Index
Page numbers with the “S:” prefix are entries from the Digital Online Companion.
accelerated relaxation, 339
Gauss-Seidel method, 339
linear, 91
quadratic, 103
animation
ant raids, S:145
atomic collision, S:228
baseball, realistic flight, 110
bouncing ball, 40
dipole radiation field, 369
electric field hockey, 332
falling tablecloth
mechanical, 316
thermal, 596
game of life, S:138
Halley's comet, S:15
laser-driven coherent state, S:94
magnetic field, 368
Mandelbrot fractals, 254
N-body simulator, 604
oscillation of slinky, S:52
plane electromagnetic wave, 371
planetary motion
Earth, 136
Mercury, 144
precession of Mercury, 148
projectile motion, 84
quantum revival, 2D, 429
quantum scattering from a barrier, S:79
quantum wavepacket
in free fall, 418
in SHO, 408
relaxation of electric potential, 336
shooting for eigenenergies, 461
simple harmonic oscillator, 272
soccer, 117
strange butterfly attractor, 246
thermal equilibrium, 626
Thomson problem, S:67
three-body motion
choreography, 168
collinear, 167
Trojan asteroids, 179
wave on a membrane, 313
ants raiding pattern, S:142, S:147
atomic form factor, S:214
elastic, S:220
atomic reaction, S:211–S:221
antiproton impact, S:247
capture, S:229, S:232
cross section, S:230, S:233
Thomas mechanism, S:234
excitation, S:162, S:214
cross section, S:215, S:216, S:230
electron impact, S:245
ionization, S:162, S:218
cross section, S:218, S:230
electron impact, S:248
free-particle model, S:220
positron impact, S:248
atomic structure, S:65, see also hydrogenic atom
atomic units, 402
dimension, see fractal
band matrix, 416
representation, 416
solver, see SciPy
baseball, 107
animation, 110
curveball, 110
drag coefficient, 89
lift coefficient, 107
basis expansion method, 479
box basis, 480
half-open space, 484
SHO basis, 481
bifurcation, 222, see also chaos
binomial
coefficient, 610
distribution, 537
Bohr model, 490, S:225, S:232
Boltzmann distribution, 576, 587
Bose-Einstein condensation, S:170
chemical potential, S:171
critical temperature, S:172
scattering length, S:206
bound states, 459
central field potentials, 486
double square well, 466
gerade and ungerade, 479
Morse potential, 485
periodic multiple wells, 469
square well, 465
boundary value problem
Dirichlet boundary condition, 297, 335, 471, 493
mixed boundary conditions, S:135
Neumann boundary condition, 296, 335, 373, 374
Brownian motion, 405, 537, 540–544, 554
simulator, 560
catenary, S:47
celestial mechanics, 136
central field approximation, 154, 486, S:165
central field motion, 139
centrifugal
potential, 141, 173, 177, 489, S:127
chaos, 227
kicked rotor, S:19, S:27
Lorenz model, 242
Lyapunov exponent, 229
nonlinear driven oscillator, 232
Poincaré surface of section, 239
stadium billiard, S:22, S:28
time scale, 236
weather system, 244
chemical potential, S:170
circle
approximation of π, 554
midpoint drawing algorithm, S:71
quantum dot, S:118
classical scattering, see scattering
classical trajectory Monte Carlo, S:221–S:236
animation, S:228
microcanonical ensemble, S:225, S:260
straightline approximation, S:229, S:248
coherent state, 429, S:88
measurement problem, 429
comet, S:15–S:18
Halley's, S:15
ISON, S:15
commutator, S:76
coupled channel method, S:86, S:90, S:110
Crank-Nicolson method, 414
Curie's law, 613
Cython, 17, 406, 631, S:110, S:193
debugging, 20
deflection function, see scattering
density of states, S:171, S:258
differentiation operator, 362, see also radial basis function
dipole selection rule, S:93
δ comb, S:129
cusp, 477
displacement of a string, 286
drag force, see also air resistance
Brownian motion, 540
coefficient, 88
empirical formula, 90
quadratic, 88
viscosity, 87
eccentric anomaly, 162, S:260
Ehrenfest theorem, 411, 438, S:96
eigenvalue problem, 283, 309, 471, 480
generalized, 281, 474, 494, 528
Jacobi transformation, 284
energy distribution, 567
temperature, 577
electric field hockey, 332
electric potentials and fields, 334
disk in a box, 364
parallel plates, 336
unit square, 356
electromagnetic waves
dipole radiation, 369
plane waves, 371
on a sphere, S:65
plum-pudding model, S:65
electrostatic potential energy, 544, 553
energy band, 470
Einstein solid, 571
heat and temperature, 586
Ising model, 586
paramagnetic system, 613
envelope function, S:92, S:96
equipartition theorem, 555, 607
error, see numerical error
Euler method, 42
Euler rotation, S:261
Euler-Cromer method, 71, 273, 326
evolution operator, 413
approximate, S:77
HD 139357, 159
modeling RV datasets, 162
radial velocity method, 158
expectation value, 409, 474, S:90, S:96
F2Py, 17, 406, 631, S:110, S:191, S:193, S:247, S:266
falling tablecloth
mechanical, 315
thermal, 596
fast Fourier transform, 250, S:30–S:44
aliasing, S:40
iterative FFT, S:38
Nyquist frequency, 305, S:44
Parseval's relation, 250
positive and negative frequencies, S:42
recursive FFT, S:34
two-dimensional, 432
wave function, 421
fast multipole method, S:193
finite difference method
displacement of a string, 288
error, 359
Laplace equation, 335
quantum dot, 493
quantum eigenenergies, 470
standing waves, 309
waves on a membrane, 311
waves on a string, 303
finite element method
accuracy, 359
building system matrix, 350
data structure, 353
Dirac δ molecule, S:129
displacement of a string, 292
error, 499
FEM library, 495
Laplace equation, 341
mesh file format, 531
mesh generation, 347, 353, 387, 495
mixed boundary conditions, S:135
nodes and elements, 344
Schrödinger equation, 472
stiffness matrix, 298
fitting, 163
fixed-point number, 23
floating point, 24
bit shifting, 26
byte string, 25
phantom bit, 25
round-off error, 26
football, 128
Fourier transform, 249, see also fast Fourier transform
fractals, 251
Cantor set, 258
correlation dimension, 254
Hausdorff dimension, 252
Julia set, 259
Koch curve, 252
Mandelbrot fractals, 254
Sierpinski carpet, 259
free fall, 42
animation, 40
Euler's method, 44
momentum profile, 423
Runge-Kutta methods, 51
game of life, S:137
Gauss elimination, S:55
Gauss-Jordan method, S:56
Gauss-Seidel method, S:56
Gaussian distribution, 551, S:140
recursion, 8
golf, 112
drag and lift, 113
Green's function, S:199
Halton sequence, 364
heat capacity, 577
Heisenberg's uncertainty principle, 305, 401, 422, 501, S:44
Hilda asteroids, see restricted three-body problem
hydrogen molecule, vibrational states, 486
l degeneracy, 487
angular probability density, 491
Hulthén potential, 512
modified potential, 1/r1+∈, 487, 511
radial equation, 487
radial probability density, 490
radial wave function, 489, S:106, S:256
screened Coulomb potential, 511
shell structure, 490
ideal gas law, 609
importance sampling, 547
integral equation, S:198
IVisual, 19
Matplotlib inline, 17
1D, 578
3D, S:179
antiferromagnetism, 578, S:178, S:187
critical temperature, 591, 594
computation, 616
exact solution in 2D, 624
ferromagnetism, 578
hysteresis, 618, S:181
staggered, 626, S:178
mean field approximation, S:177, S:180, S:181, S:184
partition function, 613
phase transition, 591
spin domains, 589
toward equilibrium, 582
Jupiter, see precession of Mercury, see also restricted three-body problem
pull on the Sun, 159
Kansa's method for PDEs, 363, see also radial basis function
Kepler's third law, 144
planets, 143
Kepler's equation, 162, S:260
Lagrange points, 175, S:233
Lambert W function, 98–103, 506, S:129
approximate formulas, S:7
Bose-Einstein condensation, S:171
Dirac δ molecule, 478
evaluation, 100
projectile motion, linear drag, 101
laminar flow, 88
Laplace equation, 335
additivity rule, 376
Laplace operator
nine-point discretization, 376
laser-electron interaction, S:92
strong fields, S:98
area-preserving, 64
space discretized, 403
time dependent, S:261
time transformation, 149
N-body system, S:222
least square fitting, 163, 204
Lennard-Jones potential, 599, S:132
Levinson theorem, 652, 663, 668, 673
lift force, 105, see also Magnus force
linear combination of atomic orbitals (LCAO), 467, 479, S:129
linear interpolation, 125, 510
Lippmann-Schwinger equation, S:200
bifurcation, 225
fixed points, 218
Lyapunov exponent, 230
period doubling, 222
renormalization, S:28
Lorenz flow, 240, see also chaos
Lyapunov exponent, 229, see also chaos
magnetic field, 366
closed loop, 368
long wire, 368
Magnus force, 106
lift coefficient, 107
Matlab, 17
3D plots
Axes3D, plot_surface, 211, 384, 389, 395
scatter, 262, S:60
animation, 451, 561, 627, S:148
axis
label, 12
off, 391
semilog scale, 265
width, 30
bitmap images, imshow, 269, 391, 455, 627, S:150
color, 30
colorbar, 391
configuration records, 31
contour
filled, contourf, 672, S:96
font size, 30
frame
off, 391
spacing, 30
histogram, 627, S:136
IPython inline, 17
frame off, S:136
line
style, 30
marker, 30
color, S:60
multiple plots, subplot, 267, 269, 455
polar plot, 668
pylab, 17
step plot, step, S:117
text label, 30
math mode, 211
triangular mesh plot, tripcolor, 379, 389, 529
triangular mesh surface, plot_trisurf, 528
vector fields, quiver, 211, 384, 392, 559
Maxwell distributions, 605
mean free path, S:161
energy dependent, S:183
Mercury, see precession of Mercury
meshfree method, 360, see also radial basis function
Metropolis algorithm, 579–581, 587–588
model building, 89
molecular dynamics, 598–605, S:222
close-neighbor interaction, 602
equipartition theorem, 607
ideal gas law, 609
Maxwell distributions, 606, 621
optimization, 631, S:191
periodic boundary condition, 601–605
pressure, 608
second virial coefficient, 609, 621
units, 600
Monte Carlo integration, 450, 544, 553
error, 547
hit-or-miss method, 554
Monte Carlo simulation
ants, S:145
Einstein solid, 564
nuclear decay, 536
particle transport, S:160
simulated annealing, 596
falling tablecloth, 596
hanging chain, S:153
hanging tablecloth, 619
traveling salesman problem, S:157, S:181
Morse potential, 485, 599, S:131
Navier-Stokes equations, see Lorenz flow
Newton
second law, 39
third law, 105
nuclear decay, 535
Numba, 16, 17, 270, 406, S:110, S:193
numerical differentiation
first order, 43
midpoint method, 48
second order, 288
numerical error, 5
global error, 45
in energy, 68
round-off, 6
truncation, 7
numerical integration, 445, see also Monte Carlo integration
Gaussian, 448, S:107
abscissa and weight, 457
multiple integral, 450
Simpson's rule, 447
trapezoid rule, 446
first derivative, 514
logarithmic scale, S:127
advanced indexing, 34
row or column swap, 36, 385, 530
with boolean array, 34, 205, 211, 604, 631
argmin, 206
array creation, 31
broadcasting, 33, 111, 330, 604, 630
concatenate, 205, 389, 421, 519, 530, S:43
conversion to list, 531
copy, 33
element insertion or deletion, 32, 291, 327, 388, 396, 523, 528
element-wise operations, 33, 37, 205, 306–307, 312–313, 330, 337, 391, 406, 455, 604, 621, 630, S:58, S:60
F2Py, 17
FFT, S:86
in 2D, 432
flatten, 388
histogram, 621, S:189
matrix multiplication, 521
maximum element, S:58
meshgrid, 210
nearest difference, 205
outer method, 205
outer product, 455, S:58
row and column convention, 211, 384
row and column insertion or deletion, 36, 388, S:130
shape, 35
slicing, 32, 211, 306–307, 312–313, 330, 337, 384, 388, 417, 518, S:60
sorting, 389
stacking, S:60
summing arrays, 329, 440, 475, 604, 621, 630
take, 389
transpose, 385
truth array, 34, 205, 211, 385, 530
universal functions (ufunc), 16, 36, 37, 210, 475
vector operations, 111
vectorizing functions, 37, 475, 517
object-oriented programming, 14, 560, 565
orbiting, S:195, see also scattering
ordinary differential equation, 39, see also Euler, leapfrog, Numerov, and Runge-Kutta methods
implicit method, 62
oscillation
damping, 273
resonance, 275
RLC circuit, 273
paramagnetic system, 576, 612, 615
partial differential equation, 271, see also Laplace, Schrödinger, and wave equations
particle transport, S:160
angular scattering, S:182
energy deposition, S:161, S:169
energy-dependent mean free path, S:183
range distribution, S:164, S:168
partition function, 576
harmonic oscillator, 577
phase transition, 591
Ising model, 591
ping pong, 114
spin effects, 116
open orbits, 144
properties, 139
simulation, 136
units, 143
Pluto, see restricted three-body problem
Poincaré map, see chaos
Poisson distribution, 536, 557, S:122
Poisson equation, 335
precession of Mercury, 146–158
by other planets, 154
oscillations, 153
scaling law, 156
probability density, 412
program profiling, S:85
programs list, 685
projectile motion, 83
linear drag, 91
quadratic drag, 103
visualizing, 84
pseudospectral method, S:78
Python, 10
2.7x vs. 3.xx compatibility, 14
assignment and type, 12
conditional, 13
deep copy, S:148
eval function, 204
exception, 528
IDLE, 29
indentation, 12
input, 12
installation, 26
list, 12
append, 12
count, 627
delete element, S:148
nested, 387
slicing, 33
sorting, 388
operator overloading, 566
pickle file handler, 527
profiling, S:109
random integer, 566
random number, 534
with-as statement, 204
quantum chaos, S:117–S:125
chaoticity, S:125
energy level statistics, S:121
nearest neighbor spacing, S:121
histogram, S:124
scars, S:125
spectrum unfolding, S:122
stadium billiard, S:117
circle, 512, S:128
degeneracy, 501
energy level distribution, S:117
hexagon, 499
isosceles right triangle, 496
stadium, S:118
triangle, 508
wave function, 498, 502, S:119, S:125
quantum mechanics, see Schrödinger equation
quantum quilt, 443
revival time, 435
semiclassical limit, 437
quantum scattering, 647
T-matrix, S:212
amplitude, 648, 654, S:200, S:211
atomic form factor, S:214
Born approximation, 663, 665, S:201, S:206, S:209, S:213
Buckingham potential, 673
cross section, 654, 658, 664, S:212
elastic, S:245
Fermi's golden rule, S:201
Gaussian potential, 673
hard sphere, 655
shadow effect, 659
Hulthén potential, 673
optical theorem, S:238
partial wave expansion, 652–655
phase shift, 649, 654, 661, 663, S:206, S:208, S:209
potential barrier, S:79
potential well, S:83
Ramsauer-Townsend effect, 674, S:205
resonance, 651, S:84
scattering length, S:203, S:242
square spherical barrier, 660, 668
square spherical well, 668
WKB approximation, S:207, S:209
Yukawa potential, 659, S:166, S:201, S:209
quantum transitions, S:86
amplitudes, S:88
dipole allowed, S:98, S:216
dipole forbidden, S:98, S:216
in hydrogen, S:105
in the SHO, S:104
laser driven, S:91
multiphoton transition, S:98
occupation probability, 425, S:88, S:96
Rabi flopping, 426
two-photon transition, S:98
two-state system, 424
Rabi flopping, 426
in hydrogen, S:106
Rabi frequency, 427
rotating wave approximation, 427
radial basis function, 361
collocation method, 362, S:71
differentiation operator, 363, 364
Gaussian and multiquadric RBF, 361
scattered data interpolation, 361
radial velocity method, see exoplanets
radial velocity transformation, 198
random number, 533
correlation and moment tests, 534
integer, 566
nonuniform distribution, 551, 556
Lorentzian, 551
rejection method, 557
transform method, 556
seed, 534
uniform range, 552
random walk, 537
binomial distribution, 537
in 2D, 538
recursion
reflection coefficient, S:81, S:83
restricted three-body problem, 171–178
Earth-Moon system, 175
Lagrange points, 175
orbital resonance, 181
Pluto libration, 183, S:14
Pluto's motion, 181
Sun-Jupiter system, 178
Sun-Neptune system, 181
units, 173
Reynolds number, 87
root finding, 94
false position, S:6
SciPy equation solver, 94, 122, 505
secant method, S:5
rotating frame, 194
rotation matrix, 197
round-off error, see numerical error
Runge-Kutta methods, 46
characteristic time, 61
non-vectorized, 60
SciPy wrapper, 81
step size control, 61
Runge-Kutta-Felhberg method, 62, 514
Rutherford scattering, 638, S:66
cross section, 639
Rydberg states, 490, S:236
scattering, 634, see also quantum scattering
cross section, 636, S:253
deflection function, 634, 675, S:208
Yukawa potential, 645
impact parameter, 634, S:197
orbiting, S:195, S:240
rainbow, 640, 644, 645, 665, 666, 669
square spherical barrier, 670
square spherical well, 670
Yukawa potential, 645
scattering length, S:203, see also quantum scattering
Schrödinger equation, time dependent, 400, see also wavepacket
boundary effects, 419
coupled channel method, S:86
direct simulation, 403
periodic boundary condition, 407
split evolution operator, S:78
split-operator method, 415
Schrödinger equation, time independent, 460, see also bound states
animated eigenstates, 461
basis expansion method, 479
integral equation, S:199
matching condition, 464
pseudo-continuum states, 471, 472, 481
shooting methods, 463
Schrödinger's cat, 429
SciPy, 16
Airy function and its zeros, 516, 523
band eigenvalue solver, 513, 521, 528
band matrix solver solve_banded, 416
Bessel function
spherical, 660
zeros, S:128
BLAS and LAPACK, 285
combination, 614
eigenvalue solver eigh, 285, 327, 521
elliptic integral, 625
gamma function, 612
Hermite polynomial, 522, S:131
integration, 457, S:243
Lambert W function, 100
least square fitting, 163, 204
Legendre polynomial, 683
linear system solver solve, 290, 291, 388
orthogonal polynomial, 457
root solver fsolve, 94, 122, 505
sparse eigenvalue solver eigsh, 471, 475, 521, 523, 528
Weave, 17
self-consistent methods, 335, S:48
relaxation error, 340
shooting methods, 118, 463, 465
simple harmonic oscillator
animation, 273
classical, 66
quantum mechanical, 408
Simpson's rule, 411
Snell's law, see scattering
snub cube, S:69
soccer, 116
space discretized leapfrog method, 403, 405
normalization error, 412
stability, 405
sparse matrix, 355
special function
Airy function, 483, 507, 515, S:130
zeros, 516
Bessel function, 500, S:128
modified spherical, 668
recurrence, 677
spherical, 652, 660, 677, S:251
zeros, S:128
Hermite polynomial, 482, S:131
Laguerre polynomial, S:127
Lambert W function, 478
Legendre polynomial, 448, S:108
in plane waves, 652
spectral staircase, S:115
spherical harmonics, 486, S:105
addition, S:251
in plane waves, S:251
orthogonality, S:251
spinning balls, 105
spin parameter, 107
split evolution operator, S:75, S:78
split-operator method, first order, 413
error, 419
stiff differential equation, 62
Stirling's approximation, 612
Stoke's law, 87
stopping power, S:162
symplectic methods, 69, see also leapfrog method
first order, 71
factorization, 257
Lambert W function, 100
physics
hydrogen atom, S:106
quantum oscillator, 482
series, 186
table tennis, 114, see also ping pong
temperature
Curie, 592
thermodynamics, 564
second law, 572
third law, 585
Thomson model, S:65
choreography, 168
dynamics, S:228, see also classical trajectory Monte Carlo
Euler's collinear motion, 165
Euler's quintic equation, 167
planar motion, 164
traffic flow, S:139
fundamental diagram, S:140
hybrid model, S:146
transmission coefficient, S:81, S:83, S:103
tridiagonal matrix, see band matrix
Trojan asteroids, 178
truncation error, see numerical error
tunneling, S:82, S:234
turbulent flow, 88
unitarity, S:76
Verlet, see leapfrog method
vibration, 277
normal modes, 280
string, 300
triatomic molecules, 277
air, 122
visualization, 3, 40, 110, 369
von Neumann stability, 304
VPython, 18
axis flip, 397
camera angle, 20
curve, 133
faces, 377
helix, 273
in GlowScript, 30
IVisual, 19
label, 133, 187, 202, 382, 455
box, 396
light source, 138
making movies, 30
opacity, 133, S:67
rate requirement, 19
retain, 85
ring, 212
sphere, 41
vector operations, 111, 202, 392, 394
VIDLE, 29
VPython modules (VPM), 14, 139, 307, 328, 330, 377, 452, 455, S:52, S:61–S:64
wave function, 400
laser driven, S:96
momentum space, 420
normalization, 412
conservation, 405
plane wave, 401
scarring, S:125
scattering, 656
broadening, 418
in 2D, 432
momentum distribution, 421, 432
optical diffraction, 433
refocusing, 409
scattering from a barrier, S:79
scattering from a well, S:85
self interference, 419
on a membrane, 311
Weave, 17, 406, 631, S:110, S:193
Weyl formula, S:116
Wigner distribution, S:122
WKB approximation, S:207, see also quantum scattering