blank
MATHEMATICAL
METHODS IN
SCIENCE
AND ENGINEERG

S.
SELCUK
BAYIN
Middle
East
Technical
University
Ankara,
Turkey
@K&CIENCE
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ISBN-I3 978-0-470-04142-0
ISBN-10 0-470-04142-0
Printed in the United States of America
10987654321
Contents
Preface xxz
Acknowledgments xxuii
1

NATURE and MATHEMATICS
1
1.1 Mathematics and Nature
3
1.2 Laws
of
Nature
4
1.3
Mathematics and Mind
5
1.4
Is Mathematics the Only Language for Nature?
6
1.5 Nature and Mankind
7
2
LEGENDRE EQUATION and POLYNOMIALS
2.1 Legendre Equation
2.2
2.3
Legendre Polynomials
2.1.1
Method
of
Separation
of
Variables
Series Solution
of

the Legendre Equation
2.2.1 Fro benius Method
2.3.1 Rodriguez Formula
2.3.2
Generating Function
2.3.3
Recursion Relations
2.3.4
Special Values
9
10
12
13
16
17
19
19
21
22
V