INDEX
null operation, as identity element of group,
1044
nullity, of a matrix, 293
numerical methods for algebraic equations,
985–992
binary chopping, 990
convergence of iteration schemes, 992–994
linear interpolation, 988
Newton–Raphson, 990–992
rearrangement methods, 987
numerical methods for integration, 1000–1009
Gaussian integration, 1005–1009
mid-point rule, 1034
Monte Carlo, 1009
nomenclature, 1001
Simpson’s rule, 1004
trapezium rule, 1002–1004
numerical methods for ordinary differential
equations, 1020–1030
accuracy and convergence, 1021
Adams method, 1024
difference schemes, 1021–1023
Euler method, 1021
first-order equations, 1021–1028
higher-order equations, 1028–1030
isoclines, 1028
Milne’s method, 1022
prediction and correction, 1024–1026
reduction to matrix form, 1030
Runge–Kutta methods, 1026–1028
Taylor series methods, 1023
numerical methods for partial differential
equations, 1030–1032
diffusion equation, 1032
Laplace’s equation, 1031
minimising error, 1032
numerical methods for simultaneous linear
equations, 994–1000
Gauss–Seidel iteration, 996–998
Gaussian elimination with interchange, 995
matrix form, 994–1000
tridiagonal matrices, 998–1000
O(x), order of, 132
observables in quantum mechanics, 277, 560
odd functions,seeantisymmetric functions
ODE,seeordinary differential equations (ODEs)
operators
Hermitian,seeHermitian operators
linear,seelinear operatorsandlinear
differential operatorandlinear integral
operator
operators (quantum)
angular momentum, 656–663
annihilation and creation, 667
coordinate-free, 648–671
eigenvalues and eigenstates, 649
physical examples
angular momentum, 658
Hamiltonian, 657
order of
approximation in Taylor series, 137n
convergence of iteration schemes, 993
group, 1043
group element, 1047
ODE, 468
permutation, 1058
recurrence relations (series), 497
subgroup, 1061
and Lagrange’s theorem, 1065
tensor, 930
ordinary differential equations (ODE),see also
differential equations, particular
boundary conditions, 468, 470, 501
complementary function, 491
degree, 468
dimensionally homogeneous, 475
exact, 472, 505
first-order, 468–484
first-order higher-degree, 480–484
soluble forp, 480
soluble forx, 481
soluble fory, 482
general form of solution, 468–470
higher-order, 490–523
homogeneous, 490
inexact, 473
isobaric, 476, 521
linear, 474, 490–517
non-linear, 518–523
exact, 519
isobaric (homogeneous), 521
xabsent, 518
yabsent, 518
order, 468
ordinary point,seeordinary points of ODE
particular integral (solution), 469, 492, 494
singular point,seesingular points of ODE
singular solution, 469, 481, 482, 484
ordinary differential equations, methods for
canonical form for second-order equations,
516
eigenfunctions, 554–573
equations containing linear forms, 478–480
equations with constant coefficients, 492–503
Green’s functions, 511–516
integrating factors, 473–475
Laplace transforms, 501–503
numerical, 1020–1030
partially known CF, 506
separable variables, 471
series solutions, 531–550, 604
undetermined coefficients, 494
variation of parameters, 508–510
ordinary points of ODE, 533, 535–538
indicial equation, 543
orthogonal lines, condition for, 12
orthogonal matrices, 270, 929, 930
general properties,seeunitary matrices
orthogonal systems of coordinates, 364