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3 0. SI : π A A-. Gauss 69 A-3. 7 A-4. Taylor 76 A A A-7. Hamiltonian 79 A-8. Hermite 8 A A A A-1. Legendre 94 A A

4 I, II Pauling and Wilson Introduction to Quantum Mechanics Dover 4

5 0 SI ; 0.1 c m s 1 ε C J 1 m 1 e C Planck h J s Boltzmann k B J K 1 m e kg m p kg m n kg m u kg Avogadro N A mol 1 Rydberg R = m e e 4 /8ε 0c h m 1 Bohr a B = ε 0 h /πm e e m 0. SI metre m 1/ s kilogram kg second s kelvin K 3 1/73.16 mole mol 0.01 kg 1 ampere A peta P femto f 10 1 tera T 10 1 pico p 10 9 giga G 10 9 nano n 10 6 mega M 10 6 micro µ 10 3 kilo k 10 3 milli m 10 hecto h 10 centi c 10 1 deca da 10 1 desi d 0.4 joule J SI 1 J = 1 kg m s electronvolt ev 1 V 1 ev = e J keiser cm 1 1 cm cm 1 = ch J calorie cal 1 cal = J cm 1 K cal ev SI 5

6 0.5 N m kg s Pa N m = m 1 kg s J N m = m kg s C A s V J C 1 = m kg s 3 A Newton Newton m x t x(t) x t x F (x) F x t v(t) x(t + t) x(t) (0.1) v(t) = lim = dx(t) t 0 t dt t a(t) v(t + t) v(t) (0.) a(t) = lim = dv(t) = d x(t) t 0 t dt dt Newton (0.3) F (x) = ma(t) p(t) (0.4) p(t) = mv(t) (0.5) F (x) = dp(t) dt (V (x)) F (x) dv (x) (0.6) F (x) = dx Newton dv (x) (0.7) = m d x(t) dx dt x V (x) x (x(0)) (v(0) x ) 6

7 0-1. SI (1) () (3) (4) (5) (6) 0-. SI (1) () (3) (4) (5) (6) 0-3. SI SI (1) cm () erg (3) dyn (4) min (5) Torr (6) mmhg 0-4. SI (1) cc () K (3) µ (4) sec (5) hr 0-5. SI (1) min () l (3) ev (4) Å (5) bar (6) atm (1) J () ev (3) cm 1 (4) cal 0-7. (1) c () e (3) Planck h (4) Boltzmann k B 0-8. Rydberg cm Bohr Å 7

8 1 1.1 y T ν λ v λ ν (1.1) v = νλ (1.) y(x, t) = A sin π T ( t x ) = A sin(ωt kx) v λ/4 λ/ A ω ω = πν k(= π/λ) λ 3λ/4 λ A x A 1. Huygens Newton E(x, t) (1.3) E(x, t) = E 0 sin(ωt kx) λ(= π/k) 400 nm 700 nm (c) (ν) (1.4) c = νλ 8

9 Young (1.5) E = E 1 + E = E 0 sin(ωt kx 1 + E 0 sin(ωt kx ( ) ( ) ωt k(x1 + x ) k(x x 1 ) = E 0 sin cos 1.3 X λ I V (1) λ > λ t I () λ < λ t I (3) (4) I I (5) V max I (6) V max λ e - e - e - W / e h / e hν h Planck ν W W hν W hν > W hν W (1.6) ev max = E max = hν W % 100 % (1.7) E = hν 9

10 p c (1.8) p = hν c = h λ Millikan Ze F = 6πηRv v R m (1.9) v = ρ ρ gr 9 η F η, ρ ρ g E (1.10) EZe = mg e e = C 1.5. Thomson e/m e + L s O P O P s (1.11) s = 1 ee m e ( ) L v (1.1) e = sv m e EL v m e H v (1.13) Ee = Hev (1.14) v = E H e/m e m e = kg 10

11 1.6 Davisson Germer de Broglie wave (1.15) λ = h p 193 de Broglie W (1) 100 W 1 s J () 1 m 100 % (3) 1 s 1 Å 100 % (4) ev Na (1) Na W 1.8 ev λ t () Na 50 nm V max 1-4. Li 300 nm V max = 1.83 V 400 nm V max = 0.80 V (1) W Planck h () λ t 1-5. K 55 nm (1) W () 300 nm (3) 400 nm V max 1-6. Cr 4.40 ev (1) λ t () 00 nm V max 1-7. Ag 30 nm V max = 0.80 V (1) W () λ t 1-8. (1) 60 kg 100 m 10 s () 150km h g 11

12 (3) m s 1 (4) J 1-9. (1) 1 ev () 100 ev (3) 10 kev c 4 1 (1) () m 0 v (1.16) m = m 0 1 (v /c ) (3) (1), () nm mw 1 s (1) He-Ne 633 nm () Ar + 59 nm (3) Ar nm (4) He + Cd + 35 nm (1) X 1 nm () 50 nm (3) 500 nm (4) 1 mm (5) 1 mm (6) 100 m ev 1

13 Schrödinger Newton Schrödinger.1 (.1) u(x, t) = u 0 sin(ωt kx) u 0 I ω k T λ (.) ω = π T = πν (.3) k = π λ v (.4) v = ω k (.5) u(x, t) = v u(x, t) t x A-14. t (.6) u(0, t) = 0 (.7) u(l, t) = 0 (.8) u(x, t) = f(x)f (t) (.9) F (t) d f(x) dx = 1 v f(x)d F (t) dt u(x, t) = f(x)f (t) (.10) 1 d f(x) 1 d F (t) f(x) dx = v F (t) dt 13

14 x t x, t x t α (.11) (.1) d f(x) dx αf(x) = 0 d F (t) dt αv F (t) = 0.3 (.13) d F (t) dt = αv F (t) α = 0 F (t) = A 1 t + A t = t = α > 0 αv = β (.14) F (t) = a 1 e βt + a e βt t = t = α < 0 αv = ω (.15) F (t) = b 1 sin ωt + b cos ωt ω (.16) u(x) = ω x v u(x) = k u(x).4 x = 0 x = L u(0, t) = u(l, t) = 0 f(0)f (t) = f(l)f (t) = 0 f(x) f(0) = f(l) = 0 (.17) α = ω v = k (.18) d f(x) dx + k f(x) = 0 (.19) f(x) = c 1 sin kx + c cos kx 14

15 c = 0 c 1 c 1 0 c 1 = 0 (.0) kl = nπ, n = 0, 1,, 3, n = 0 c 1 n = 4 n = 3 n = n = 1 0 a.5 (.1) x u(x) = k u(x) de Broglie (.) p = h λ = hk V (x) (.3) E = p mv + V (x) = + V (x) m u(x) ψ(x) d (.4) h ψ(x) + V (x)ψ(x) = Eψ(x) m dx Schrödinger III IV -1. x(t) = cos ωt x(t) = A cos ωt + B sin ωt -. (.5) d X(x) dx + k X(x) = 0 (1) (.6) X(x) = c 1 sin kx + c cos kx 15

16 () λ k k (3) u k ν -3. (.7) d f(x) = ω dx u f(x) (1) u = νλ () ω = πν (3) Schrödinger Schrödinger (.8) i h h Ψ(x, t) = t m Ψ(x, t) + V (x)ψ(x, t) x Ψ(x, t) = ψ(x)f(t) ψ(x) f(t) ψ(x) Schrödinger 16

17 3 Schrödinger Schrödinger 3.1 m { 0 at 0 < x < a (3.1) V (x) = at x 0 or x a dv (x) F = 0 < x < a dx F = 0 x = a E V(x) 1 1 (3.) p = me E 3. Schrödinger 0 0 a d (3.3) h ϕ(x) + V (x)ϕ(x) = Eϕ(x) m dx x 0 x a V (x) = ϕ(x) = 0 0 < x < a V (x) = 0 (3.4) d dx ϕ(x) = k ϕ(x), me h = k V (x) = V (x) = 0 (3.5) ϕ(0) = ϕ(a) = 0 (3.6) ϕ(x) = A sin kx + B cos kx A, B ϕ(0) = 0 (3.7) B = 0 ϕ(a) = 0 (3.8) A sin ka + B cos ka = 0 17

18 (3.9) A sin ka = 0 A A = 0 ϕ(x) = 0 A 0 B = 0 (3.10) sin ka = 0 (3.11) ka = πn (n = 1,, 3, ) ( nπx ) (3.1) ϕ(x) = A sin a n = 0 A = 0 (3.13) E n = h k m = n π h, n = 1,, 3, ma sin( x) = sin(x) n < 0 n > 0 1 n < 0 n > 0 A φ φ E 4 E 3 E E 1 a 0 x a a a 0 x a a 3.3 Hamiltonian 90 (3.14) ϕ n(x)ϕ m (x)dx = a a ϕ n(x)ϕ m (x)dx ϕ n (x) ϕ m (x) = 0 for n m ϕ n (x) ϕ m (x) x n = m (3.15) ϕ n (x) ϕ n (x) = 1 18

19 A A (3.16) ϕ n (x) ϕ n (x) = A a (3.17) A = a 0 ( sin nπx ) dx = 1 a Hamiltonian ϕ n ψ(x) ϕ n (x) (3.18) ψ(x) = c n ϕ n (x) n 3.4 (3.19) (3.0) d dx ϕ(x) = k ϕ(x) me h = k (3.1) ϕ(x) = Ce ikx + De ikx Euler (3.) e ix = cos x + i sin x ϕ(0) = 0 (3.3) C + D = 0 ϕ(a) = 0 (3.4) Ce ika + De ika = 0 D = C (3.5) ϕ(x) = ic sin kx (3.6) ic sin ka = 0 A = ic 3-1. m 0 at a (3.7) V (x) = < x < a at x a or x a 19

20 (1) Schrödinger () ψ(a/) = ψ( a/) = π k x π k (3.8) sin kx, cos kx, e ikx, e ikx (1) () m { 0 at a < x < a (3.9) V (x) = at x a or x a (1) x = 0 (3.30) ψ n (x) = ψ n ( x) or ψ n (x) = ψ n ( x) () x x (3.31) Ĥ(x) = Ĥ( x) (3) () Schrödinger (3.3) Ĥ(x)ψ n ( x) = E n ψ n ( x) (4) c (3.33) ψ n ( x) = cψ n (x) (3) (1) nm 3-5. m { 0 at 0 < x < a (3.34) V (x) = at x 0 or x a (1) () (3) de Broglie 3-6. m 0 c m 0 c L 3-7. (1) de = F dl F L m n ( E ) () F = 1 N kg 1 m 1 J (1) () 1 (3) 0

21 4 4.1 E v E (4.1) v = m T (4.) T = a m v = a E x x + dx P (x)dx dx (4.3) P (x)dx = dx v 1 T = dx a at 0 < x < a (4.4) P (x)dx = 0 at x 0 or x a P (x) x = x x x + dx P (x)dx dx P (x) 1 1 (4.5) P (x)dx = 1 a a 0 dx = 1 A A (4.6) A = AP (x)dx (4.7) x = xp (x)dx = 1 a a 0 xdx = a (4.8) (x x ) = x x = ( a ) x 1 P (x)dx = a a 0 x dx a 4 = a 1 1

22 p x x p > 0 P (x)/ p < 0 P (x)/ (4.9) p = p P (x) dx + ( p) P (x) dx = 0 p = 0 (4.10) p = p P (x)dx = p 4. x x p x (4.11) x p x > h x p x E t θ L (4.1) E t > h (4.13) θ L > h x y p y ψ(x) 4.3 x x + dx ψ(x) (4.14) P (x) = ψ (x)ψ(x) ψ (x) ψ(x) ψ(x) ψ (x) = ψ(x) (4.15) ψ (x)ψ(x)dx = 1 ϕ(x) x x + dx (4.16) P (x) = ϕ (x)ϕ(x) ϕ (x)ϕ(x)dx N (4.17) P (x) = N ϕ (x)ϕ(x)

23 1 (4.18) N = ϕ (x)ϕ(x)dx N 4-1. x ϕ(x) m x ˆx = x p ˆp = i h x h T ˆT = m x V (x) ˆV (x) = V (x) Hamiltonian Ĥ Hamiltonian H ψ ˆpψ ˆp ψ ψ x i h ˆx ˆV (x) x V (x) T (4.19) T = p m ˆpˆp (4.0) ˆT = m = h m x p ˆp (4.1) ˆxˆpψ(x) = i hx ψ(x) x (4.) ˆpˆxψ(x) = i hψ(x) i hx ψ(x) x Q ˆQ ψ(x) Q (4.3) Q = ψ (x) ˆQψ(x)dx 3

24 Q (4.4) Q = ψ (x) ˆQ ˆQψ(x)dx ϕ(x) Q Q (4.5) Q = ϕ (x) ˆQϕ(x)dx ϕ (x)ϕ(x)dx (4.6) Q = ϕ (x) ˆQ ˆQϕ(x)dx ϕ (x)ϕ(x)dx (4.7) x = (4.8) x = (4.9) p = (4.30) p = ϕ (x)ˆxϕ(x)dx = a a ϕ (x)ˆxˆxϕ(x)dx = a ϕ (x)ˆpϕ(x)dx = i h a 0 a ( x sin nπx ) dx = a a 0 a ϕ (x)ˆpˆpϕ(x)dx = h a ( x sin nπx ) dx = a a 3 a π n sin 0 a 0 ( nπx ) [ ( nπx ) ] a x sin dx = 0 a sin ( nπx ) [ ( nπx ) ] a x sin dx = n π h a a x p p x n n 4.5 Â (4.31) Âψ(x) = Aψ(x) Â A Â A x ψ(x) Schrödinger Hamiltonian ψ Â ψ A A A A A (4.3) A = ψ (x)[âψ(x)]dx = ψ (x)aψ(x)dx = A ψ (x)ψ(x)dx = A Â A (4.33) A = ψ(x) Â ψ(x) = ψ(x) Aψ(x) = A ψ(x) ψ(x) = A 4

25 (4.34) i h dψ(x) dx = pψ(x) (4.35) ψ(x) = Ce ikx (4.36) p = k h p x ψ(x) (4.37) h d ψ(x) m dx = T ψ(x) (4.38) ψ(x) = Ce ikx (4.39) T = k h m = p m (4.40) ϕ(x) = D sin kx (4.41) T = k h m 4-1. (1) e αx ( α > 0, 0 x ) () e αx ( α > 0, 0 x ) (3) e imϕ ( 0 ϕ π, m, ϕ ) (4) e iαϕ ( 0 ϕ π, α, ϕ ) (5) e αx ( α > 0, x ) (6) e αx ( α > 0, x ) 4-. x V (x) = 1 kx m (k ) (1) C (4.4) ϕ(x) = Cx exp () p (3) x (4) T ) ( αx km, α = h 5

26 (5) E (6) (7) 0 x 4-3. ψ 1 (x) ψ (x) x = x = (1) ˆp (4.43) ϕ 1(x)ˆpϕ (x)dx = [ˆp ϕ 1 (x)] ϕ (x)dx Hermite Hermite () Hermite 4-4. ψ(x) Â Hermite A 4-5. Â ˆB [Â, ˆB] (4.44) [Â, ˆB] = Â ˆB ˆB Â (1) ˆp x ˆx () ˆp y ˆx m { 0 at a < x < a (4.45) V (x) = at x a or x a (1) 0 x () 0 x a/3 (3) a/3 x a/3 (4) () (3) (5) p (p p ) (6) x (x x ) (7) x p (8) T (9) V (10) (11) x x + dx (3) 6

27 5 π 5.1 molecular orbital Ψ i E i Ψ f E f E (5.1) E = E f E i E > 0 E < 0 ν E (5.) hν = E Bohr λ (5.3) λ = ch E c λ < 800 nm HOMO (highest occupied molecular orbital) LUMO (lowest unoccupied molecular orbital) HOMO LUMO 5. π π π HOMO N LUMO N + 1 LUMO HOMO C=C-C=C-C=C-C=C-C=C 7

28 L (5.4) E(HOMO) = N h 8m e L (5.5) E(LUMO) = (N + 1) h 8m e L (5.6) E = E(LUMO) E(HOMO) = (5.7) λ = ch E = 8cm el h(n + 1) N (CH CH) n CH N + (N + 1)h 8m e L N N (n + ) (n + 4) π HOMO N = (n + ) L (5.8) L = (n + )R + L 0 n λ(obs)/nm λ(cal)/nm R = nm, L 0 = nm 5-1. π 6 68 nm (1) π () HOMO LUMO cm 1 E = hν ν = 1/λ λ 5-. a/ a 5-3. m (1) () (1) L (3) () L (4) (3) 8

29 6 Schrödinger 6.1 Schrödinger Ψ(x, y, z) r ˆr x y z (6.1) ˆr = ˆx ŷ ẑ = x y z V (r) = V (x, y, z) ˆV (6.) ˆV = V (x, y, z) p ˆp x p x y p y z p z (6.3) ˆp = pˆ x ˆp y ˆp z = i h / x i h / y i h / z = i h / x / y / z = i h nabla (6.4) = i x + j y + k z = / x / y / z i, j, k x, y, z Ψ ˆp (6.5) ˆpΨ = i h Ψ/ x Ψ/ y Ψ/ z m p K (6.6) K = p m = p p m = p x + p y + p z m ˆK ˆK = ˆp ˆp ˆp = m m = pˆ ( ) xpˆ x + ˆp y ˆp y + ˆp z ˆp z = ( i h) (6.7) m m x + y + z ( ) = h m x + y + z = h m 9

30 (6.8) = = x + y + z Ψ (6.9) Ψ = Ψ x + Ψ y + Ψ z m V (x, y, z) Schrödinger (6.10) h m Ψ(x, y, z) + V (x, y, z)ψ(x, y, z) = EΨ(x, y, z) Ĥ (6.11) ĤΨ(x, y, z) = EΨ(x, y, z) (6.1) Ĥ = h m + V (x, y, z) 6. { 0 at 0 < x < a, 0 < y < b, 0 < z < c (6.13) V (x, y, z) = at (6.14) Ψ(x, y, z) = 0 at x = 0, y = 0, z = 0, x = a, y = b, z = c (6.15) Ψ(x, y, z) = X(x)Y (y)z(z) Schrödinger Ψ(x, y, z) = X(x)Y (y)z(z) (6.16) 1 d X(x) X(x) dx = 1 d Y (y) Y (y) dy 1 d Z(z) Z(z) dz me h x x me x / h (6.17) 1 d X(x) X(x) dx = me x h = 1 d Y (y) Y (y) dy 1 d Z(z) Z(z) dz me h 3 (6.18) (6.19) (6.0) d X(x) dx d Y (y) dy d Z(z) dz = me x h X(x) = me y h Y (y) = me z h Z(z) (6.1) E = E x + E y + E z 30

31 (6.) X(0) = X(a) = 0, Y (0) = Y (b) = 0, Z(0) = Z(c) = 0 3 (6.3) ( X(x) = a sin nx πx ) a (6.4) ( Y (y) = b sin ny πy ) b (6.5) ( Z(z) = c sin nz πz ) c (6.6) X(x) = 0, Y (y) = 0, Z(z) = 0 (6.7) E x = π h n x ma (6.8) E y = π h n y mb (6.9) E z = π h n z mc (6.30) a b c x=0 y=0 z=0 X(x)Y (y)z(z) dxdydz = 1 dτ (6.31) dτ = dxdydz (6.3) Ψ(x, y, z) dτ = 1 8 ( (6.33) Ψ(x, y, z) = abc sin nx πx ) ( ny πy ) ( nz πz ) sin sin a b c ( ) (6.34) E = π h n x m a + n y b + n z c a = b = c N = n x + n y + n z = 6 (1, 1, ), (1,, 1), (, 1, 1)

32 Ω N Ω N Ω V (x, y) m (6.35) V (x, y) = (1) Schrödinger () { 0 at 0 < x < a, 0 < y < b at (3) (4) a = b 5 (5) (6) x, y (7) (8) x, y (9) (10) 0 < x < a/3, 0 < y < b/3 (11) a/3 < x < a/3, 0 < y < b/3 (1) a/3 < x < a/3, b/3 < y < b/3 (13) (10), (11), (1) (14) 1 x = a/ y (15) 1 y = b/ x 6-. (1) n x + n y + n z FORTRAN () (1) 6-3. q 1 q Ĥ(q 1, q ) = Ĥ1(q 1 ) + Ĥ(q ) Ĥ1ψ 1 (q 1 ) = E 1 ψ 1 (q 1 ), Ĥ ψ (q ) = E ψ (q ) (1) ψ(q 1, q ) = ψ 1 (q 1 )ψ (q ) () E = E 1 + E 6-4. π 10 (1) 1: π a () ( 30 nm ) a 3

33 7 7.1 m 1, m r 1, r (7.1) v 1 = dr 1 dt = ṙ 1 (7.) v = dr dt = ṙ (7.3) p 1 = m 1 v 1 (7.4) p = m v (7.5) r G = m 1r 1 + m r M (7.6) M = m 1 + m 1 V (r) r (7.7) r = (x x 1 ) + (y y 1 ) + (z z 1 ) 1 F 1 1 F 1 (7.8) F 1,x = r dv (r) = x x 1 dv (r) = r dv (r) = F 1,x x 1 dr r dr x dr y z (7.9) F 1 = F 1 (7.10) m 1 d r 1 dt = F 1 = F 1 (7.11) m d r dt = F 1 = F 1 d r 1 (7.1) m 1 dt + m d r dt = M d r G dt = 0 r m r 1 m 1 d r (7.13) dt d r 1 dt = F 1 + F ( 1 1 = + 1 ) F 1 m m 1 m 1 m 33

34 (7.14) µ d r dt = F (7.15) F = F 1 (7.16) r = r r 1 (7.17) µ = m 1m m 1 + m µ V (r) µ 1 (7.18) H = 1 m 1ṙ 1 ṙ m ṙ ṙ + V (r 1, r ) = 1 Mṙ G ṙ G + 1 µṙ ṙ + V (r) 7-1. (7.18) 34

35 8 8.1 x x e x (8.1) x = x (t) x 1 (t) x e Hooke (8.) F (x) = kx k (8.3) V (x) = 1 kx Newton (8.4) µ d x(t) dt = kx(t) µ ω = k/µ (8.5) d x(t) dt = ω x(t) (8.6) x(t) = A sin(ωt + δ) x(t) = A cos(ωt + δ) A ω δ t = 0 E v (8.7) E = p m + 1 kx A p = 0 (8.8) A = E k (8.9) v(x) = p(x) k m = ± m (A x ) 35

36 8.1.1 x x + dx P (x)dx 1 (8.10) P (x)dx = 1 T v dx = 1 π A x dx 8. Schrödinger d (8.11) h µ dx ϕ(x) + k x ϕ(x) = Eϕ(x) d (8.1) dξ ϕ(ξ) + (λ ξ )ϕ(ξ) = 0 k (8.13) ω = µ (8.14) λ = E hω (8.15) ξ = α 1 x (8.16) µk α = h (8.17) ϕ( ) = ϕ( ) = 0 A n = 0, 1,, ( (8.18) E n = hν n + 1 ) ( = hω n + 1 ) (n = 0) 8.4 N n (8.19) ϕ(ξ) = N n e ξ / H n (ξ) 36

37 (8.0) H n (ξ) = ( 1) n e ξ dn e ξ (8.1) N = 1 ( α ) 1/4 n n! π dξ n H n (ξ) Hermite ξ n (8.) H n+1 (ξ) ξh n (ξ) + nh n 1 (ξ) = 0 (8.3) H n(ξ) ξh n(ξ) + nh(ξ) = 0 (8.4) H n(ξ) = dh n(ξ) dξ (8.5) H m (ξ)h n (ξ)e ξ dξ = 0 Hermite (8.6) H 0 (ξ) = 1 (8.7) H 1 (ξ) = ξ (8.8) H (ξ) = 4ξ (8.9) H 3 (ξ) = 8ξ 3 1ξ (8.30) H 4 (ξ) = 16ξ 4 48ξ + 1 (8.31) ( α ) 1/4 ϕ 0 (x) = e αx / π ( ) 4α 3 1/4 (8.3) ϕ 1 (x) = xe αx / π ( α ) 1/4 (8.33) ϕ (x) = (αx 1)e αx / 4π ( ) α 3 1/4 (8.34) ϕ 3 (x) = (αx 3 3x)e αx / 9π n = 0 n = 1 n = n = 3 n = 4 φ(ξ) ξ 37

38 8.5 n = 0 n = 1 n = n = 3 n = 4 φ(ξ) ξ n n 8-1. m 1 z 1 m z z (1) () k R (3) (4) 8-. Morse (8.35) V (x) = D (1 e β(x R)) D, β, R (1) () (3) (4) (5) (6) Taylor 8-3. Lennard-Jones [ (σ ) 1 ( σ ) ] 6 (8.36) V (x) = 4ε x x ε, σ (1) () (3) (4) (5) (6) Taylor 8-4. x = a U(x) U(x) 38

39 8-5. m V (x) = 1 kx k (1) () (3) (4) t = 0 x = A (5) (4) 8-6. m, k (1) () A, B (8.37) x(t) = A sin ωt + B cos ωt, ω = k m (3) C, ϕ (8.38) x(t) = C sin(ωt + ϕ) (4) (3) C, ϕ () A, B 8-7. m, k (1) 1 x (x x ) () 1 p (p p ) (3) (1) n = 0 () 8-9. (1) n = 0 n = 1 () (3) (4) n = 0 n = Hermite (1) () (3) (4) (5) (6) (3), (4) (7) T V (8) x n T = n V 8-1. R mn (8.39) R mn = ϕ mxϕ n dx = ϕ m x ϕ n 39

40 m n ϕ n ( mω (8.40) â = ˆx + i ) ( mω h mω ˆp, â + = ˆx i ) h mω ˆp m V (x) k { 1 (8.41) V (x, y) = kx at x > 0 at x 0 (1) Schrödinger () (3) m V (x, y) k (8.4) V (x, y) = 1 k(x + y ) (1) Schrödinger () Schrödinger (3) (4) m V (x, y, z) k (8.43) V (x, y, z) = 1 k(x + y + z ) (1) Schrödinger () Schrödinger (3) (4) m V (x) k, α (8.44) V (x) = 1 kx + αx (1) Schrödinger () (3) (8.45) y = x + α k V (x) = 1 kx m (1) x x (8.46) Ĥ(x) = Ĥ( x) () Schrödinger (8.47) Ĥ(x)ψ n ( x) = E n ψ( x) 40

41 (3) c (8.48) ψ n ( x) = cψ n (x) x = 0 (8.49) ψ n (x) = ψ n ( x) or ψ n (x) = ψ n ( x) 41

42 z m R (9.1) x = R cos ϕ R y r y x φ x (9.) y = R sin ϕ (9.3) ẋ = ωr sin ϕ (9.4) ẏ = ωr cos ϕ (9.5) ω = ϕ ω (9.6) E = 1 mv = 1 m(ẋ + ẏ ) = 1 m(x + y )ω = 1 mr ω = 1 Iω (9.7) I = mr I (9.8) E = p m = L I (9.9) L = Iω L x, y, z p x, p y, p z ϕ L I 4

43 p p y p R y p x φ x p x R y φ x p y z xp y yp x (9.10) xp y yp x = mxẏ myẋ = mr ω = L 3 (9.11) L = r p = i j k x y z p x p y p z = yp z zp y zp x xp z xp y yp x 9.1. r G (9.1) mi (r i r G ) = 0 (9.13) L = r i p i x, y, z (9.14) E = 1 (I xω x + I y ω y + I z ω z) = 1 (9.15) I x = m i (x i x G ) ( ) L x + L y + L z I x I y I z x I x = 0, I y = I z = R µ R µ µ R

44 (9.16) L z = Iω = pr p (9.17) p = h λ (9.18) L z = hr λ (9.19) πr = m l λ m l = 0, ±1, ±, (9.0) L z = hr 1 m l πr = m l h (9.1) E = L z I = m l h I 9.. ( ) (9.) Ĥ = h m x + y + V (x, y) V (x, y) = 0 r R) Ĥψ(x, y) = Eψ(x, y) (9.3) (9.4) x = r cos ϕ y = r sin ϕ 0 r < 0 ϕ < ϕ (9.5) (9.6) r = x ( + y ϕ = tan 1 y ) x r, ϕ x, y (A-11 (9.7) x + y r + 1 r r + 1 r ϕ r (R) ϕ (9.8) ψ(ϕ) r = 0 44

45 (9.9) Ĥ = h 1 d m R dϕ = h d I dϕ Schrödinger (9.30) h d ψ(ϕ) I dϕ = Eψ(ϕ) (9.31) ψ ml (ϕ) = Ae im lϕ (9.3) ψ ml (ϕ + π) = ψ ml (ϕ + π) (9.33) ψ ml (ϕ + π) = Ae im l(ϕ+π) = ψ(ϕ)(e iπ ) m l = ψ ml (ϕ)( 1) m l m l = 0, ±1, ±, A (9.34) π 0 π ψ ml (ϕ) ψ ml (ϕ)dϕ = A e imlϕ e imlϕ dϕ = 1 0 A = 1/ π Schrödinger (9.1)) 9..3 x, y (9.35) L z = xp y yp z (9.36) ˆLz = xˆp y yˆp z = h i ( x y y ) x (9.37) ˆLz = h i ϕ ψ ml (ϕ) ˆL z (9.38) ˆLz ψ ml (ϕ) = h d 1 e imlϕ = m l hψ ml (ϕ) i dϕ π m l = π π n = 3 n = n = 1 n = 0 LUMO HOMO 45

46 9-1. π 6 55 nm (1) π () π π 46

47 Schrödinger R m ( (10.1) Ĥ = h m x + ) y z + V (x, y, z) V (x, y, z) = 0 r R) Ĥψ(x, y, z) = Eψ(x, y, z) A-5 A-6 z r cos θ (10.) (10.3) x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ 0 r < 0 θ < π 0 ϕ < π r sin θ cos φ φ θ r (x,y,z) r sin θ r sin θ sin φ (x,y,0) y x (10.4) = x + y + = 1 r r ( r r Schrödinger z ) + 1 r sin θ ( ( (10.5) h r ψ ) + 1 m r r sin θ ( sin θ ) + θ θ ( sin θ ψ ) + 1 θ θ sin θ 1 r sin θ ϕ ) ψ ϕ + V (r, θ, ϕ)ψ(r, θ, ϕ) = Eψ(r, θ, ϕ) r R) ψ(θ, ϕ) r ( (10.6) Ĥ = h 1 mr sin θ ( sin θ ) + 1 θ θ sin θ Λ (10.7) Λ = 1 sin θ ( sin θ ) + 1 θ θ sin θ (10.8) h I Λ ψ(θ, ϕ) = Eψ(θ, ϕ) ϕ 47 ) ϕ = h I Λ

48 ( (10.9) θ + cos θ sin θ (10.10) λ = IE h θ + 1 sin θ (10.11) ψ(θ, ϕ) = Θ(θ)Φ(ϕ) ) ϕ ψ(θ, ϕ) + λψ(θ, ϕ) = 0 Y (θ, ϕ) θ ϕ m l (10.1) (10.13) Φ(ϕ) ϕ + m lφ(ϕ) = 0 Θ(θ) θ + cos θ Θ(θ) + sin θ θ ( λ ) m l sin Θ(θ) = 0 θ 10. Φ(ϕ) (10.14) Φ(ϕ) ϕ + m lφ(ϕ) = 0 (10.15) Φ ml (ϕ) = 1 π e im lϕ m l = 0, ±1, ±, m = 0 m = 1 m = 0.5 Re Im(+1) Im(+) Re Φ m (φ) 0.0 Im Re Im( 1) π π 0 Im( ) π π 0 π π φ 10.3 Θ(θ) l = 0, 1,, l m l (10.16) l, l + 1, l +,, 0,, l, l 1, l Y l,ml (θ, ϕ) (10.17) Y l,ml (θ, ϕ) = ( 1) (m l+ m l )/ (l + 1) (l m l )! 4π (l + m l )! P m l l (cos θ)e im lϕ 48

49 l = 0, 1,,, l m l l (10.18) Y 0,0 (θ, ϕ) = 1 4π (10.19) 3 Y 1,0 (θ, ϕ) = 4π cos θ (10.0) 3 Y 1,±1 (θ, ϕ) = sin θe±iϕ 8π (10.1) 5 Y,0 (θ, ϕ) = 16π (3 cos θ 1) (10.) 15 Y,±1 (θ, ϕ) = sin θ cos θe±iϕ 8π (10.3) 15 Y,± (θ, ϕ) = 3π sin θe ±iϕ 10.4 l l J (10.4) E = J(J + 1) h I = λ h, J = 0, 1,, I (J + 1) 10.5 m l = 0 l l 1 0 4π π cos θ π (3 cos θ 1) π (5 cos3 θ 3 cos θ) i j k yp z zp y (10.5) L = r p = x y z p x p y p z = zp x xp z xp y yp x (10.6) ˆL = ˆr ˆp = i h y z z y z x x z x y y x 49

50 (10.7) ˆL = ˆ L x ˆ Lx + ˆL y ˆLy + ˆL z ˆLz (10.8) ˆL = h [ 1 sin θ z ( (10.9) ˆLz = i h x y y ) x ( sin θ ) + 1 θ θ sin θ = i h ϕ ] ϕ = h Λ 10.7 Y l,ml (θ, ϕ) = NP m l l (cos θ)φ ml (ϕ) ˆL z ˆL z (10.30) ˆL Y l,ml (θ, ϕ) = L Y l,ml (θ, ϕ) (10.31) ˆLz Y l,ml (θ, ϕ) = L z Y l,ml (θ, ϕ) L, L z (10.3) L = h l(l + 1) 1 (10.33) L z = m l h (L = h l(l + 1)) z (l z = hm l ) z θ cos θ (10.34) cos θ = m l l(l + 1) cos θ (l + 1) z x, y (1) () x, y, z ˆl x, ˆl y, ˆl z (3) (10.35) [Â, ˆB] = Â ˆB ˆBÂ ˆlx, ˆl y, ˆl z 50

51 (4) ˆl (5) ˆl ˆl x, ˆl y, ˆl z 10-. (1) () (3) L = r p r p (1) x, y, z () (3) (4) Y l,ml (θ, ϕ) z (5) Y l,ml (θ, ϕ) Schrödinger Φ(ϕ) (1) 4 (10.36) e im lϕ, e im lϕ, sin m l ϕ, cos m l ϕ () 4 (3) 4 z (4) 4 z Y 11 Y 0 Schrödinger Y l,ml (θ, ϕ) z m l l Legendre (10.37) (10.38) d dz d dz [(1 z ) dp m l k dz [(1 z ) dp m l l dz ] [ + k(k + 1) ] [ + l(l + 1) ] m l 1 z P m l k = 0 m l 1 z ] P m l l = 0 P m l l z -1 1 P m l k z -1 1 Legendre 51

52 Lambert-Beer Lambert dx I di (11.1) di = κi dx I' x x+dx I' di' l I 0 I (11.) I di I 0 I = l 0 κdx I 0 I (11.3) I = I 0 e κl 0 l A I (11.4) A = log 10 = κ I l al Beer c (11.5) a = εc Lambert Beer Lambert-Beer (11.6) A = εcl ε c mol dm 3 l cm Ψ n E n Ψ m E m Bohr hν (11.7) hν = E m E n = E R mn (11.8) R mn = Ψ m µ Ψ n µ (11.9) µ = e i r i 5

53 ν ν λ ν (11.10) ν = 1 λ = ν c c ν k = π/λ λ nm ν cm 1 λ ν ν 1 mm 100 cm cm GHz 300 MHz 800 nm 1 mm cm THz 300 GHz nm cm THz 400 THz nm cm 1 10 PHz 800 THz E J (11.11) E J = h J(J + 1) = BJ(J + 1) I J 8 l m J + 1 B (11.1) B = h I E J E J J = J J J = ±1 J = Bohr (11.13) hν = E = B[J (J + 1) J(J + 1)] = B[(J + 1)(J + ) J(J + 1)] (11.14) ν = = B(J + 1) = h (J + 1) I h(j + 1) πi 53

54 (11.15) ν = ν c = h(j + 1) πci = B(J + 1) B (11.16) B = h 4πcI = B ch I ν ν (11.17) ν = B = h πci J = 10 9 A E J B B - B ν E v,j (11.18) E v,j = h k µ ( ) ( ) 1 + v + h 1 J(J + 1) = hν I + v + BJ(J + 1) v 9 n ν (11.19) ν = 1 k π µ 54

55 E v,j E v,j v = v v, J = J J v = ±1 J = ±1 v = +1 J J = 0 Q J = 1 P P Bohr (11.0) hν = E = hν (v v) + B[J (J + 1) J(J + 1)] = hν (v + 1 v) + B[(J 1)J J(J + 1)] = hν BJ = 1 k π µ h I J J 1 (11.1) ν = ν hj πi (11.) ν = ν c = 1 k πc µ hj πci = ν BJ ν (11.3) ν = 1 k πc µ J = +1 R R Bohr (11.4) hν = E = hν (v + 1 v) + B[(J + 1)(J + ) J(J + 1)] = hν + B(J + 1) = 1 k π µ + h (J + 1) I J 0 (11.5) ν = ν + h(j + 1) πi (11.6) ν = ν c = 1 k h(j + 1) + = ν + πc µ πci B(J + 1) I k - ν = ν ν ν (11.7) ν = B = h πci ν = ν 55

56 v = 1, J = E v,j v = 0, J = P Q R A B - 4B - - B ν 11.4 Raman Raman ν I Rayleigh ν I + ν R ν I ν R Raman ν I (ν R < 0) Stokes (ν R > 0) anti-stokes ν R - hν I - E v,j E v,j E = E v,j E v,j ν I + ν R Bohr (11.8) hν R = E v = 0, ±1, J = 0, ± Raman Raman Raman - v = 0 J = anti-stokes (11.9) hν R = E = B[J (J + 1) J(J + 1)] = B[(J )(J 1) J(J + 1)] = B(J 1) = h (J 1) I 56

57 J (11.30) ν R = h(j 1) πci = B(J 1) J = + Stokes (11.31) hν R = E = B[J (J + 1) J(J + 1)] = B[(J + )(J + 3) J(J + 1)] J 0 (11.3) ν R = = B(J + 3) = h (J + 3) I h(j + 3) πci = B(J + 3) 4 B Rayleigh Raman 6 B Raman - v = +1 Stokes v = 1 anti-stokes v 1 v = 0 anti-stokes Stokes J = 0 Q J = O J = + S Stokes Q ν R = ν ν R = ν Stokes O Bohr (11.33) hν R = E = hν (v v) B[J (J + 1) J(J + 1)] = hν B[(J )(J 1) J(J + 1)] J (11.34) ν R = ν + = hν + B(J 1) = hν + h (J 1) I h(j 1) πci Stokes R = ν + B(J 1) (11.35) hν R = E = hν (v v) B[J (J + 1) J(J + 1)] = hν B[(J + )(J + 3) J(J + 1)] J 0 (11.36) ν R = ν = hν B(J + 3) = hν h (J + 3) I h(j + 3) πci = ν B(J + 3) 4 B Q O R 6 B 57

58 I v = 1, J = E v,j v = 0, J = Stokes lines S Q O Rayleigh (incident light) anti-stokes lines S Q O 4B - 6B - 4B - 4B - 6B - 4B - 4B - 6B - 4B - ν I ν nm mm 81.0 % (1) () 8 mm nm MnO m mol g MnO ml 540 nm 1.00 cm 3.5 % % g mol g 100 ml 1.00 ml 50.0 ml 1.00 cm 55.0 % (1) () k B T λ (1) λ = 50 nm () λ = 500 nm (3) λ = 1 mm (4) λ = 100 mm (5) λ = 10 mm ν 1 H 17 F 1 H 35 Cl 1 H 81 Br 1 H 17 I ν/cm (1) () 58

59 (3) (4) Raman (5) 1 H D k 17 F 35 Cl 79 Br k/n m (1) () (3) Raman ν R 1 H 14 N 16 O 3 S ν/cm R/nm Morse (11.37) V (x) = D (1 e β(x R)) D, β, R (1) () ν (3) Morse (11.38) G(v) = ( v + 1 ) ( ν + v + 1 ) x e ν, x e = ν 4chD x e v H 35 Cl /cm Raman 306 cm 1 99 cm nm Ar-Kr Raman H 35 Cl D 35 Cl 1 H D D /cm H (11.39) H 35 Cl(v = 0) + D (v = 0) D 35 Cl(v = 0) + HD(v = 0) Born-Oppenheimer H D 1 H D ν /cm B (11.40) E J ch = BJ(J + 1) B cm 1 (1) 59

60 () B H 14 C 16 O 35 Cl B/cm H 35 Cl H 79 Br 1 C 16 O 14 N 16 O B/cm J (11.41) E J ch = BJ(J + 1) DJ (J + 1) (1) () H 35 Cl (11.4) ν = 0.794(J + 1) (J + 1)3 cm 1 D (1) 3 () (3) CO C=O nm P Q R Raman O Q S C 14 N nm, 1630 N m 1 Raman H 35 Cl nm, cm 1 (1) () 1 H 37 Cl, D 35 Cl, D 37 Cl 60

61 (1.1) ϕ(x) = Ae iαx (1.) E = mα h (1.3) p = ᾱ h L (1.4) α = π L n (1.5) A = 1 L L 1 A p p (1.6) ϕ(x) = Ae iαx + Be iαx (1.7) A + B p A p B 1.1. p α m h A p α m h B m x = E < V 0 x = 0 { V0 > 0 at x 0 (1.8) V (x) = 0 at x < 0 x 0 61

62 1.. Schrödinger (1.9) d V (x)] ϕ(x) = m[e dx h ϕ(x) x < 0 (1.10) ϕ(x) = Ae iαx + Be iαx x a (1.11) ϕ(x) = Ce βx + De βx me (1.1) α = h m(v 0 E) (1.13) β = h ϕ(x) dϕ(x) x = 0 dx C 0 x C = 0 (1.14) A + B = D (1.15) iα(a B) = βd 3 (1.16) (1.17) B A = iα β iα + β D A = iα iα + β A = B D m x = E < V 0 x = 0 { V0 > 0 at 0 x a (1.18) V (x) = 0 at x < 0 or x > a x 0 6

63 1.3. Schrödinger (1.19) d V (x)] ϕ(x) = m[e dx h ϕ(x) x < 0 (1.0) ϕ(x) = Ae iαx + Be iαx 0 x a (1.1) ϕ(x) = Ce βx + De βx x > a (1.) ϕ(x) = F e iαx x = x > a Ge iαx ϕ(x) dϕ(x) x = 0 x = a dx x = 0 (1.3) A + B = C + D (1.4) iα(a B) = β(c + D) x = a (1.5) Ce βa + De βa = F e iαa (1.6) β(ce βa De βa ) = iαf e iαa 5 4 A (1.7) (1.8) (1.9) (1.30) B A = [(iα) β ](1 e βa ) (iα + β) (iα β) e βa F A = 4iαβe (iα β)a (iα + β) (iα β) e βa C (β + iα)e(iα β)a F = A β A D (β iα)e(iα β)a F = A β A x < 0 (1.31) (1.3) α m h A α m h B x > a (1.33) α F m h 63

64 R T (1.34) R = B A (1.35) T = F A (1.36) sinh x = ex e x [ = 1 + 4E(V 0 E) V0 sinh βa = [1 + V 0 sinh βa 4E(V 0 E) (1.37) R + T = 1 ] 1 ] E > V 0 x < 0 (1.38) ϕ(x) = Ae iαx + Be iαx 0 x a (1.39) ϕ(x) = Ce iβx + De iβx x > a (1.40) ϕ(x) = F e iαx (1.41) β = m(e V 0 ) h [ (1.4) R = 1 + 4E(E V 0) V0 βa (1.43) T = ] 1 [1 + V 0 sin ] 1 βa 4E(E V 0 ) E > V

65 1-1. ψ m j (1.44) j = i h m ( ψ d dx ψ ψ d ) dx ψ (1) ψ = Ae ikx () ψ = Ae ikx + Be ikx 1-. V (x) m x = (1.45) V (x, y) = { V0 > 0 at x 0 0 at x < 0 (1) E V 0 > E > 0 () E > V 0 (1) (3) 1 (4) (1), () 1-3. V (x) m x = (1.46) V (x, y) = { V0 > 0 at a x 0 0 at x < 0 or x > a (1) E V 0 > E > 0 () E > V 0 (1) (3) 1 (4) (1), () 1-4. V (x) m x = (1.47) V (x, y) = { V0 < 0 at a x 0 0 at x < 0 or x > a (1) E E > 0 () 1 (3) () 1-5. (1) () (3) 1-6. (1) Arrhenius () Arrhenius (3) Arrhenius 65

66 A df(x)/dx = f (x) = f (A-1.1) f(x) = c ( ), f (x) = 0 (A-1.) f(x) = x α, f (x) = αx α 1 (A-1.3) f(x) = e x, f (x) = e x (A-1.4) f(x) = ln x, f (x) = 1 x (A-1.5) f(x) = sin x, f (x) = cos x (A-1.6) f(x) = cos x, f (x) = sin x (A-1.7) f(x) = tan x, f (x) = sec x (A-1.8) f(x) = sin 1 x, f (x) = 1 1 x (A-1.9) f(x) = cos 1 x, f 1 (x) = 1 x (A-1.10) f(x) = tan 1 x, f (x) = x (A-1.11) (f + g) = f + g (A-1.1) (f g) = f g (A-1.13) (fg) = f g + fg (A-1.14) (cf) = cf (c ) (A-1.15) (A-1.16) ( ) f = f g fg g g, g 0 ( ) 1 = g g g, g 0 y = f(z), z = g(x) (A-1.17) dy dx = dy dz dz dx 66

67 1. (A-1.18) cdx = cx (c ) (A-1.19) (A-1.0) (A-1.1) (A-1.) (A-1.3) x α dx = xα+1 α dx = ln x x e x dx = e x 1 1 x dx = sin 1 x x dx = tan 1 x (α 1) (A-1.4) [f(x) + g(x)]dx = f(x)dx + g(x)dx (A-1.5) (A-1.6) [f(x) g(x)]dx = cf(x)dx = c f(x)dx f(x)dx (c ) x = ϕ(t) (A-1.7) f(x)dx = f(ϕ(t)) dϕ(t) dt dt g(x)dx (A-1.8) f(x)g (x)dx = f(x)g(x) f (x)g(x)dx (A-1.9) sin xdx = cos x (A-1.30) cos xdx = sin x (A-1.31) (A-1.3) sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y sin x sin y x = y (A-1.33) sin x = sin x cos x (A-1.34) cos x = cos x sin x = 1 sin x = cos x 1 67

68 (A-1.35) (A-1.36) (A-1.37) sin xdx = 1 (1 cos x)dx = x cos xdx = 1 (1 + cos x)dx = x sin x cos xdx = 1 sin x 4 + sin x 4 cos x sin xdx = = sin x = x cos t = cos x dt = sin xdx (A-1.38) sin x cos xdx = s = sin x ds = cos xdx (A-1.39) sin x cos xdx = tdt = t = cos x = sin x sds = s = sin x = 1 cos x ( ) ( ) x (A-1.40) x sin sin x x sin x xdx = x dx = x x 4 sin x 1 cos x 8 1 f(x) = x, g (x) = sin x x sin xdx (A-1.41) x sin xdx = x3 6 x 1 sin x x cos x 8 4 a > 0, n (A-1.4) 0 x n e ax dx = n! a n+1 e ax 68

69 A- Gauss.1 xe ax (A-.1) I 1 = 0 xe ax dx (A-.) t = ax (A-.3) dt = axdx (A-.1) (A-.4) I 1 = 1 a 0 e t dt = 1 a (, ). e ax (A-.5) I 0 = 0 e ax dx (A-.6) I 0 = e ax dx e ay dy = e a(x +y ) dxdy xy (A-.7) (A-.8) x = r sin θ y = r cos θ (A-.9) r = x + y (A-.10) dxdy = rdθdr (A-.9), (A-.10) (A-.6) (A-.11) I 0 = π/ 0 dθ 0 re ar dr = π (A-.4) (A-.1) I 0 = π 4a (A-.13) I 0 = 1 0 π a e t dt = π 4a 0 re ar dr (, ) I 0 69

70 y r sin θ dθ θ r dθ dr r r cos θ x.3 x n e ax (A-.14) I n = (A-.15) I n = 0 0 = 1 a [ = 1 = n 1 a x n e ax dx x (n 1) xe ax dx 0 ( x (n 1) axe ax) dx a x(n 1) e ax 0 = n 1 a I n ] 0 x (n ) e ax dx ( 1 ) a 0 (n 1)x (n ) e ax dx (A-.14) n (A-.5) n (A-.1) (, ) n I n n A--1. n n a > 0 (A-.16) 0 x n e ax dx = (n 1) π (n 1)!! π n+1 a n a n+1 a n a A--. n n + 1 a > 0 (A-.17) 0 x n+1 e ax dx = n! a n+1 A--3. a (A-.18) 0 e ax dx = 1 π a 70

71 A--4. a (A-.19) 0 x 1 e ax dx = 1 a 71

72 A y = g(x) (A-3.1) dy dx = g(x) (A-3.) dy = g(x)dx + c c (A-3.3) y = g(x)dx + c c 3. y + ay = g(x) (A-3.4) dy + ay = g(x) dx (A-3.5) d dx (eax ax dy y) = e dx + aeax y (A-3.4) e ax (A-3.5) (A-3.6) d dx (eax y) = e ax g(x) (A-3.7) d (e ax y) = e ax g(x)dx + c (A-3.8) e ax y = e ax g(x)dx + c (A-3.9) y = e ax e ax g(x)dx + ce ax g(x) = 0 (A-3.10) y = ce ax 7

73 3.3 y + ay + by = 0 (A-3.11) d y dx + a dy dx + by = 0 d y dx + a dy dx + by = d ( ) ( ) dy dy (A-3.1) dx dx py q dx py p, q = d y dy (p + q) dx dx + pqy = 0 (A-3.13) (A-3.14) p + q = a pq = b p, q (A-3.15) (r p)(r q) = r (p + q)r + pq = r + ar + b = 0 (A-3.16) (A-3.17) r 1 = a + a 4b r = a a 4b = p = q r 1 = q, r = p (A-3.1) (A-3.18) ( ) ( ) d dy dy dx dx py q dx py = 0 (A-3.19) w = dy dx py (A-3.18) (A-3.0) dw dx qw = 0 (A-3.10) (A-3.1) w = c 1 e qx (A-3.19) (A-3.) dy dx py = c 1e qx (A-3.9) (A-3.3) y = c 1 e px e px e qx dx + c e px p = q p q 73

74 p = q (A-3.4) y = c 1 e px dx + c e px = c 1 xe px + c e px p q (A-3.5) y = c 1 e px e (q p)x dx + c e px = c 1 q p eqx + c e px c 1, c (A-3.6) y = Ae px + Be qx p, q A, B (A-3.16) a 4b > 0 p, q (A-3.6) a 4b = 0 p = q (A-3.4) a 4b < 0 p, q p, q (A-3.7) (A-3.8) p = α + iβ q = α iβ (A-3.9) α = a (A-3.30) 4b a β = (A-3.6) (A-3.31) y = Ae αx e iβx + Be αx e iβx (A-3.11) (A-3.11) 3.4 y ± β y = 0 (A-3.11) a = 0 β (A-3.3) d y dx β y = 0 β > 0 p = β, q = β (A-3.6) (A-3.33) y = Ae βx + Be βx (A-3.34) d y dx + β y = 0 p = iβ, q = iβ (A-3.31) (A-3.35) y = Ae iβx + Be iβx 74

75 Euler (A-3.36) e ±iβx = cos βx ± i sin βx (A-3.35) (A-3.37) y = (A + B) cos βx + i(a B) sin βx A, B (A-3.38) y = C cos βx + D sin βx (A-3.34) (A-3.35) (A-3.38) (A-3.35) C, D (A-3.38) 75

76 A-4 Taylor 4.1 Taylor f(x) a x b n 1 f(x) n 1 f (n 1) (x) f (n) (x) a < x < b n (A-4.1) f(b) = f(a) + f (a)(b a) + f (a) (b a) + + f (n 1) (a)! (n 1)! (b a)n 1 + R n (A-4.) R n = f (n) (c) n! (b a) n = f (n) (a + θ(b a)) (b a) n n! a < c < b, 0 < θ < 1 c θ Taylor a = 0 Maclaurin 4. Taylor Taylor f(x) x = a a = b (A-4.3) f(b) = f(a) + f (a)(b a) + f (a)! (b a) + + f (n) (a) (b a) n + n! Taylor a = 0 Maclaurin b a Taylor Maclaurin (A-4.4) (A-4.5) (A-4.6) (A-4.7) (A-4.8) e x = 1 + x 1! + x! + + xn n! + sin x = x x3 3! + x5 5! + + ( 1)n 1 x n 1 (n 1)! + cos x = 1 x! + x4 xn + + ( 1)n 4! (n)! + ln(1 + x) = 1 x + x3 xn + + ( 1)n 1 3 n x = 1 + x + x + + x n + 76

77 A (A-5.1) i = 1 a, b c (A-5.) c = a + ib c Re(c) = a c Im(c) = b c c = a ib c = a + ib z = s + it a = s b = t c = z 5. i i 1 (A-5.3) c ± z = (a + ib) ± (s + it) = (a ± s) + i(b ± t) (A-5.4) (A-5.5) cz = (a + ib)(s + it) = (as bt) + i(at + bs) c z = a + ib s + it as + bt at = s + ibs + t s + t (A-5.6) c + c = (a + ib) + (a ib) = a 5.3 (A-5.7) c = a + b (A-5.8) c = c (A-5.9) c = c = cc 5.4 Euler Taylor (A-5.10) e ix = cos x + i sin x A-5-1. Taylor Euler (A-5.11) e ix = cos x + i sin x A-5-. Euler 77

78 A f 1, f, f 3,, f n c 1, c, c 3,, c n (A-6.1) c 1 f 1 + c f + c 3 f c n f n 6. f 1, f, f 3,, f n (A-6.) c 1 f 1 + c f + c 3 f c n f n = 0 c 1 = c = c 3 = = c n = 0 f 1, f, f 3,, f n 6.3 n y y, y, y (3),, y (n) (A-6.3) y (n) + P 1 (x)y (n 1) + +P (x)y (n ) + + P n 1 (x)y + P n (x)y = R(x) n P 1 (x), P (x),, P n (x), R(x) R(x) = 0 u 1 (x), u (x),, u n (x) n n (A-6.4) y = c 1 u 1 (x) + c u (x) + + c n u n (x) c 1, c, c 3,, c n Schrödinger A-6-1. (1) sin x, sin x, sin 3x, sin 4x () cos x, cos x, cos 3x, cos 4x (3) sin x, cos x, sin x, cos x (4) e ix, e ix, e 3ix, e 4ix (5) e ix, e ix, e ix, e ix (6) sin x, cos x, e ix, e ix 78

79 A-7 Hamiltonian 7.1 Lagrangian m 1 V (x, y, z) F V (x, y, z) (A-7.1) F x = x V (x, y, z) (A-7.) F y = y V (x, y, z) (A-7.3) F z = z Lagrangian L (A-7.4) L = T V = 1 m(ẋ + ẏ + ż ) V (x, y, z) ẋ = v x L Newton F = ma ( ) d L (A-7.5) = L dt ẋ x y, z Lagrange t 1 x 1 t x I (A-7.6) I = t t 1 Ldt Hamilton Lagrange Hamilton 7. Lagrangian L x ẋ = v x L dl (A-7.7) dl = ( ) L dẋ + ẋ p x = mẋ (A-7.8) (A-7.9) p x = L ẋ L x = dp dt ( ) L dx x (A-7.10) dl = p x dẋ + ṗ x dx L x p x H (A-7.11) H = p x ẋ L = T T + V = T + V 79

80 (A-7.1) dh = p x dẋ + ẋdp x dl = p x dẋ + ẋdp x p x dẋ ṗ x dx = ẋdp x ṗ x dx ( H (A-7.13) dh = p x ) dp x + ( ) H dx x (A-7.14) dx dt = H p x (A-7.15) dp x dt = H x x p x H Hamiltonian H Hamilton 7.3 Hamiltonian q r p r r Hamiltonian (A-7.16) H = 1 ( p r + 1 ) 1 m r p θ + r sin θ p ϕ + V (r) (A-7.17) (A-7.18) (A-7.19) p r = mṙ p θ = mr θ p ϕ = mr sin θ ϕ r p r θ p θ ϕ p ϕ Hamiltonian ϕ ϕ (A-7.0) dp ϕ dt = H ϕ = 0 p ϕ 7.4 t Hamiltonian E F (A-7.1) F = H E (A-7.) (A-7.3) dx dt = F = H p x p x dp x dt = F x = H x 80

81 t (A-7.4) (A-7.5) d( E) dt = F t dt dt = F ( E) = 1 t E A-7-1. Hamilton Lagrange A-7-. y p x A-7-3. r Hamiltonian A-7-4. r p r θ p θ ϕ p ϕ A-7-5. ˆp x = i h x Ê 81

82 A-8 Hermite 8.1 Schrödinger (A-8.1) (A-8.) (A-8.3) (A-8.4) (A-8.5) d h mω ϕ(x) + m dx x ϕ(x) = Eϕ(x) d dξ ϕ(ξ) + (λ ξ )ϕ(ξ) = 0 λ = E hω ξ = αx α = mω h (A-8.6) ϕ( ) = ϕ( ) = 0 8. x x (A-8.7) d dξ ϕ(ξ) ξ ϕ(ξ) = 0 (A-8.8) ϕ(ξ) = Ae ξ / + Be ξ / A = ( N ) (A-8.9) ϕ(ξ) = NH(ξ)e ξ / (A-8.10) (A-8.11) dϕ(ξ) dξ d ϕ(ξ) dξ [ = Ne ξ / ξh(ξ) + dh(ξ) ] dξ Schrödinger (A-8.1) [ d H(ξ) dξ [ = Ne ξ / (ξ 1)H(ξ) ξ dh(ξ) ] + d H(ξ) dξ dξ ξ dh(ξ) dξ ] + (λ 1)H(ξ) e ξ / = 0 8

83 [ ] (A-8.13) d H(ξ) dξ ξ dh(ξ) dξ + (λ 1)H(ξ) = 0 Hermite Hermite H(ξ)e ξ / λ (A-8.14) λ = n + 1 n = 0, 1,, (A-8.15) H n (ξ) = ( 1) n exp(ξ ) dn exp( ξ ) dξ n H n n Hermite n n 8.4 Hermite Hermite (A-8.16) d H(x) dx x dh(x) dx + (λ 1)H(x) = 0 H(x) (A-8.17) H(x) = a 0 + a 1 x + a x + a 3 x a m x m + H(x) (A-8.18) (A-8.19) dh(x) dx = a 1 + a x + 3a 3 x + + ma m x m 1 + d H(x) dx = a x + 3 a 3 x + + m(m 1)a m x m + a +6a 3 x +1a 4 x + +(m )(m 1)a m+ x m + a 1 x 4a x + ma m x m +(λ 1)a 0 +(λ 1)a 1 x +(λ 1)a x + +(λ 1)a m x m = 0 m x m a m (A-8.0) a m+ = 1 + m λ (m + 1)(m + ) a m a 0 a 1 H(x) m n n λ (A-8.1) λ = n + 1, n = 0, 1,, 3, n a 0 = 1, a 1 = 0 m m n + a m = 0 n a 0 = 0, a 1 = m m n + a m = 0 Hermite 83

84 8.5 Hermite Taylor (A-8.) e ξ = 1 + ξ 1 + ξ4! + + b nξ n + b n+ ξ n+ +, b n = 1 (n/)! Hermite (A-8.3) H(ξ) = a 0 + a 1 ξ + a ξ + + a n ξ n + n (A-8.4) a n+ = b n+ = a n b n n ϕ(x) = e x / a m x m m=0 e x / x 8.6 Hermite Hermite (A-8.5) H n (ξ) = ( 1) n exp(ξ ) dn exp( ξ ) dξ n Hermite (A-8.6) (A-8.7) (A-8.8) H n+1 (ξ) = ξh n (ξ) nh n 1 (ξ) d H n (ξ) d ξ = ξ dh n(ξ) dξ nh n (ξ) H n (ξ)h m (ξ)e ξ dξ = n πn!δ nm 84

85 A N n (A-9.1) ϕ(x) = N n e ξ / H n (ξ) (A-9.) N n = 1 ( α ) 1/4 n n! π H n (ξ) Hermite (A-9.3) ξ = α 1 x µk (A-9.4) α = h 9. (A-9.5) (A-9.6) h d ϕ(x) µ dx + 1 kx ϕ(x) = h d ϕ(x) µ dx + h α µ x ϕ(x) ( = h d ) ϕ(x) µ dx α x ϕ(x) d ϕ(x) dx ( ) = d dx N n H n (α 1/ x)e αx / dϕ(x) = N n (α ) 1/ H dx n(α 1/ x)e αx / αxh n (α 1/ x)e αx / = N n (αh n(α 1/ x)e αx / α 3/ xh n(α 1/ x)e αx / αh n (α 1/ x)e αx / α 3/ xh n(α 1/ x)e αx / + α x H n (α 1/ x)e αx / ) ξ = α 1/ x (A-9.7) (A-9.8) d ϕ(x) dx α x ϕ(x) = N n αe ξ / (H n(ξ) ξh n(ξ) H n (ξ) + ξ H n (ξ) ξ H n (ξ)) ( E n = h α(n + 1) = hω n + 1 ) µ = N n αe ξ / (H n(ξ) ξh n(ξ) H n (ξ)) H n(ξ) ξh n(ξ) + nh n (ξ) = 0 (A-8.7) = N n αe ξ / ( nh n (ξ) H n (ξ)) = α(n + 1)ϕ(x) 9.3 (A-9.9) ϕ mϕ n dx = δ mn = { 0 m n 1 m = n 85

86 (A-9.10) ϕ mϕ n dx = N m H m (α 1/ x)e αx / N n H n (α 1/ x)e αx / dx = N m N n H m (α 1/ x)h n (α 1/ x)e αx = N m N n α 1/ H m (ξ)h n (ξ)e ξ dξ (A-8.8) (A-9.) 9.4 n (A-9.11) x = ϕ mxϕ n dx = = Nnα 1 ξh n (ξ)h n (ξ)e ξ dξ N n H n (α 1/ x)e αx / xn n H n (α 1/ x)e αx / dx ξh n = nh n 1 + H n+1 / (A-8.6) = Nnα 1 nh n 1 H n e ξ dξ + Nn/α 1 H n+1 H n e ξ dξ = 0 (A-9.1) x = ϕ mx ϕ n dx = = Nnα 3/ (nh n 1 (ξ) + H n+1 (ξ)/) e ξ dξ = N nα 3/ = 1 n n! ( α π N n H n (α 1/ x)e αx / x N n H n (α 1/ x)e αx / dx (nh n 1 ) e ξ dξ + (H n+1 ) e ξ dξ ) 1/ α 3/ [n π 1/ n 1 (n 1)! + (1/4)π 1/ n+1 (n + 1)!] = α 1 [n 1 + (1/4)(n + 1)] = α 1 ( n + 1 ) V = 1 kx = 1 k x = 1 ( kα 1 n + 1 ) (A-9.13) (A-9.14) α = kµ/ h = 1 ( h k n + 1 ) = 1 hω ( n + 1 ) µ ( T = E n V = hω n + 1 ) 1 hω ( n + 1 ) = 1 hω ( n + 1 ) V (x) = ax b (A-9.15) T = b V 86

87 A (x, y, z) (q 1, q, q 3 ) x, y, z i, j, k x q 1, q, q 3 x = x(q 1, q, q 3 ) q 1 = q 1 (x, y, z) y = y(q 1, q, q 3 ) q = q (x, y, z) z = z(q 1, q, q 3 ) q 3 = q 3 (x, y, z) a i (i = 1,, 3) q i = const. q i dx dy, dz (A-10.1) dx = x q 1 dq 1 + x q dq + x q 3 dq 3 ds (A-10.) (A-10.3) ds = dx + dy + dz = h ijdq i dq j h ij = x q i x q j + y q i y q j + z q i z q j i j h ij h ii = h i = 0 (A-10.4) ds = (h 1 dq 1 ) + (h dq ) + (h 3 dq 3 ) ds 1 + ds + ds 3 dτ (A-10.5) dτ = dxdydz = ds 1 ds ds 3 = h 1 h h 3 dq 1 dq dq 3 = Jdq 1 dq dq 3 J Jacobian x x x q 1 q q 3 y y y (A-10.6) J = q 1 q q 3 z z z q 1 q q ψ(x, y, z) gradψ (A-10.7) gradψ = ψ = i ψ x +j ψ y +k ψ z = a ψ ψ ψ ψ ψ ψ 1 +a +a 3 = a 1 +a +a 3 s 1 s s 3 h 1 q 1 h q h 3 q 3 V(x, y, z) divv (A-10.8) div V = V = V x x + V y y + V [ z z = 1 (V 1 h h 3 ) + h 1 h h 3 q 1 V = gradψ Laplacian q (V h 3 h 1 ) + ] (V 3 h 1 h ) q 3 (A-10.9) ψ = ψ = ψ x + ψ y + ψ z [ ( ) 1 h h 3 ψ = + ( ) h3 h 1 ψ + ( )] h1 h ψ h 1 h h 3 q 1 h 1 q 1 q h q q 3 h 3 q 3 87

88 V(x, y, z) rotv (A-10.10) rotv = V = i j k a 1 1 h 1 a h a 3 h 3 = x y z h V x V y V 1 h h 3 q 1 q q 3 z h 1 V 1 h V h 3 V 3 A (1) () Laplacian A-10-. (x, y, z) (q 1, q, q 3 ) a i (i = 1,, 3) q i = const. q i (1) dx, dy, dz dq i () ds = dx + dy + dz dq i h i (A-10.11) h ij = x q i x q j + y q i y q j + z q i z q j i j h ij = 0 h ii = h i (3) dxdydz dq i h i (4) (5) Laplacian 88

89 A z r cos θ (x,y,z) θ r r sin θ r sin θ sin φ r sin θ cos φ φ (x,y,0) y x r r z r θ (A-11.1) z = r cos θ r xy r xy r xy (A-11.) r xy = r sin θ x r xy ϕ (A-11.3) (A-11.4) x = r xy cos ϕ = r sin θ cos ϕ y = r xy sin ϕ = r sin θ sin ϕ (A-11.5) (A-11.6) (A-11.7) 0 r < 0 θ π 0 ϕ π r, θ, ϕ x, y, z (A-11.8) r = x + y + z (A-11.9) θ = tan 1 x + y z (A-11.10) ϕ = tan 1 y x (A-11.11) h 1 = h r = 1 89

90 (A-11.1) (A-11.13) h = h θ = r h 3 = h ϕ = sin θ (A-11.14) dτ = r sin θdrdθdϕ dτ dτ (dr, dθ, dϕ) dr dr dθ rdθ θ dϕ sin θdϕ dτ = dxdydz (A-11.14) z r cos θ dr r dθ r dθ dφ y r sin θ r sin θ dφ x (A-11.15) ψ ψ = a r r + a 1 ψ θ r θ + a 1 ϕ r sin θ 1 (A-11.16) V = r sin θ (A-11.17) ψ = (A-11.18) V = 1 r sin θ 1 r sin θ ψ ϕ [ sin θ r (r V r ) + r θ (sin θv θ) + r [ sin θ r ( r ψ ) + ( sin θ ψ r θ θ a r ra θ r sin θa ϕ r θ ϕ V r rv θ r sin θv ϕ 11. ] ϕ (V ϕ) ) + 1 sin θ ] ψ ϕ (A-11.19) (A-11.0) (A-11.1) x = r cos ϕ y = r sin ϕ z = z 90

91 xy x, y r, ϕ f(x, y) f(r, ϕ) (A-11.) f(x, y) = x + y x + y (A-11.3) f(r, ϕ) = cos ϕ + sin ϕ r (A-11.4) (A-11.5) f(x, y) x f(x, y) y = r f(r, ϕ) x r = r f(r, ϕ) y r + ϕ f(r, ϕ) x ϕ + ϕ f(r, ϕ) y ϕ r x ; ϕ x ; r y ; ϕ y (A-11.6) r = x + y (A-11.7) tan ϕ = y x (A-11.8) (A-11.9) (A-11.30) (A-11.31) r x = 1 x x + y = r cos ϕ = cos ϕ r r y = 1 y x + y = r sin ϕ = sin ϕ r tan ϕ x tan ϕ x = ( y ) = y x x x = ϕ tan ϕ x ϕ = ϕ x 1 cos ϕ (A-11.3) ϕ x = y x cos ϕ = r sin ϕ r cos ϕ cos ϕ = sin ϕ r (A-11.33) ϕ y = cos ϕ r (A-11.34) f(x, y) = x ( cos ϕ r sin ϕ r ϕ ) f(r, ϕ) 91

92 (A-11.35) ( f(x, y) = sin ϕ y r + cos ϕ r ϕ ) f(r, ϕ) f(x, y) = x x ( ) ( f(x, y) = cos ϕ x r sin ϕ ) (( cos ϕ r ϕ r sin ϕ r ϕ (( = cos ϕ r sin ϕ ) ( cos ϕ r ϕ r sin ϕ )) f(r, ϕ) r ϕ ( = cos ϕ r r + cos ϕ ( sin ϕ ) sin ϕ ( cos ϕ ) r r ϕ r ϕ r + sin ϕ ( ) ) sin ϕ f(r, ϕ) r ϕ r ϕ ( = (cos ϕ 1 + cos ϕ sin ϕ r r ϕ 1 ) r r ϕ ( sin ϕ r + cos ϕ ) + sin ϕ ( ϕ r r cos ϕ + sin ϕ ϕ ϕ = (cos ϕ cos ϕ sin ϕ cos ϕ sin ϕ + r r ϕ r r ϕ + sin ϕ r r sin ϕ r sin ϕ cos ϕ r y f(x, y) = ( y = + (sin ϕ r sin ϕ cos ϕ r ϕ r f(x, y) y + sin ϕ cos ϕ r ) ( = cos ϕ sin ϕ r ϕ ϕ r sin ϕ + cos ϕ = 1 (A-11.36) ( x + y ) ( f(x, y) = r + 1 r sin ϕ r + cos ϕ r sin ϕ cos ϕ r ϕ + sin ϕ r ϕ + cos ϕ sin ϕ r ) (( ϕ ϕ + cos ϕ r ϕ r + 1 ) r ϕ f(r, ϕ) z (A-11.37) ˆLz = xˆp y yˆp x = h ( x i y y ) x A A ˆL z = h ( ( r cos ϕ sin ϕ i r + cos ϕ r ϕ = h ( cos ϕ i ϕ + sin ϕ ) = h ϕ i ) ( r sin ϕ ϕ ) f(r, ϕ) sin ϕ r + cos ϕ r r ϕ + cos ϕ r r ) f(r, ϕ) cos ϕ r sin ϕ r )) ϕ ) ) f(r, ϕ) ) ) f(r, ϕ) ϕ ) ) f(r, ϕ) 9

93 A J x x x r θ ϕ y y y (A-11.38) J = r θ ϕ z z z r θ ϕ A-11-. (A-11.39) 1 ( x/ r), r x A e r A (A-11.40) (A-11.41) (A-11.4) x = r x r + θ x θ + ϕ x ϕ y = r y r + θ y θ + ϕ y ϕ z = r z r + θ z θ + ϕ z ϕ (A-11.17) A x = ρ cos ϕ (A-11.43) y = ρ sin ϕ z = z (1) r, θ, ϕ () dτ (3) (4) Laplacian 93

94 A-1 Legendre 1.1 Θ(θ) N Θ(θ) = NP (θ) (A-1.1) P (θ) θ (A-1.) z = cos θ, 1 z 1 + cos θ ) P (θ) + (λ m sin θ θ sin P (θ) = 0 θ (A-1.3) dz = sin θdθ (A-1.4) sin θ = 1 z (A-1.5) (A-1.6) θ = θ sin θ z = sin θ z cos θ z = (1 z ) z z z [ ] ) d (1 z dp (z) ) + (λ m dz dz 1 z P (z) = 0 Legendre Legendre A-7 λ = l(l + 1) l = 0, 1,, Legendre (A-1.7) l (z) = (1 z ) m / d m Pl 0(z) P m (A-1.8) Pl 0 = 1 l l! d l dz l (z 1) l dz m Pl 0 (z) l l (A-1.9) m l l m l N (A-1.10) 4 N π 0 [ P m l (cos θ)] sin θdθ = 1 0 m P (cos θ) l P (cos θ) 0 0 P (cos θ) 1 1 P (cos θ) P (cos θ) P (cos θ) 4 0 π/ π 0 P (cos θ) π/ π 0 π/ π θ 94

95 1. Legendre Legendre [ ] ) d (A-1.11) (1 z dp (z) ) + (λ m dz dz 1 z P (z) = m = 0 m = 0 Legendre (A-1.1) (1 z ) d P 0 (z) dz z dp 0 (z) dz (A-1.13) P 0 (z) = a 0 + a 1 z + a z + P 0 (z) (A-1.14) (A-1.15) + λp 0 (z) = 0 dp 0 (z) = a 1 + a z + 3a 3 z + + na n z n 1 + dz d P 0 (z) dz = a + 3 a 3 z + + n(n 1)a n z n + a +6a 3 z +1a 4 z + +(n + )(n + 1)a n+ z n + a z n(n 1)a n z n a 1 z 4a z na n z n +λa 0 +λa 1 z +λa z + +λa n z n = 0 (A-1.16) (n + )(n + 1)a n+ z n [n(n + 1) λ]a n z n = 0 n=0 n=0 z z n (A-1.17) (n + )(n + 1)a n+ [n(n + 1) λ]a n = 0 (A-1.18) (n + )(n + 1)a n+ = [n(n + 1) λ]a n z 1 z 1 λ (A-1.19) λ = l(l + 1), l = 0, 1,, l l a 0 = 1, a 1 = 0 (A-1.0) P 0 l (z) = 1 + a z + + a l z l l a 0 = 0, a 1 = 1 (A-1.1) P 0 l (z) = z + a 3 z a l z l Pl 0 (z) l Pl 0 (z) Rodrigues (A-1.) Pl 0 = 1 l l! d l dz l (z 1) l 95

96 1.4 m 0 m 0 m = 0 Legendre m (A-1.3) (A-1.4) d m dz m [(1 z ) d P 0 d m [ z dp 0 dz m l (z) dz l (z) dz ] = (1 z ) d m + P 0 l (z) m ( m 1) ] = z d m +1 P 0 l (z) z m d m +1 Pl 0(z) dz m + dz m +1 d m +1 P 0 l (z) dz m +1 m d m Pl 0(z) dz m +1 dz m (A-1.5) (1 z ) d u(z) dz (A-1.6) u(z) = d m P 0 l (z) dz m ( m + 1)z du(z) dz (1 z ) m / (A-1.7) (1 z ) d w(z) dz z dw(z) dz (A-1.8) w(z) = (1 z ) m / d m P 0 l (z) dz m + [l(l + 1) m ( m + 1)]u(z) = 0 ] + [l(l + 1) m 1 z w(z) = 0 m 0 (A-1.9) l (z) = (1 z ) m / d m Pl 0(z) P m dz m Pl 0 (z) l l (A-1.30) m l l m l A-1-1. Rodorigez P l m = 0 Legendre 96

97 A-13 Bohr Bohr 13.1 Rydberg 1 ( 1 λ = R n 1 ) 1 n, n > n 1 > 0, R = m 1 Rydberg 13. J. J. Thomson H. Nagaoka 13.3 α Rutherford m m Nagaoka-Rutherford 13.4 { x = r cos θ y = r sin θ r θ dθ dt = θ = ω = const. { { ẋ = vx = rω sin θ vx = rω (A-13.1) cos θ ẏ = v y = rω cos θ v y = rω sin θ (A-13.) v = v = vx + vy = rω, v = v x + v y = rω = v r 1 mv = 1 mr ω 1 Iω I = mr 13.5 Bohr Bohr Bohr L = mvr 97

98 A A-13-. (A-13.3) ν = E n E n1 h Bohr h(= h/π) (A-13.4) L = mvr = n h, n = 1,, 3, e m e e e Coulomb 4πε 0 r = m ev r Bohr r = 4πε 0n h Bohr a B = 4πε 0 h 13.6 Sommerfeld Bohr m e e, E = p m e m e e, Rydberg R = m ee 4 ( e 4πε 0 r = me e 4 ) 1 3π ε 0 h n 64π 3 ε 0 h3 c p q (A-13.5) J = p dq = nh J : A r m (1) x () (3) (4) (5) A-13-. e e, m e Bohr (1) () (3) (4) A Bohr (1) n () (3) A Bohr A Bohr G = m 3 s kg 1 98

99 A kg kg m G = m 3 s kg 1 Bohr (1) () (3) 1 A Rydberg ( 1 1 (A-13.6) λ = R n 1 ) 1 n, n > n 1 > 0 1 R Rydberg m 1 (1) Lyman n 1 = 1 4 () Balmer n 1 = 4 (3) Paschen n 1 = 3 4 (4) Bracket n 1 = 4 4 (5) Pfund n 1 = 5 4 (6) e e, m e Bohr Rydberg (7) Rydberg A A He + e Bohr (1) () (3) A He + Bohr (1) Rydberg () n = 4 Pickering 4 (3) A Ze 1 Bohr A r m 1, m (1) m 1 r 1 m r () ω m 1, m (3) ω (4) K (A-13.7) K = 1 Iω I (5) r µ A Bohr 99

100 (1) Bohr () Rydberg (3) A Bohr A (1) () nm (3) ω (4) Bohr (5) A HCl (1) H 1.008, Cl () nm (3) A Bohr Sommerfeld A s A V (x) m (A-13.8) V (x) = { 0 at 0 < x < a at x 0 or x a Sommerfeld A V (x) = 1 kx m k Sommerfeld 100

101 A ρ T T(x) θ T (x+dx) y T (x) y φ T(x+dx) x x+dx x y x x + dx dx Newton F m a (A-14.1) F = ma dx y ma (A-14.) ma = ρdx u(x) t F θ at x = x, ϕ at x = x + dx y T y (A-14.3) (A-14.4) T y (x) = T sin θ T y (x + dx) = T sin ϕ F (A-14.5) F = T y (x) + T y (x + dx) u(x) θ, ϕ sin θ cos θ θ = 0 Taylor ϕ (A-14.6) sin θ = θ θ3 3! + θ (A-14.7) cos θ = 1 θ! + 1 θ (A-14.8) sin θ sin θ u = tan θ = cos θ x 101

102 ( u(x) F = T x u(x) (A-14.9) x=x+dx x = T u(x) x dx x=x Newton (A-14.10) ρ u t = T u(x) x v ) u(x) t = 0 t = t vt x t = 0 u(x) t = t u(x vt) (A-14.11) (A-14.1) u(x vt) t u(x vt) x = v u (x vt) = u (x vt) (A-14.13) 1 u(x, t) v t = u(x, t) x T/ρ u(x, t) λ k T ω ν (A-14.14) (A-14.15) u(x, t) = u(kx ωt) k = π λ (A-14.16) ω = π T = πν (A-14.17) (A-14.18) u(x, t) = v u(x, t) t x v = ω k 10