Appendix A Vector Relations

A.I Vector Identities

a . (b x c) = (a x b) . c

a x (b x c) = (a· c)b - (a . b)c

V(F· G) = (G . V) F + (F· V) G + G x (V x F) + F x (V x G)

V . (ljrF) = ljrV . F + F· Vljr

V x (ljrF) = ljrV x F + (Vljr) x F

V . (F x G) = G . (V x F) - F . (V x G)

V x (F x G) = (G· V) F + F (V· G) - (F . V) G - G (V . F)

V· (V x F) = 0

V x (V x F) = -V2F + V(V· F)

V . (Vljr) = V2ljr

V X (Vljr) = 0

A.2 Integral Theorems

Divergence theorem: Is F . dA = Iv V . F d 3r

Stokes's theorem: iF, dl = Is (V x F) . dA

228 A. Vector Relations

A.3 The Functions of Vector Calculus

A.3.1 RECTANGULAR COORDINATES

A.3.2 SPHERICAL POLAR COORDINATES

A alj! A 1 alj! ~ 1 alj! Vlj! =r- +0-- +¢---

ar r ae rsine a¢ 1 a 2 1 a 1 a F¢

V· F = --(r F) + ---(sineFe) + ----r2 ar r r sin e ae r sin e a¢

V x F = r-l-(~(SineF¢) _ aFe) r sine ae a¢

~ 1 ( 1 a Fr a ) A 1 ( a a Fr ) +0- -.-----(rF¢) +¢- -(rFe)--r Sill e a¢ ar r ar ae

V2lj! = ~~(r2alj!) + _l_~(sinealj!) + 1 a2lj! r2 ar ar r2 sine ae ae r2 sin2 e a¢2

A.3.3 CYLINDRICAL COORDINATES

Appendix B Fundamental Equations of Physics

B.1 Poisson's Equation

B.2 Laplace's Equation

B.3 Maxwell's Equations

\7·D=p(r,t) (Gauss's law)

\7·B=O aB

\7 x E + - = 0 (Faraday's law) at aD

\7xU--=j(r,t) (Ampere's law) at

with the auxiliary relations D = E E and B = fLU

230 B. Fundamental Equations of Physics

B.4 Time-Dependent Schrodinger Equation

[--li2 a2 ] a --2 + Vex) ljf(x, t) = iJi-ljf(x, t) 2m ax at (one dimension)

_\72 + VCr) ljf(r, t) = iJi-ljf(r, t) [ --li2 ] a 2m at (three dimensions)

Appendix C Some Useful Integrals and Sums

In the following expressions, m and n are nonnegative integers and a and b are constants.

C.I Integrals

100 2n -a'I'd (2n)!.J]r t e t = ----::----:---

o (2a)2n+!n!

100 ,

2n+! -a'I' _ _ n_._ t e dt - 2 2 o 2a n+

-----.:- dt = ----100 tm m!n!

o (l + t)m+n+2 (m + n + 1)!

232 C. Some Useful Integrals and Sums

11 m!n! tm (1 - tr dt = ----­

o (m + n + I)!

/1 dy (x - 1) -- -log --

-I Y - X X + 1 for Ixl > 1.

/1 ydy (X-I) -- =2+xlog --

-I Y - X X + 1 for Ixl > l.

/1 yndy = n(x+l)-I+(-lt[n(x-l)+I] +x2/1 yn-2 dy

-1 y - x n(n - 1) -1 y - x

for Ix I > 1 and n ~ 2.

f -----r=""dx=~ = log [x + -Jx 2 ± a2]

Jx 2 ± a2 a

f xdx = Ja 2 +x2

Ja 2 + x2

12][ sin mx sin nx dx = { ~8mn for m I- 0

for m = 0 or n = 0,

cosmx cosnx dx = n mn 12][ { 8 for m I- 0 form = n = 0 o 2n

C.l Integrals 233

{2:rr 10 sinmxcosnxdx = 0

for m i= 0 {2:rr mx nx {o 10 sin 2 sin 2 dx = ~ mn for m = 0 or n = 0,

12:rr mx nx {no cos - cos - dx = mn o 2 2 2n

for m i= 0 form = n = 0

f sin2 dx = Hx - 4 sin2x)

f cos2 dx = 4(x + 4 sin 2x)

f sec x dx = log (sec x + tan x)

f -1 cosn x sinx dx = -- cosn+1 X

n+l

f sinn x cosx dx = _1_ sinn+l x n+l

f dx

1 + cosx

x = tan-

2

smaxsmbxdx = - - -----f .. 1 [Sin[(a-b)X] Sin[(a+b)X]] 2 a-b a+b

234 C. Some Useful Integrals and Sums

cos ax cos bx dx = - + -----f 1 [Sin[(a-b)X] Sin[(a+b)X]]

2 a-b a+b

SIll ax cos bx dx = - - + -----f . 1 [COs[(a - b)x] cos[(a + b)X]]

2 a-b a+b

C.2 Sums

00 1 n 2

~ (2n + 1)2 = 8

00 (-l)n 7n 4 "'---­~ n4 - 720 n=1

Appendix D Algebraic Equations

D.l Quadratic Equation

It is well known that any quadratic algebraic equation can be expressed in the form

x 2 +bx + c = 0,

which has solutions

-b ± Jb2 - 4c x = ------------

2

D.2 Cubic Equation

The general cubic algebraic equation can be written,

x 3 + ax 2 + bx + c = o.

(D.I)

(D.2)

(D.3)

For our purpose, we assume the coefficients a, b, and c are all real. By a sequence of redefinitions of the unknown quantity, the solution of this equation can be reduced to the solution of a quadratic equation, as we shall now show. 1

First, define a new unknown according to x = y + s and choose s so that the resulting cubic equation in y has no y2 term. With this substitution in Eq. (D.3)

1 See, for example, Raymond W. Brink, College Algebra, 2nd ed., Appleton-Century­Crofts, Inc., New York, 1951, p. 333.

236 D. Algebraic Equations

we have,

l + (3s + a)i + (3s 2 + 2as + b)y + s3 + as2 + bs + c = O.

Choosing s = -a/3, reduces this equation to

y3 + ty + u = 0, (D.4)

where we have defined t = b _a2 /3 and u = (2a 3 -9ab+27c)/27. The next step is to express y in terms of a new unknown z so that Eq. (D.4) is transformed into a quadratic equation. For this purpose, we write y = z + v / z. With this substitution for y, Eq. (D.4) becomes,

Z6 + (3v + t)Z4 + uz3 + v(3v + t)Z2 + v3 = O.

Choosing v = -t /3 reduces this equation to

t 3

i+uz3 - - =0 27 '

(D.5)

which is a quadratic equation for Z3. From Eq. (D.2), we see that the solution is

-u + lu2 + 413

3 V 27 Z = 2 ' (D.6)

where we have taken the upper sign in Eq. (D.2). Equation (D.6) is of the form Z3 = a discussed in Section 4.2 and, in general, has three solutions,

1

u + J u2 + 4/3

1

1 Z = - 2 27 3 e i (<p+2mrr )/3 with m = 0,1,2. (D.7)

The phase <p is defined by,

-u + J u2 + ~ I-u + J u2 + ~ 1 i<P 2 = 2 e . (D.8)

Working backward from Eq. (D.7), we use our results for z to construct y (of Eq. (D.4», and from y we obtain x, the solutions to the original cubic equation, Eq. (D.3).

References

1. G. Arfken, Mathematical Methods for Physicists, 2nd ed., Academic Press, London, 1970.

2. M. L. Boas, Mathematical Methods in the Physical Sciences, 2nd ed., John Wiley and Sons, New York, 1983.

3. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, 1928.

4. A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua, McGraw-Hill Book Company, New York, 1980.

5. H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950. 6. D. J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, Englewood

Cliffs, NJ, 1981. 7. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw­

Hill Book Company, New York, 1966. 8. W. Kaplan, Advanced Calculus, Addison-Wesley, Reading, MA, 1953. 9. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry,

Van Nostrand, Princeton, 1956. 10. J. B. Marion, Classical Dynamics of Particles and Systems, 2nd ed., Academic

Press, New York, 1970. 11. P. T. Matthews, Introduction to Quantum Mechanics, 3rd ed., McGraw-Hill

Book Company (UK), Ltd., London, 1974. 12. E. Merzbacher, Quantum Mechanics, John Wiley & Sons, New York, 1961. 13. J. B. Seaborn, Hypergeometric Functions and Their Applications, Springer­

Verlag, New York, 1991. 14. E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford,

1939.

Bibliography

1. G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed., Harcourt! Academic Press, San Diego, 2001.

2. S. Axler, Linear Algebra Done Right, Springer-Verlag, New York, 1996. 3. V. D. Barger and M. G. Olsson, Classical Mechanics: A Modem Perspective,

2nd ed., McGraw-Hill, Inc., 1995. 4. R. A. Becker, Introduction to Theoretical Mechanics, McGraw-Hill Book

Company, New York, 1954. 5. R. W. Brink, College Algebra, 2nd ed., Appleton-Century-Crofts, Inc., New

York,195l. 6. R. V. Churchill, Fourier Series and Boundary Value Problems, McGraw-Hill

Book Company, New York, 1941. 7. H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics, Cambridge

University Press, Cambridge, U.K. 1956. 8. P. B. Kahn, Mathematical Methods for Scientists and Engineers: Linear and

Nonlinear Systems, John Wiley and Sons, New York, 1990. 9. E. A. Kraut, Fundamentals of Mathematical Physics, McGraw-Hill Book

Company, New York, 1972. 10. N. N. Lebedev, Special Functions and Their Applications, Dover Publications,

Inc., New York, 1972. 11. J. D. Logan, Applied Partial Differential Equations, Springer-Verlag, New

York,1998. 12. 1. Mathews and R. Walker, Mathematical Methods of Physics, Benjamin, New

York, 1965. 13. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill

Book Company, New York, 1953.

240 Bibliography

14. J. F. Randolph, Calculus, Macmillan, New York, 1952. 15. 1. R. Reitz, F. 1. Milford, and R. W. Christy, Foundations of Electromagnetic

Theory, 4th ed., Addison-Wesley, Reading, MA, 1993. 16. D. S. Saxon, Elementary Quantum Mechanics, Holden-Day, San Francisco,

1968. 17. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge

University Press, Cambridge, 1922. 18. E. T. Whittaker and G. N. Watson, A Course in Modem Analysis, Cambridge

University Press, Cambridge, 1927. 19. C. W. Wong, Introduction to Mathematical Physics: Methods and Concepts,

Oxford University Press, New York, 1991.

Index

Absolute value, 72 adjoint, 184-187 Ampere's circuital law, 58 Ampere's law, 56, 59 amplitude, 5, 10, 11 analytic function, 80 angular frequency, 167,223 angular momentum, 7, 11,21-23,26,

27,195,196,200,223 angular velocity, 7, 21, 23, 27,196,197,

216 atom, 7,103,138,170 average value, 140, 185 axis of rotation, 7,196,203,216

Bead,216, 224 beam current, 3, 9 Bessel function, 118, 130, 147 Bessel's equation, 130 Biot-Savart law, 47 black body, 138 boundary condition, 84, 86, 87, 97,

99, 107, 110, 111, 114, 115, 121-123,130-135,143,159, 160,165,168,172,174,216

boundary value problem, 127 brachistochrone, 216, 226

Cable, 218-220 cartesian coordinate system, 24 catenary, 214, 220 central force, 60 classical harmonic oscillator, 7, 82 classical physics, 137, 138 cofactor, 181 column matrix, 184 column vector, 181, 183, 184, 188 complete set, 139, 141, 148, 151,

153-155,172,193 completeness, 141, 142, 150, 151, 155 complex conjugate, 72, 146, 183, 184 complex function, 74, 119 complex numbers, 69, 71 complex plane, 70 complex variable, 70, 72 conducting sphere, 109, 110, 122, 123,

127 conservation of probability, 142, 167 conservative force, 39, 40, 142,214,

215,222 constraint, 215, 218, 219 Coulomb constant, 1, 169 Coulomb's law, 1,48, 110 coupled harmonic oscillators, 193,202 Cramer's rule, 182

242 Index

cross product, 18, 19,23,33,53,54 cube, 44, 196, 197 cubic equation, 184, 185, 198,235 curl, 33, 54, 56, 59, 60, 110 cycloid,218 cylinder, 175 cylindrical coordinates, 12, 128, 129,

228

Damped harmonic oscillator, 84, 93, 101, 162, 176

degeneracy, 190, 204 degenerate, 190 delta function, 148, 150-152, 170 density, 3, 12,23,27,41,50, 58, 170,

196,203 dependent variable, 77 derivative, 81,220 determinant, 181-184, 193, 196, 197 differential, 31, 38, 214 diffusion equation, 121 directional derivative, 35 disk, 8, 12, 225 divergence, 33, 34, 46, 60 divergence theorem, 42, 44, 46-48, 50,

60-62,66,109 dot product, 16, 17,33 driving force, 162, 163, 176, 177 drumhead, 128-130 dumbbell, 27 dynamical variable, 139, 141, 185-188

Earth's magnetic field, 10 eigenfrequency, 131, 135, 194, 202 eigenfunction, 140, 141, 143, 144, 148,

154,166,168,169,172,180,200 eigenstate, 140, 142, 144, 169 eigenvalue, 132-134, 140, 143, 166,

168, 180, 183, 184, 186-188, 190-192,196-200,202-205

eigenvalue equation, 139, 142, 143, 154, 180,199,200

eigenValue problem, 127, 128, 180, 182, 193, 196

eigenvector, 180, 183, 184, 186, 187, 190, 191, 194, 196-199, 203-205

elastic force, 82, 84, 85, 92, 93, 98, 162, 167

elastic membrane, 128, 134 elastic potential energy, 5, 7 elastic spring, 6, 10, 11, 193,202 elastic string, 132, 173 electric charge, 1-3, 8,9,19,26,49,50,

58,65,67, 100, 109, 115, 122, 170,226

electric current, 3, 9, 47, 57, 110 electric field, 1,8,48,50,85, 109-111,

115, 122, 123, 127, 135, 170, 175,226

electric flux, 50, 51 electric flux density, 109 electric monopole, 109 electric potential, 9, 115, 122, 123, 135,

158, 160, 175 electromagnetic force, 85, 226 elementary function, 90, 97, 151 emf,4,l0 energy conservation, 217 Euler's equation, 71, 82, 91 Euler-Lagrange equation, 211, 212,

215,217,219,221,222

Faraday's law, 4 Fermat's principle, 207, 209, 221, 222 fluid current, 42 flux,4,40-44,109 force constant, 6, 7, 105,202 forced harmonic oscillator, 162 Fourier amplitude, 153, 155, 170, 171 Fourier coefficient, 158, 163, 164, 173,

177 Fourier integral, 153, 155, 170, 171 Fourier series, 156, 157, 161-163, 165,

172-176 frequency, 85

Gauss's law, 170 Gauss's theorem, 44 general solution, 70, 74, 75, 78, 84, 86,

93,95,97,99,100,102,113, 118, 124, 129, 131-134, 159, 173,176,194,202,216

gradient, 32, 33, 110,215 gravitational field, 38, 217, 218, 224 gravitational force, 37, 39, 99 gravitational potential energy, 37, 39

Hamilton's principle, 215, 218, 222 heat flow equation, 121 Hermite polynomial, 147 hennitian, 146, 147, 168, 186, 188, 190,

199,205 higher transcendental function, 147 homogeneous equation, 176 homogeneous partial differential

equation, 104 Hooke's law,S, 10,69,82, 105 hydrogen atom, 169,200

Imaginary number, 70 independent variable, 77 index of refraction, 209, 222 indicial equation, 112 infinite series, 79, 106, 108, 112, 134,

135 inhomogeneous differential equation,

162, 163, 165, 176 initial condition, 83, 85, 94, 98, 99, 101,

131, 134, 194, 202, 225 integral theorems, 227 inverse, 189, 190

Kinetic energy,S, 7,131,143,165,167, 215

Kronecker delta, 141, 149

Lagrange's equation, 216, 222, 223, 226 Lagrangian, 215, 216, 223-226 Laguerre function, 147 Laplace's equation, 110, 111, 122, 124,

135, 158, 159, 175 Laplacian, 33, 34 law of cosines, 17 law of reflection, 208 law of refraction, 209 Legendre function, 147 Legendre polynomial, 113-115, 127 Legendre's equation, 112, 127, 177 light, 207, 221, 222 line integral, 51-53, 55, 58 linear differential equation, 78

Magnet, 40, 50, 57 magnetic charge, 47, 50 magnetic dipole, 65

Index 243

magnetic field, 4, 10, 20, 26, 40, 41, 50, 56-58,65,67,85,100,109,226

magnetic flux, 4, 40, 41, 47 magnetic flux density, 4, 41, 47, 65, 109 magnetic force, 100 magnetic force constant, 65 magnetic monopole, 56, 109 magnetic pole, 40, 47, 56, 58 magnitude of a vector, 15 matrix, 180, 181, 183, 185-190, 193,

196,198-200,202,204,205 matrix mechanics, 179 Maxwell's equations, 29, 32, 47, 59, 66,

226,229 mean value theorem, 31 moment of inertia, 7, 8,11,12,196,

197,203,204 moment of inertia tensor, 196, 197,203,

204 momentum, 21, 26, 119, 142, 154, 165,

225

Natural angular frequency, 82, 105 natural frequency, 85 Newton's law of universal gravitation,

37 Newton's second law, 21, 29, 60, 69,

77,82,84,86,93,97,100,103, 193,215,225,226

Nonlinear differential equation, 79 nonlinear partial differential equation,

105 nonnal, 41, 48, 53 nonnal mode, 193, 194,202 nonnalization, 113, 148,201,202 nonnalization constant, 143, 165, 167,

168 nonnalize, 140

Observable, 138-141 one-dimensional box, 131, 168 operator, 32, 33, 103, 116, 139, 140,

142, 143, 147, 166, 188, 199 optics, 207 ordinary derivative, 104 ordinary differential equation, 77, 78,

103, 105, Ill, 116, 119, 120, 122, 128, 129, 143, 159

ordinary point, 92

244 Index

orthogonal, 27,148,177,183,184,186, 191,204,205

orthogonal function, 137, 145, 147 orthogonality, 146, 201, 202 orthogonality relation, 177 orthonormal, 148, 166, 180, 184, 187,

201,202,204,205

Parity, 134 partial derivative, 29-31, 34, 52, 68,104 partial differential equation, 103-105,

109, 110, 116, 119-121, 124, 129, 133, 142, 143, 179

Pauli spin matrices, 200, 201 pendulum, 222 period, 10, 11, 175 periodic force, 164 periodic functions, 161 photoelectric effect, 138 Planck's constant, 103, 169 polynomial, 88-90, 94-96, 98, 108,

109,112,113,119-122,147, 148, 177

polynomial solution, 88, 94-96,148 potential energy, 5, 7, 38-40, 60,

105, 116, 119, 131, 133, 134, 142-145,166,173,215-219, 222,224,225

principal axis, 7,196,197,203,204 probability, 144, 145, 154, 166--168,

170 pulley, 224 Pythagorean theorem, 15

Quadratic equation, 70, 235 quantum harmonic oscillator, 105, 106,

143, 144, 147, 166 quantum physics, 137, 188

Real numbers, 70, 74 rectangular components, 195 rectangular coordinates, 36, 54, 59, 60,

65--67, 110, 123, 152, 188, 196, 204,228

rectangular coordinate system, 15, 60--62,158,174,203

rectangular pulse, 153, 161, 162 recursion formula, 94-96,106--109,112 reflection, 207

refraction, 207, 209 relativistic energy, 119 relativistic mass, 225 right-handed screw, 18, 19,41,57, 188 rigid body, 195 rod, 11,27 root, 73, 75, 183-185, 194, 196--198 rotation axis, 196 row matrix, 184

Scalar, 1,4, 13, 17,26,32-34,60, 110, 124,196,215,226

scalar product, 16, 17,23,25,34,183, 184

Schmidt orthogonalization method, 192, 201,204

SchrOdinger's equation, 29, 103, 108, 115,117,119,120,142,143, 147,179,200,230

Schwarz inequality, 26 self-adjoint, 146 separation constant, 105, 120, 122, 123,

127, 129, 131, 133, 134, 159 separation of variables, 104, 111, 119,

120, 122, 123, 128, 129, 133, 134, 143, 158

series, 83, 88, 94-97, 102, 107, 108, 112, 113, 117, 118, 135

series solution, 107, 112, 121, 174 sheet, 203 simple harmonic motion, 5, 6, 10, 11 simple harmonic oscillator, 6, 7, 143,

166,176 singular point, 92, 94, 95 singularity, 91, 92 Snell's law, 209 solid angle, 50 special function, 147 sphere, 23, 27, 46, 61, 110, 115, 123,

124,135 spherical Bessel function, 118 spherical cap, 55, 64 spherical Neumann function, 118 spherical polar coordinates, 22, 36, 38,

48, 55, 56, 67, 111, 116, 124, 228

spherical shell, 168 spike, 150 spin, 200

spring, 6, 11, 84, 85, 10 1, 193, 202 standard form, 91 state function, 103, 139, 141 stationary value, 209, 212, 218, 219, 221 steady-state solution, 176, 177 Stokes's theorem, 50, 53-56, 59, 62, 63 string, 132, 133, 173, 223, 224 Sturm-Liouville equation, 147, 148, 177 Sturm-Liouville theory, 145 surface integral, 44, 48, 53, 55, 56 surface of revolution, 213

Taylor series, 81, 82, 92, 98 torque, 21, 26 transpose, 183 triple cross product rule, 20, 26

Unitary, 190, 199

Index 245

Variational principle, 207, 209, 214-216

vector, 1,4, 13-15, 18,21,23-26, 32-34,40,44,53,56,59,61-63, 65, 110, 122, 180, 187-190, 193

vector differential operator, 32, 33 vector identities, 34, 227 vector product, 18,25 vector relations, 227 vectors, 67,184,191,201,202,204 viscous fluid, 39 viscous force, 84, 85, 93, 99, 162, 177

Wave function, 103, 131, 154, 165 wave mechanics, 179 weighting factor, 141 weighting function, 146, 168 work, 5, 17,39,58, 105,214