A 2008 10 (2010 4 )

1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2.................................. 4 1.3.3................................... 4 1.3.4............................ 4 1.3.5.................... 6 2 7 2.1................................. 7 2.2 Taylor................................... 8 2.2.1........................ 8 2.2.2 Taylor............................... 11 2.2.3.......................... 13 2.3................................... 15 2.3.1................................. 20 2.3.2 [ ]............................ 21 2.4................................ 23 2.4.1 (chain rule)...................... 23 2.4.2 (Leibniz rule)........................ 24 2.4.3.............................. 25 2.5........................ 25 2.5.1 x................................ 25 2.5.2................................. 26 2.5.3................................. 27 2.6.................................. 29 2.6.1 AM................................. 29 2.6.2................................... 30 2.6.3............................. 31 2.7....................................... 32 iii

iv 2.7.1............................... 32 2.7.2....................... 33 2.7.3.......................... 34 2.8 Fourier............................... 36 3 Fourier 38 3.1 Fourier.............. 38 3.1.1.............................. 38 3.1.2.............................. 40 3.2 Fourier.............. 41 3.2.1.............................. 41 3.2.2.............................. 42 3.3 Fourier.......................... 43 3.4........................... 45 3.4.1......................... 45 3.4.2.......................... 46 3.4.3.................................. 47 3.4.4............................... 48 3.5 Dirac............................... 49 3.5.1 Kronecker Dirac................. 49 3.5.2................... 53 3.5.3 Fourier..................... 55 3.5.4 Fourier.................. 57 3.5.5......................... 58 3.6................................ 58 3.6.1.................................. 58 3.6.2................................. 59 3.6.3.......... 59 4 62 4.1................................... 62 4.1.1............................ 63 4.2................................. 64 4.2.1......................... 65 4.3..................................... 66 4.3.1........................ 66 4.3.2............................... 67 4.3.3 Fourier............................ 69 4.4........................... 72

v r = ix + jy + kz, f(r) = f(x, y, z), r = i x + j y + k z = grad =, 2 r = 2 r r = 2 = = 2 x + 2 2 y + 2 2 z, 2 df(r) = f(r + dr) f(r) = gradf(r) dr = f(r) dr = f(r) r dv = d 3 r = dx dy dz ( = dτ), dr, i = 1, ( ) z = x + iy = r e iθ, z = x iy = r e iθ, z = x 2 + y 2 = r, arg z = tan 1 y x = θ, ( ) ( ) ( ) Re z = x, Im z = y, ( ) ( )

vi A α a alpha B β b beta Γ γ g gamma δ d delta E ɛ, ε e epsilon Z ζ z zeta H η e eta Θ θ, ϑ th theta I ι i iota K κ k kappa Λ λ l lambda M µ m mu N ν n nu Ξ ξ x xi O o o omicron Π π p pi P ρ r rho Σ σ s sigma T τ t tau Υ υ u, y upsilon Φ φ, ϕ ph phi X χ ch chi Ψ ψ ps psi Ω ω o omega

( ) vii

viii ( ) (ζ) (ξ) http://kscalar.kj.yamagata-u.ac.jp/~endo/greek/zeta_xi.html

1 1.1 1.1 e iθ θ e iθ = 1 e iθ = cos θ + i sin θ (1.1.1) z = x + iy z = x 2 + y 2 e iθ = cos 2 θ + sin 2 θ = 1 e iθ 1 1.2 1.2 z z 2 = z 2 1

2 A ( ) 1.2 or z z = x + iy ; 2 z, z (modulus) r = z, r = z (argument) θ = arg z, θ = arg z z z z z = z z, arg(z z ) = arg(z) + arg(z ) z = x + iy i (arg i = π/2 = 90 ) 90 i i = i 2 = 1 1 = 180 90

3 1.3 1.3.1 z = 1 z θ = arg z z θ θ z = z 1 z θ z = f(θ) f(θ) f(θ ) = f(θ + θ ) θ θ θ + θ f(0) = 1 f(θ) z = f(θ) = e cθ c c θ z = e cθ ( ) 1 θ 0 2π z z = 1 z = 1 2π 2π 1 z = 1 (θ = 0) 1 θ = 0 ( ) i de cθ dθ = c = i, θ=0 c = i 1 θ z z = e iθ e iθ e iθ = cos θ + i sin θ ( )

4 A 1.3.2 1 z r (= z ), θ (= arg z) z r (> 0) θ r e iθ z z = re iθ ( ) 1.3.3 e iθ = 1 e iθ ( e i(α+β) = e iα e iβ ) i e iθ 0 θ 1 1.3.4 1.3 z = 3 e iπ/6 z

5 1.4 x f(x) = e ikx k f(x) 1.5 z = i 1.6 z = 1 + i 1.7 z = 1 + i z 12 1.8 z 7 = 1 1.9 z 7 = 1 7 ω 0 = 1, ω 1 ω 2,..., ω 6 1 + ω 1 + ω 2 + + ω 6 = 0, 1.10 z 11 = 1 11 ω 0 = 1, ω 1 ω 2,..., ω 10 1 + ω 1 + ω 2 + + ω 10 = 0, 1.11 z z = z 2 z z 1.12 z = e a+ib a, b 1.13 z = e a+ib a, b 1.14 z = e iθ θ 0 π z 1.15 z = e iθ θ 0 π z 1.16 z = e iθ θ 0 3π/2 z 1.17 z = 2e iθ θ 0 2π z 1.18 θ 0 2π z (1) z = 1 + e iθ (2) z = i + e iθ (3) z = 1 + i + e iθ (4) z = 1 + i + 2 e iθ 1.19 θ 0 2π ( z 1 = 1 + θ ) ( e iθ z 2 = 2 + θ ) ie 2iθ 2π π

6 A 1.20 ( ) θ 0 2π 4 z = 3 + cos θ eiθ 1.21 ( ) θ z 1 = θ 2π eiθ (θ > 0) z 2 = e θ+iθ ( < θ < ) 1.22 ( ) z 2 α e α 1.3.5 1.23 ( e iθ) 2 = e 2iθ cos, sin 1.24 ( e iθ) 3 = e 3iθ cos, sin 1.25 e i(α+β) = e iα e iβ cos, sin 1.26 sin θ, cos θ 1.27 α = α + β 2 β = α + β 2 + α β, 2 α β 2 e iα + e iβ, e iα e iβ

2 2.1 2.1 x = 0 x Taylor x 3 (1) e x (2) (1 + x) α 2.2 ( ) dy (1) dx = y d 2 y (2) dx = 2 ω2 y (ω > 0) 2.3 y = Ae x2 d 2 y/dx 2 A 2.4 x (i.e. x ) x 10, x 10, x 1/100, 1, e x, e x, ln x. 2.5 y = f(x) y = f(x) cos 6x y y = f(x) π 2 0 π 2 π 3π 2 2π 5π 2 x 2.6 e ax2 dx. (a > 0) 7

8 A e ξ2 dξ = π 2.7 f(x) [a, b] [a, b] N x n x n = a + (b a)n/n (n = 0, 1,, N) f(x) [a, b] lim N lim N 1 N N f(x n ), n=1 N f(x n ), n=1 lim N lim N N f(x n )(b a), n=1 1 N N f(x n )(b a). n=1 2.8 x-y (r, ϕ) (x = r cos ϕ, y = r sin ϕ.) 2.9 x-y-z (r, θ, ϕ) (x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ.) 2.2 Taylor 2.2.1 2.10 1.01 2 3 2.11 1.01 3 3 1.01 2 1.01 2 = (1 + 0.01) 2 = 1 + 2 0.01 + 0.01 2 = 1 + 0.02 + 0.0001 = 1.0201 1.01 3 4 1.02

9 0.01 2 1.01 3 3 1.01 3 = (1 + 0.01) 3 = 1 + 3 0.01 + 3 0.01 2 + 0.01 3 1 + 3 0.01 = 1.03 3 0.01 2 3 4 1.01 3 = (1 + 0.01) 3 = 1 + 3 0.01 + 3 0.01 2 + 0.01 3 1 + 3 0.01 + 3 0.01 2 = 1.0303 0.01 x f(x) x x f(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + (2.2.1) f(x) x = 0.01 x 2 = 0.001, x 3 = 0.000001 x 1 1.01 f(x) = 1/(1 + x) x (2.2.1) f(x) f(x)(1 + x) = 1 1 = (c 0 + c 1 x + c 2 x 2 + c 3 x 3 + )(1 + x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + + c 0 x + c 1 x 2 + c 2 x 3 + c 3 x 4 + = c 0 + (c 1 + c 0 )x + (c 2 + c 1 )x 2 + (c 3 + c 2 )x 3 +

10 A c n c 0 = 1, c 1 + c 0 = 0, c 2 + c 1 = 0, c 3 + c 2 = 0, c 0 = 1, c 1 = c 0 = 1, c 2 = c 1 = 1, c 3 = c 2 = 1, 1 1 + x = 1 x + x2 x 3 +. (2.2.2) x = 0.01 1 1.01 = 1 0.01 + 0.012 0.01 3 + 1 0.01 + 0.01 2 = 0.0901 2.12 1 + x e x e ( e = lim 1 + 1 n. (2.2.3) n n) x > 0 ( e x = lim 1 + 1 ) nx n n = lim N ( 1 + x N ) N. N = nx N ( 1 + x ) N N x N(N 1) ( x ) 2 N(N 1)(N 2) ( x ) 3 = 1 + + + + N 1! N ( 2! N 3! N = 1 + x + 1 1 ) ( 1 N 2! x2 + 1 1 ) ( 1 2 ) 1 N N 3! x3 + N e x = 1 + x + 1 2! x2 + 1 3! x3 + (2.2.4) x > 0 x 0 2.13 e 0.1 3

11 2.2.2 Taylor 1/(1 + x) e x Taylor f(x) x = a Taylor f(x) = f(a) + f (a)(x a) + f (a) (x a) 2 + f (a) (x a) 3 + (2.2.5) 2! 3! f (n) (a) = (x a) n, n! n=0 a = 0 f(x) = f(0) + f (0)x + f (0) x 2 + f (0) x 3 + (2.2.6) 2! 3! f (n) (0) = x n. n! n=0 Taylor x f(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + (2.2.7) ( ) c n (2.2.7) x = 0 f(0) = c 0. (2.2.7) f (x) = c 1 + 2c 2 x + 3c 3 x 2 + (2.2.8) x = 0 f (0) = c 1. (2.2.8) f (x) = 2c 2 + 3 2c 3 x + 4 3c 4 x 2 + (2.2.9) x = 0 f (0) = 2c 2. (2.2.9) x = 0 f (0) = 3 2 c 3.

12 A f (n) (0) = n! c n, c n = f (n) (0) n! (2.2.7) (2.2.6). f(x) = e x f (n) (x) = e x f (n) (0) = 1. (2.2.4) e x = 1 + x + 1 2! x2 + 1 3! x3 + 1 = n! xn (2.2.10) n=0 (1 + x) α ( ) (1 + x) α α(α 1) = 1 + αx + x 2 α(α 1)(α 2) + x 3 + 2! 3! ( ) α = x n. (2.2.11) n n=0 α n ( ) α α(α 1)(α 2) (α n + 1) = n n! (2.2.12) α α = m m C n α α Taylor ( ) f(x) = f(0) + f (0) + 1 2! f (0) + 1 3! f (0) +, ( ) f(x) = f(x) + f (x)x + 1 2! f (x)x 2 + 1 3! f (x)x 3 +. 2.14 1 2.15 2

13 2.2.3 2.16 x = 0 (1 + x) 2 2.17 x = 0 (1 + x) 3 2.18 x = 0 (1 + x) 4 2.19 x = 0 (1 + x) 1/2 2.20 x = 0 (1 + x) 1/2 2.21 x = 0 (1 x) 1/2 2.22 (1 x 2 ) 1/2 x = 0 2.23 m v E E = mc 2 1 v2 /c 2 c v c E (v 2 /c 2 ) 2.24 f(x) = (1 x) 1 f(0), f (0), f (0), f (0), 2.25 (1 x) 1 x = 0 Taylor 2.26 (1 x) 1 x

14 A 2.27 n 1 x k = 1 xn 1 x k=0 x < 1 n 2.25 2.26 2.28 f(x) = ln(1 x) f(0), f (0), f (0), f (0), 2.29 2.28 ln(1 x) x = 0 2.30 2.27 x 2.29 2.31 e x f(x) = e x2 x = 0 2.32 f(x) = 1 1 + x x = 0 g(t) = 1 2 1 + t f(x) = 1 Taylor 1 + x2 2.33 ( ) x F x F = f(x) a x = a F = 0 f(a) = 0 F = f(x) x = a Taylor F = f(a) + f (a)(x a) + f (a) (x a) 2 + 2 = f (a)(x a) + f (a) (x a) 2 +, 2 (x a) (x a) (x a) F f (a)(x a). k = f (a)

15 2.3 2.34 x (2x 1) 2 = 4x 2 4x + 1, 4x 2 4x + 1 = 2, (x + 1)(x 1) + 1 = 4x 2. 3x 9 = 0, x x x dy dx = F (x, y), y = y(x) F (x, y) y y = f(x) df(x) dx = F (x, f(x)), x ( ) f(x) (or ) (or ) x 2 = 6 1 x 2 + 0 x 6 = 0

16 A x = 0 ± 0 2 4 1 ( 6) 2 1 = ± 6 x = ± 6 ax 2 + bx + c = 0, (a 0) x 2 + b a x + c a = 0 ( x + b ) 2 b2 2a 4a + c 2 a = 0 ( x + b ) 2 = b2 4ac ( ) 2a 4a 2 x + b 2a = ± b2 4ac ( ) 2a x = b 2a ± b2 4ac = b ± b 2 4ac. 2a 2a ( ) ( ) x 2 = A x = ± A x 2 = 6 x = ± 6 x 2 = 6 2.2 (1) dy dx = y (y = e x ) de x /dx = e x y = Ae x (A: ) y = e x + A dy/dx = d(e x + A)/dx = e x = e x + A A = y A y

17 ( ) dy y = dx dy y = dx ln y = x + C y = e C+x y = C e x C C = ±e C x 2 = 6 dy/y = ln y d ln y/dy = 1/y de x /dx = e x 2 dy = 2xy + x + 6y + 3 dx dy dx = x(2y + 1) + 3(2y + 1) = (x + 3)(2y + 1) dy 2y + 1 = (x + 3)dx 1 2 log 2y + 1 = 1 2 (x + 3)2 + C (C : ) 2y + 1 = ±e 2C e (x+3)2 y = Ae (x+3)2 1 2 (A = ±e 2C /2)

18 A ( ) a a 2 3 5 2.35 R, ϕ a [ ] R l a/l = ϕ/2π a = l ϕ 2π = 2πR ϕ 2π = Rϕ. 2.36 2 log 2 5 [ ] x x = 2 log 2 5. log 2 log 2 x = log 2 2 log 2 5 = log 2 5 log 2 2 = log 2 5 log 2 x = 5, 2 log 2 5 = 5 2.37 a < b 2 a < 2 b [ ] 2 a 2 b a b 2 a 2 b log 2 2 a log 2 2 b a log 2 2 b log 2 2 a b. log 2 x (x > 0) 2.2 (2) d 2 y dx 2 = ω2 y

19 x t y x d 2 x dt = 2 ω2 x m k = mω 2 m d2 x dt = kx 2 k m (sin cos ) x = sin αt α 2 = k/m = ω 2 x = sin ωt y = sin ωx cos y = A sin ωx + B cos ωx, y = A sin(ωx + ϕ), (A, B : ) (A, ϕ : ) y = Ae iωx + Be iωx. (A, B : ) ( ) y = e αx α 2 = ω 2, α = ±iω y = e ±iωx 2.38 d 2 y dx 2 = κ2 y. (κ > 0) 2.39 y = e 2x y = 2e x y = e x + 2 d 2 y dx 3dy + 2y = 0. 2 dx

20 A 2.40 m F = mgz z = 0 z g ( ) 2.3.1 [ ] dy dx = 0 y = A (A = const.) d2 y dx = 0 dy 2 dx = A, (A = const.) y = Ax + B (B = const.). u(x, y) u(x, y) x = 0 u(x, y) = f(y), (f(y) y ) 2 u(x, y) u(x, y) = 0 = f(y), (f(y) y ) x 2 x u(x, y) = f(y)x + g(y). (g(y) y ) 2 1

21 2 1 3 2 2.41 u(x, t) = f(x vt) u t + v u x = 0, (v = const.) f 2.42 u(x, t) = f(x vt) + g(x + vt) 2 u t 2 = v2 2 u x 2, (v = const.) f, g 2.3.2 [ ] dy = F (x, y), dx f(x) df(x) dx = F (x, f(x)), x ( ) f (x n ) f(x n+1) f(x n ). x x n x x = x n+1 x n a < x < b N n x n x n = a + b a N n = a + n x. (n = 0, 1,, N) a = x 0 x 1 x 2 x 3 x n x n+1 x N 1 x N = b x

22 A N x = (b a)/n f(x n+1 ) f(x n ) x F (x n, f(x n )), f(x n+1 ) f(x n ) + F (x n, f(x n )) x x = x n f(x n ) x = x n+1 f(x n+1 ) x = x 0 f(x 0 ) f(x 1 ), f(x 2 ) f(x n ) x = x 0 f(x 0 ) f(x) d 2 y dx 2 = F ( x, y, dy dx ). f (x) = F (x, f(x), f (x)) x f (x n ) f (x n+1 ) f (x n ) x f(x n+2) 2f(x n+1 ) + f(x n ) x 2. f(x n+2 ) 2f(x n+1 ) f(x n ) + F ( x n, f(x n ), f(x ) n+1 f(x n ) ( x) 2, x 2 x = x n x n+1 f(x n ), f(x n+1 ) x n+2 f(x n+2 ) x = x 0, x 1 f(x 0 ), f(x 1 ) f(x 2 ) f(x 2 ) f(x 1 ) f(x 3 ) f(x n ) f(x 0 ), f(x 1 ) f(x 0 ) f(x 1 ) f(x 0 ) f (x 0 ) = [f(x 1 ) f(x 0 )]/ x

23 u x = u y. x y u = u(x, y) f(x, y) f(x, y) f(x, y) = x y x x f x (x n, y) f(x n+1, y) f(x n, y). ( x = x n+1 x n ) x f(x n+1, y) f(x n, y) + f(x n, y) y x = x 0 f(x 0, y) g(y): y f(x n, y) f(x 0, y) y f(x 0, y)/ y f(x 0, y) f(x 1, y) f(x 1, y) y f(x 1, y) f(x 1, y)/ y f(x 2, y) n f(x n, y) x = x 0 f(x 0, y) = g(y) x 2.4 2.4.1 (chain rule) y = f(x), z = g(y) dz dx = dz dy dy dx, (2.4.1) { g ( f(x) )} = g ( f(x) ) f (x), (2.4.2) { e f(x) } = f (x) e f(x),

24 A {ln f(x)} = f (x) f(x). y = f(x), z = g(y), u = h(z) du dx = du dz dz dy dy dx, { ( h g ( f(x) ))} ( = h g ( f(x) )) g ( f(x) ) f (x). 2.43 2.44 2.45 x d dx ef(x) = f (x) e f(x) d dx exp [ g(x) 2] = 2g(x) g (x) exp [ g(x) 2] [ 2 sin 3 (5x + 1) + 7 ] 4 2.46 x ln { sin ( e x2 )} 2.4.2 (Leibniz rule) {f(x)g(x)} = f (x)g(x) + f(x)g (x), (2.4.3) {f(x)g(x)} = f (x)g(x) + 2f (x)g (x) + f(x)g (x). (2.4.4) 2.47 {f(x)g(x)} 2.48 {f(x)g(x)h(x)} 2.49 { e f(x)} { 2.50 e +x+1} 2x2

25 2.4.3 y = f 1 (x) x = f(y). dy dx = 1 dx dy = 1 f (y). 2.51 ln x e x (ln x) = 1/x (e x ) = e x 2.52 x y dy/dx (1) y = tan 1 x (2) y = sin 1 x (3) y = cos 1 x 2.5 2.5.1 x x > 0 a a > 0 x a dx a dx = axa 1 > 0. (a > 0, x > 0) a a < 0 x a a > 0 a x a a < 0 a (i.e. ) x a a, b x a < b x a < x b x 100 x 10 x 1/100 x 1/10 x 0 = 1

26 A 2.5.2 e x x 1 x e x x e x 2 e x e x e x2 e ex, as x 2.53 A 1000 A 1000 1 2 2 2 4 1 2 A 1 (1) 2 3 4 A (2) n (3) 11 A 1 ( 2 10 = 1024 1 10 3 ) 1 x 10 10 e x 10 e x x 10 < e x (x ) 2 1 + r + r 2 + r 3 + + r n 1 = rn 1 r 1

27 (4) 21 A 1 ( 2 20 = 2 10 2 10 ) (5) 4 3 51 A 1 (6) A 2.54 9 9 81 1 1 2 2 3 4 21 41 61 81 kg 2 10 = 1024 10 3 2 20 = 2 10 2 10 10 6. 10kg 50 1 1000 6 10 24 kg 2.55 B 18.9 10 4 8 40 80 ( 1.189 4 2.0) 2.5.3 ln x (= log e x) ε > 0 x ε ln x x +0 y = e x x y x = ln y y x e x ln x ln x ln x ln x 1/100 = 1 ln x ln ln x, as x 100

28 A 2.56 x 1/10000 x > 0 2.57 x x 2, e x2, (x 2 + 1)e x2, 1, 0. 2.58 x ( ln 2 0.693, ln 10 2.30) e x, e x 2, 2 x, 10 x. 2.59 x e x, e x2, x x ( x = e ln x ) 2.60 20cm 1cm ( ) ( ) 10cm 10cm 10cm 2 10cm 5cm 15cm 15cm 15cm = 10cm + 5cm = ( 1 + 1 ) 10cm 2 3 2 3 3 3 2 1 10cm 2 10cm 1 3 15cm + 10cm 3 = ( 1 + 1 2 + 1 3 ) 10cm 2 1 : 1 3 1 2m 10cm m 10cm 1/2 2 : 1 10cm 1/3

29 1m ( m) 5m 6m n 1 + 1 2 + 1 3 + + 1 n n 1 1 dx = ln n x e 10 2.2 10 4, e 50 5.2 10 21, e 60 1.1 10 26, 1 10 16 m, 130 10 28 m 2.6 2.6.1 AM 2.5 y = f(x) cos 6x cos 6x f(x) AM f(x) y y = A y = A cos 6x 0 π 2π x y = A y y = f(x) y = f(x) cos 6x 0 π 2π x y = f(x) f(x)

30 A cos 6x = ±1, 0 y = ±f(x). x. 2.61 y = 2.62 1 cos 4x. x 2 + 22 y = sin 10x. x x = 0 1/x x 0 sin x/x 1 2.63 y = cos 10x + cos 8x ( cos α + cos β = 2 cos α β cos α + β ) 2 2 2.6.2 y = xe x2 x e x2 = 1/e x2 x ± y = xe x2 0 y = 0 (x = 0), y ( ) (x < 0 y < 0 x > 0 y > 0) y y = x y = e x2 y = xe x2 O x 2.64 y = (x 1)e x2 2.65 y = (x 2 1)e x2 2.66 y = x(x 2 1)e x2

31 2.67 y = x(x 2 1)(x 2 4)e x2 2.68 y = xe x 2.69 y = (x 2 1)e x 2.70 y = 1 + x (x > 0). x 2.71 y = 1 x 2 + x2 (x > 0) 2.6.3 y = f(x) x f (x) y = f(x) y = f (x) 2.72 y = f(x) f = f (x) y y = f(x) y y = f(x) y = 1x 2 O x x O (a) (b) y y = f(x) y O x O y = f(x) x (c) (d)

32 A 2.7 2.7.1 f(x) [a, b] [a, b] N x n x n = a + (b a)n/n (n = 0, 1,, N) x = x n+1 x n = (b a)/n b a f(x)dx = lim N N f(x n ) x. (2.7.1), n=1 x dx, x ( ) dx ( ) [ ] b { b 2 {f(x)} 2 dx = f(x) dx}. a a 2 + b 2 + c 2 + = (a + b + c + ) 2 2.73 2.6 e ax2 dx = a (e x2) ( a a dx = e dx) x2 = ( π ) a.

33 2.7.2 2.6 ax = ξ ( ) I = e ax2 dx = e dξ π ξ2 = a a. (dimensional analysis) x [ x ] = [ L ]. I [ I ] = [ e ax2 ] [ dx ] = 1 [ L ] = [ L ]. dx (2.7.1) x = x n+1 x n [dx] = [L] e e n [ e ax2 ] = 1. e e [ ax 2 ] = 1 a [ a ] = [ L 2 ]. x I x a I a I = const. a 1/2 const. a = 1 a = 1 I = π const. = π I = π a. 2.74 f(x) = 1 (k 2 + a 2 ) 2 eikx dk, [ x ] = [ L ] [ df(x) [ k ], [ a ], [ f(x) ], dx ].

34 A 2.75 I = x 2 e x2 /b 2 dx, [ x ] = [ L ] [ ] e x 2 /b 2, [ b ], [ I ]. 2.76 2.77 2.78 0 x 2 e x2 /b 2 dx = const. b 3. (b > 0) dx (x 2 + a 2 ) 2 = const. a 3. (a > 0) x 3 e x2 /a 2 dx = const. ( a ) (x 2 + a 2 ) 2 2.7.3 2 f(x, y) dxdy { x = f(u, v) S y = g(u, v) ( ) ds = dxdy dudv dxdy = (x, y) (u, v) dudv (2.7.2) (x, y)/ (u, v) (Jacobian) x x (x, y) (u, v) = u v y y (2.7.3) u v

35 3 V f(x, y, z) dxdydz, x = f(u, v, w) y = g(u, v, w) z = h(u, v, w) ( ) dv = dxdydz dudvdw dxdydz = (x, y, z) (u, v, w) dudvdw (2.7.4) (x, y, z)/ (u, v, w) x u (x, y, z) (u, v, w) = y u z u x v y v z v x w y w z w (2.7.5) 2 (r, ϕ) (x = r cos ϕ, y = r sin ϕ) (x, y) (r, ϕ) = x/ r x/ ϕ y/ r y/ ϕ = cos ϕ r sin ϕ sin ϕ r cos ϕ = r. dxdy = rdrdϕ, (2.7.6) 3 (r, θ, ϕ) (x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ) (x, y, z) x/ r x/ θ x/ ϕ sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ (r, θ, ϕ) = y/ r y/ θ y/ ϕ = sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ = r 2 sin θ. z/ r z/ θ z/ ϕ cos θ r sin θ 0 dxdydz = r 2 sin θdrdθdϕ (2.7.7)

36 A [ ] z r sin θ dϕ dr r dϕ rdϕ θ dϕ r sin θ 0 ds = dr rdϕ dθ 0 2.79 2 (x, y) x = u + v 2 y = u v 2 r rdθ dr dv = dr rdθ r sin θdϕ ds = dxdy (u, v) 2.80 R S = πr 2 2.81 R V = 4 3 πr3 2.82 xy e x2 y 2 dxdy 2.8 Fourier 2.83 x L 2 u t 2 = T σ 2 u x 2

37 T σ u = u(x, t) t x (x = 0, L) x = 0 L u = 0 u x Fourier u(x, t) = q n (t) sin nπx L n=1 sin (u t t- Fourier q n ) t Fourier q n (t) (n = 1, 2, 3,... ) n=1 A n sin nπx L = n=1 B n sin nπx L A n = B n (n = 1, 2, 3,... ) Fourier 2.83 Fourier q n (t) d 2 q n (t) dt 2 = T σ ( nπ L ) 2 qn (t), (n = 1, 2, ) (2.8.1) [ ] n q n (t) n ω n q n T ω n = σ nπ L. u(x, t)

3 Fourier 3.1 Fourier 3.1.1 3.1 e ikx L k L f(x) ( L ) f(x) = c n e iknx. n= k n = 2πn L (3.1.1) c n { 1 L/2 e ikmx e iknx 1 (n = m) dx = δ mn = L L/2 0 (n m). (3.1.2) (3.1.1) e ik mx x L/2 L/2 (3.1.2) 1 L/2 e ikmx f(x) = 1 L L/2 L n= = δ mn c n 3.1 f(x) [ π, π] f(x) = n= L/2 L/2 e ikmx e iknx dx c n = c m. (3.1.3) { 1 (0 < x < π) 1 ( π < x < 0), (3.1.4) 2π f(x) 38

39 c n (3.1.3) c n = 1 2π = 1 2π π π { 0 e inx f(x) dx π 1 e inx dx + = 1 iπn [1 ( 1)n ] = f(x) = 2 iπ odd n 1 n einx = 2 iπ π 0 } 1 e inx dx 0 (n: even) 2 iπn (n: odd) m=. (3.1.5) 1 2m + 1 ei(2m+1)x. (3.1.6) f(x) = 2 ( 1n iπ e inx + 1n ) einx odd n>0 = 4 1 sin(2m + 1)x (3.1.7) π 2m + 1 m=0 = 4 (sin x + 13 π sin 3x + 15 ) sin 5x + (3.1.8) y = 4 π sin x 1 y = 4 π ( sin x + 1 sin 3x) 3 π 0 π x -1 y = 4 1 sin 3x π 3 3.2 3.1 f(x) 3.3 (3.1.2) Parseval 1 L/2 f(x) 2 dx = c n 2. (3.1.9) L L/2 n=

40 A 3.4 3.3 Riemann-Lebesgue 3.5 Riemann-Lebesgue L/2 lim e 2iπnx/L f(x) dx = 0, (3.1.10) n ± L/2 3.1.2 x, y, z L f(x, y, z) = f(r) f(x, y, z) = c nx,n y,n z e ikxx e ikyy e ikzz. n x = n y = n z = k i = 2πn i L (i = x, y, z) (3.1.11) f(r) = n cne ik r, k = 2πn L (3.1.12) n = (n x, n y, n z ) (3.1.2) { 1 e ik r e ik r d 3 r = δ L 3 kk = δnn = 1 (n = n ) 0 (n n ), (3.1.13) : cn = 1 e ik r f(r) d 3 r. (k = 2πn L 3 L ) (3.1.14) 3.6 (3.1.13) Parseval 1 f(r) 2 d 3 r = cn 2. (3.1.15) L 3 n

41 3.2 Fourier 3.2.1 f(x) (3.1.1) L k n k n+1 k k = k n+1 k n = 2π/L 0 k n f(x) = 1 2π c(k) (3.1.3). c(k) = (3.1.1) L f(x) = lim c(k)e ikx dk, (3.2.1) e ikx f(x) dx, (3.2.2) L n= c n L (3.1.3) c n = 1 L L/2 L/2 e ik nx f(x) dx 1 L k n = 2πn/L L c n e iknx. (3.2.3) e ik nx f(x) dx 1 L c(k n) (3.2.4) k k n+1 k n = 2π L 0, (3.2.5) L n k n = 2π/L f(x) = lim L n= = 1 2π lim = 1 2π L n= c(k n )e iknx 1 L c(k n )e iknx k c(k)e ikx dk, (3.2.6)

42 A f(x) f(x) = 1 2π c(k)e ikx dk, (3.2.7) c(k) = e ikx f(x) dx. (3.2.8) c(k) f(k) = c(k)/ 2π x f(x) k f(k) f(x) = 1 2π f(k)e ikx dk, (3.2.9) f(k) = 1 2π e ikx f(x) dx. (3.2.10) f(k) f(x) (Fourier transform) (3.2.10) (3.2.9) 3.7 i) f 1 (x), f 2 (x) f 1 (k), f 2 (k) c 1 f 1 (x)+c 2 f 2 (x) c 1 f1 (k) + c 2 f2 (k) ii) f(x) f(k) f(x)/ x ik f(k) 3.8 L 3.3 Paraseval 3.9 f(x) = e x2 /2σ 2 3.2.2 f(x, y, z) f(x, y, z) = 1 (2π) 3/2 e i(k xx+k y y+k z z) f(kx, k y, k z ) dk x dk y dk z, (3.2.11)

43 f(k x, k y, k z ) = 1 e i(k xx+k y y+k z z) f(x, y, z) dxdydz. (3.2.12) (2π) 3/2 1 f(r) = e ik r f(k) d 3 k, (3.2.13) (2π) 3/2 f(k) = 1 (2π) 3/2 e ik r f(r) d 3 r, (3.2.14) d 3 k = dk x dk y dk z, d 3 r = dxdydz k- r- 3.3 Fourier ψ(x) = Ae ik 0x, (k 0 = 2π/λ 0 = p 0 / ) ψ(x) = λ 0 = 2π/k 0 k 0 ψ 2 ψ(x) = λ 2π/k 0 ψ(x) ψ(x) = 1 A(k)e ikx dk (3.3.1) 2π Fourier, k 0 k k 0 A(k) k = k 0 A(k) = 0, if k k 0 > k.

44 A 3.2 k 0 ψ(x) { e ik 0x ( x x 0 a) ψ(x) = (3.3.2) 0 ( x x 0 > a) 2π/k 0 =. x 0 a x 0 + a ψ(x) = 1 2π A(k)e ikx dk Fourier 1 A(k) = ψ(x)e ikx dx 2π = = 1 x0 +a e ik0x e ikx dx 2π x 0 a 1 e i(k k 0)(x 0 +a) e i(k k 0)(x 0 a) 2π i(k k 0 ) sin(k k 0 )a e ikx 0. 2π (k k 0 ) = 2eik 0x 0 ψ(x) = eik 0x 0 π sin(k k 0 )a k k 0 e ik(x x 0) dk. (3.3.3) sin(k k 0 )a/(k k 0 ) k k k 0 π/a sin(k k 0 )a a k k 0 0 k 0 k 0 π/a k 0 + π/a k k π/a x a

45 x k π (3.3.4) 3.10 ψ(x) Fourier A(k) { e ikx 0 ( k k 0 k) A(k) = 0 (otherwise) (3.3.5) ψ(x) (3.3.4) 3.4 3.10 ψ(x) Fourier A(k) Fourier 3.4.1 I = r(k)e iφ(k) dk. (r(k), Φ(k) r(k) > 0) (3.4.1) r(k) k k 0 k r(k) Φ(k) Φ k (k 0) k π I 0, Φ k (k 0) k π I 0.

46 A k k 0 k k 0 + k n = Φ(k 0 + k) Φ(k 0 k) 2π n 1 n 1 n 1 n 1 r(k) r(k) k k r(k) r(k) n n 1 2π Φ k (k 0) 2 k 3.4.2 Fourier ψ(x), ψ(x) = 1 2π A(k) A(k)e ikx dk, (3.4.2) A(k) = r(k)e iθ(k), (r(k), θ(k) r(k) > 0) (3.4.3) r(k) k k 0 k r(k) θ(k) ψ(x) Φ(k, x) ψ(x) = 1 2π r(k)e iφ(k,x) dk, (3.4.4) Φ(k, x) = kx θ(k) (3.4.5)

47 ψ(x) Φ k (k 0, x) k = x θ k (k 0) k π (3.4.6) Φ k (k 0, x) = 0, (3.4.7) x = θ k (k 0) (3.4.8) x π k (3.4.9) x k x k π (3.4.10) 3.4.3 ψ(x, t) = 1 A(k)e i[kx ω(k)t] dk. (3.4.11) 2π k 0 k k 0 + k A(k) = 0 for k k 0 > k. ω = ω(k), (3.4.12) v ϕ (k) = ω(k) k. (3.4.13) v g A(k) A(k) = r(k)e iθ(k), (r(k), θ(k) r(k) > 0) (3.4.14) ψ(x, t) ψ(x, t) = 1 r(k)e iφ(k,x,t) dk, (3.4.15) 2π

48 A Φ(k, x, t) Φ(k, x, t) = kx ω(k) t θ(k) (3.4.16) Φ k (k 0, x, t) = 0, (3.4.17) x = ω k (k 0) t + θ k (k 0) v g t + x 0, (3.4.18) v g = ω k (k 0) (3.4.19) 3.4.4 (3.4.6) x π k Φ(k, x) k t 0 t Φ (3.4.16) Φ(k, x, t) ω(k) t t or t = 0 x = x 0 x 1/ k t > 0 t = 0 t = 0 3.11 ω(k) = k2 2m (3.4.20)

49 ψ(x, t) = e (k k 0) 2 /(2b 2) e i[kx ω(k)t] dk. (3.4.21) 3.5 Dirac 3.5.1 Kronecker Dirac δ ij = { 1 (i = j) 0 (i j) (3.5.1) N f i δ ij = f j. (3.5.2) i=1 δ ij - δ(x a) = { (x = a) 0 (x a) (3.5.3) x = a δ(x a) dx = 1. (3.5.4) f(x) δ(x a) dx = f(a). (3.5.5)

50 A δ(x a) - 3.3 e 1, e 2, e 3 e i e j = δ ij (3.5.6) V 3 V = V i e i. (3.5.7) i=1 V j = e j V (3.5.8) e j V = e j 3 V i e i = i=1 3 V i e j e i = i=1 3 V i δ ji = V j. i=1 3.4 2π f(x) Fourier f(x) = c n e inx, (3.5.9) n= Fourier c m c m = 1 π e imx f(x) dx. (3.5.10) 2π 1 2π π π π 1 π e imx e inx dx = δ mn 2π π e imx f(x) dx = 1 2π = = n= π e imx π n= π n= = c m. c n 1 2π c n δ mn π c n e inx dx e imx e inx dx

51 3.5 a parametrize {ϕ a (x)} ϕ a (x) ϕ b (x) dx = δ(a b) (3.5.11) ψ(x) ψ(x) = c(a) c(b) = c(a) ϕ a (x) da. (3.5.12) ϕ b (x) ψ(x) dx (3.5.13) ϕ b (x) ψ(x) dx = = = = c(b). ϕ b (x) c(a)ϕ a (x) da dx da c(a) da c(a)δ(b a) ϕ b (x) ϕ a (x) dx 3.12 n n P = (P ij ) P T P = I (= ) P (P T P ) n P ij P ik = δ jk i=1 n x i = P ij x j, y i = j=1 n x iy i = i=1 n P ik y k, k=1 n x i y i i=1

52 A N f i δ ij i=1 i (dummy index) j free index [ 1] N f i δ ij = i=1 N f k δ kj. k=1 [ 2] ( N ) 2 f i = i=1 N N f i f j = i=1 j=1 N N f i f j. i=1 j=1 [ 3] N N n=1 m=1 n 3 m 3 n 2 + m 2 = 0. n m m n = N N m=1 n=1 m 3 n 3 m 2 + n = N 2 N m=1 n=1 f(x)δ(x y) dx n 3 m 3 = = 0. n 2 + m2 x y f(x)δ(x y) dx = f(z)δ(z y) dz. 3.13 ( b 2 f(x) dx) = a b b a a f(x)f(y)dxdy.

53 [ 1] i j N f i δ? ij = i=1 N f j δ jj j=1 [ 3] N N f i f i i=1 i=1 3.5.2 ρ(x) M M = ρ(x) dx (3.5.14) ρ x = a { (x = a) ρ(x) = (3.5.15) 0 (x a) (3.5.14) ρ(x) = Mδ(x a). (3.5.16)

54 A, or } 1 (x a)2 δ(x a) = lim exp {. (3.5.17) σ 0 2πσ 2σ 2 σ (standard deviation) σ σ a x 1 ε δ(x a) = lim ε 0 π (x a) 2 + ε. (3.5.18) 2 2ε ε ε a x

55 step δ(x a) = lim ε 0 1 2ε ( x a ε) 0 ( x a > ε) (3.5.19) 1 2ε 0 a ε a a + ε x 3.5.3 Fourier Fourier δ(x a) = 1 e ik(x a) dk. (3.5.20) 2π f(x) Fourier f(x) = 1 2π f(k) = 1 2π f(x) = 1 2π = dk dx f(x ) 1 2π f(k) e ikx dk, (3.5.21) f(x )e ikx dx, (3.5.22) dx f(x ) e ikx e ikx e ik(x x ) dk. (3.5.23) f(x) k δ(x x )

56 A (i) Fourier (5.22) f(x) = δ(x a) f(k) = (1/ 2π)e ika Fourier (5.21) (ii) (3.5.20) lim ε 0 1 2π e ik(x a) 1 e εk2 dk = lim e (x a)2 /4ε. (3.5.24) ε 0 4πε σ = 2ε exp ( εk 2 ) (damping factor) (iii) (3.5.20) exp ( ε k ) lim ε 0 1 2π e ik(x a) e ε k 1 dk = lim ε 0 π (iv) (3.5.20) ε (x a) 2 + ε 2. (3.5.25) lim L 1 L e ik(x a) L dk = lim 2π L L π sin L(x a). (3.5.26) L(x a) L (3.5.3), (3.5.4) ( 3.15)

57 L π a + π L 0 a x a π L 3.5.4 Fourier Fourier f(x), g(x) Fourier f(k), g(k) f(x) = 1 2π g(x) = 1 2π f(k) e ikx dk, (3.5.27) g(k) e ikx dk, (3.5.28) f (x) g(x) dx = f (k) g(k) dk. (3.5.29) Fourier f (x) g(x) dx = = = = dx 1 2π dk dk f (k) e ikx dk dk f (k) g(k ) 1 2π 1 2π dk f (k) g(k )δ(k k) e i(k k)x dx g(k ) e ik x dk f (k) g(k) dk. (3.5.30) f(x) g(x) (3.5.27), (3.5.28) k k k

58 A 3.14 (3.5.25) 3.15 (3.5.26) (3.5.3) sin ξ dξ = π. ξ lim sin nx 0 n 1 (3.5.26) (3.5.4) 3.5.5 δ( x) = δ(x), (3.5.31) δ(ax) = 1 δ(x), a (3.5.32) (x a)δ(x a) = 0, (3.5.33) f(x)δ(x a) = f(a)δ(x a), (3.5.34) δ(x y)δ(y a) dy = δ(x a). (3.5.35) 3.6 3.6.1 Schwartz (distribution) δ(0) = ϕ(x) ϕ(x) δ(x a) dx = ϕ(a), (3.6.1) p.49 1

59 3.6.2 ϕ(x) (test function) 1/x (principal value) P/x : ϕ(x) P ϕ(x) = lim x ε 0 ( ε + +ε ) 1 x ϕ(x) dx. (3.6.2) f ϕ = fϕ + 0 [ ] F (x) x F x x ε x x 0 < ε F (x) x 0 F (x 0 ) F (x 0 ) = F (x) ϕ(x) dx, (3.6.3) ϕ(x) x x 0 < ε ϕ(x) dx = ε ε ϕ(x) dx = 1, (3.6.4) x 0 ϕ(x) ϕ(x) (3.6.3) F (x) 3.6.3

60 A F (x) ϕ(x) F (x) ϕ(x) dx (3.6.5) F (x) F (x) F (x) ϕ(x) dx = F (x) ϕ (x) dx. (3.6.6) F (x) δ (x a) ϕ(x) dx = δ (x a) Fourier δ(x a) ϕ (x) dx = ϕ (a). (3.6.7) δ ( x) = δ (x), (3.6.8) (x a) 2 δ (x a) = 0, (3.6.9) (x a)δ (x a) = δ(x a). (3.6.10) δ (x a) = 1 2π ik e ik(x a) dk, (3.6.11) (3.5.31) (3.5.35) (3.5.31) ϕ(x) δ( x)ϕ(x) dx = δ(y)ϕ( y) dy = ϕ( y) y=0 = ϕ(0) = y = x (3.5.32) ϕ(x) δ(ax)ϕ(x) dx = δ(y)ϕ( y a ) dy a = ϕ(0) a = y = ax δ(x)ϕ(x) dx. (3.6.12) 1 δ(x)ϕ(x) dx. (3.6.13) a

61 (3.5.34) ϕ(x) = f(x)δ(x a)ϕ(x) dx = δ(x a) (f(x)ϕ(x)) dx = f(a)ϕ(a) f(a)δ(x a)ϕ(x) dx. (3.6.14) 3.16 (3.5.33) 3.17 (3.6.8), (3.6.9), (3.6.10) 3.18 θ(x) θ(x) = { 0 (x < 0) 1 (x > 0) (3.6.15) ϕ(x) θ(x)ϕ(x) dx = 0 ϕ(x) dx, (3.6.16) θ (x) = δ(x) (3.6.17) 3.19 ε ( ε +0 ) (i) x x 2 + ε 2 = P x. (3.6.18) (ii) 1 x iε = P x + iπδ(x). (3.6.19)

4 4.1 n x 1, x 2,..., x n n ( ) x = x 1 x 2 : x n (4.1.1) n ( ) n n ( ) x = cx = x 1 x 2 : x n cx 1 cx 2 : cx n, y =. y 1 y 2 : y n, x + y = c : n ( ) R n (R ) x 1 + y 1 x 2 + y 2 : x n + y n. 62

63 4.1.1 R n x = x 1 x 2 :, y = y 1 y 2 :, x n y n (x, y) = x 1 y 1 + x 2 y 2 + x n y n = n x k y k (4.1.2) k=1 x y (inner product) ( 1) ( 2) x y (x, y) x, y ( ) (i) (x, y) = (y, x) (ii) (x 1 + x 2, y) = (x 1, y) + (x 2, y), (x, y 1 + y 2 ) = (x, y 1 ) + (x, y 2 ). (iii) (x, cy) = c(x, y), (cx, y) = c(x, y). (iv) (x, x) 0, x = 0 (= ) (Remark 1) x = (x, x) x x = 0 x (Remark 2) R n n n (Remark 3) (Remark 4) (x, y) = 0 x y x, y θ (x, y) = x y cos θ. (Remark 5) (ii),(iii), (x, y) x (linear) y (linear) x, y (bilinear)

64 A 4.1 x (4.1.1) x T = (x 1, x 2,, x n ) x y (x, y) (x, y) = x T y 4.2 n z 1, z 2,..., z n n z = z 1 z 2 : z n (4.2.1) n n n z = λz = z 1 z 2 : z n, w = λz 1 λz 2 : λz n. w 1 w 2 : w n, z + w = λ : n C n (C ) z 1 + w 1 z 2 + w 2 : z n + w n (Remark 1) C n z z z (λz) = λ z, (z + w) = z + w.

65 4.2.1 C n z = z 1 z 2 : z n, w = w 1 w 2 : w n, (z, w) = z 1w 1 + z 2w 2 + z nw n = n zk w k (4.2.2) k=1 z w ( 1) ( 2) ( (iv) Remark 2 ) 1 z = (z) = (x + iy) 2 (x, y) (z, z) = z z = x 2 + y 2 2 ( 3) (z, w) z, w z w ( ) (i) (z, w) = (w, z) (ii) (z 1 + z 2, w) = (z 1, w) + (z 2, w), (z, w 1 + w 2 ) = (z, w 1 ) + (z, w 2 ). (iii) (z, λw) = λ(z, w), (λz, w) = λ (z, w). (iv) (z, z) 0, z = 0 (= ) (Remark 2) z = (z, z) z z = 0 z (Remark 3) (z, w) = 0 z w (Remark 4) (i) (iii) z w iz w i(z, w) (iz, w)

66 A (Remark 5) (ii),(iii), (z, w) w (linear) z (anti-linear) 4.2 (4.2.1) z z = (z 1, z 2,, z n) z w (z, w) (z, w) = z w ( ) ( ) 1 1 4.3 z =, w = z 2, w 2, (z, w) 1 i 4.4 z = ( 1 i ), w = ( 1 i ) z w 4.5 z = 1, w = 2, (z, w) = 3i (z + 2 iw, z) 3 4.3 4.3.1 ψ(x), φ(x) [0, 1] [0, 1] N N (x 1 = 1/N, x 2 = 2/N,..., x N = 1) ψ, φ ψ(x 1 ) φ(x 1 ) ψ(x 2 ) ψ z N = :, φ w φ(x 2 ) N = :. ψ(x N ) φ(x N ) N ψ(x), φ(x) z N, w N C N ψ(x), φ(x) C [0, 1] [a, b] (, ) = R ψ(x), φ(x) C (Remark 1) (ψ + φ)(x) = ψ(x) + φ(x), (λψ)(x) = λψ(x),

67 4.3.2 [0, 1] N (z N, w N ) = N znw n = n=1 N ψ (x n )φ(x n ) N N n=1 1 (ψ, φ) lim N N (z N, w N ) = lim N 1/N = x n+1 x n = x (ψ, φ) = 1 0 N n=1 ψ (x n )φ(x n ) 1 N ψ (x)φ(x) dx (4.3.1) ( ) [0, 1] (, ) = R ψ(x), φ(x) ψ 0, φ 0 (ψ, φ) = ψ (x)φ(x) dx. (4.3.2) (Remark 2) ψ 0 (Remark 3) ( ) (Hilbert space) (Remark 4) (i),(ii),(iii),(iv) (Remark 5) ψ (ψ, ψ) = 1 ψ. (Remark 6) ψ a (x) x = a ψ b (x) x = b

68 A x = a, x = b ψ a (x) ψ b (x) (Remark 7) f(x) g(x) [a, b] (f, g) = b a f(x)g(x) dx, (4.3.3) (i),(ii),(iii),(iv) Schwarz (ψ, φ) 2 (ψ, ψ)(φ, φ), (4.3.4) ψ, φ 2 ψ(x) φ(x) dx ψ(x) 2 dx φ(x) 2 dx, ψ = λφ (Remak 2) (ψ, ψ) = ψ(x) 2 dx < + ψ(x) ψ φ Schwarz (ψ, φ) 4.6 (i),(ii),(iii),(iv) Schwarz (4.3.4) ( ) (iv) Φ(x) (Φ, Φ) 0 (φ, φ) 0 (φ, ψ) Φ = ψ λφ, λ = (φ, φ) Φ

69 thechapter.1 [ π, π] (f, g) f g (f, g) = f(x) = sin x, g(x) = cos x, π π f(x)g(x) dx = π sin x cos x f 2 = (f, f) = g 2 = (g, g) = π π π π sin 2 x dx = cos 2 x dx = π sin x cos x dx = 0. π π π π 1 cos 2x dx = π, 2 1 + cos 2x dx = π. 2 f = g = π. 4.7 [ π, π] f(x) = cos mx, g(x) = cos nx, (f, g) f g m, n m n 4.8 [ π, π] ϕ n (x) = 1 2π e inx, (n = 0, ±1, ±2, ±3, ) (ϕ n, ϕ m ) 4.3.3 Fourier Fourier

70 A [ π, π] f(x) (Fourier ) f(x) = a 0 2 + (a n cos nx + b n sin nx). (4.3.5) n=1 a n, b n {1, cos x, sin x, cos 2x, sin 2x,, cos nx, sin nx, } [ π, π] 4.1 4.7 (cos nx, cos nx) = (sin nx, sin nx) = π π π (1, cos nx) = (1, sin nx) = (cos nx, cos mx) = (sin nx, sin mx) = (cos nx, sin mx) = (1, 1) = π cos 2 nx dx = π, sin 2 nx dx = π, π π π π π π π π π π π π 1 cos nx dx = 0, 1 sin nx dx = 0, 1 cos nx cos mx dx = 0, (n m) 1 sin nx sin mx dx = 0, (n m) 1 cos nx sin mx dx = 0 1 1 dx = 2π, (1, f) = a 0 2 (1, 1) + [ an (1, cos nx) + b n (1, sin nx) ] n=1 = πa 0, (4.3.6) (cos mx, f) = a 0 2 (cos mx, 1) + [ an (cos mx, cos nx) + b n (cos mx, sin nx) ] n=1 = πa m, (4.3.7) (sin mx, f) = a 0 2 (sin nx, 1) + [ an (sin mx, cos nx) + b n (sin mx, sin nx) ] n=1 = πb m, (4.3.8)

71 a n = 1 (cos nx, f), (n = 0, 1, 2, 3, ) (4.3.9) π b n = 1 (sin nx, f), (n = 1, 2, 3, ) (4.3.10) π V {e 1, e 2, e 3 } 3 V = V 1 e 1 + V 2 e 2 + V 3 e 3 = V n e n, e n e m = { n=1 0 (n m) 1 (n = m) V V n V n = e n V V f(x) e n cos nx sin nx V n a n, b n Fourier 3 Fourier [ π, π] ( ) f(x) e inx f(x) = c n e inx, (4.3.11) n= c n [ π, π] 5.8 { π (e imx, e inx ) = e i(n m)x 0 (m n) dx = (4.3.12) 2π (m = n) π (e imx, f) = c n n= c n = 1 2π (einx, f) = 1 2π c n (e imx, e inx ) = 2π c m. π π e inx f(x) dx, (4.3.13)

72 A 4.4 ψ(x) 1 = ψ(x) 2 dx = ψ (x)ψ(x) dx ψ ψ (ψ, ψ) = ψ 2 = 1 ψ. ψ 1, ψ 2 ψ 1 + ψ 2 c 1 ψ 1 + c 2 ψ 2 ψ, ϕ ψ ϕ ϕ (x)ψ(x) dx = (ϕ, ψ)

73 or ˆp x = i / x ψ(x) : ψ(x n 1 ) ψ ψ(x n ), ψ(x n+1 ) : ˆp x ψ i 2 x 0 1 0 1 0 1 0 1 0 : ψ(x n 1 ) ψ(x n ) ψ(x n+1 ) : ψ ψ ψ (x n ) = 1 { ψ(xn+1 ) ψ(x n ) + ψ(x } n) ψ(x n 1 ) 2 x x 4.9 ˆxψ = xψ ˆx x n 1 0 0 ˆx 0 x n 0, 0 0 x n+1 ψ ψ Â Â = ψ (x)âψ(x) dx = (ψ, Âψ),

74 A ψ  Âψ = aψ ψ  c a a  ψ  a (, eigenstate) 4.10 ˆp = ( /i)d/dx p 0 4.11 4.10 ˆp