Transcription

1 m = m

2

3 nm 486nm 434nm 410nm D D 1 D Σ E 3

4 c = ms 1 h = Js = Js e = C ϵ 0 = J 1 m 1 C e 4πϵ 0 = Jm m e = kg m p = kg µ B = e = 9.74 m e c 10 4 JT 1 = evt 1 µ B = e = m 10 7 pc JT 1 g g eV = J 4

5 x ẋ p x t ψ(x, t) = x ψ(t) (1) x ˆx = x, p ˆp = i x. () n n q i (i = 1,,..., n) p i = L/ q i q i q 1, q,..., q n (3) ψ(q i ; t) = q i ψ(t) (4) q i ˆq i = q i, p i ˆp i = i. (5) q i E H(ˆq i, ˆp i )ψ(q i, t) = i t ψ(q i, t) (6) ψ(q i, t) = e iet/ ψ(q i ) (7) H(ˆq i, ˆp i )ψ(q i ) = Eψ(q i ) (8) 1 5

6 1. 3 x = (x 1, x, x 3 ) = (x, y, z) 3 U(x) H = 1 m ˆp + U(x) (9) ˆp ˆp = (ˆp x, ˆp y, ˆp z ) = ( i x, i y, i z ) = i (10) x, y, z nabla ( x, y, z ) (11) Laplacian = = x + y + z (1) ) ( m + U(x) ψ(x, t) = i ψ(x, t) (13) t ρ(x) ρ(x, t) = ψ(x, t) (14) 3 ψ ρ(x, t) = ψ t t + ψ t ψ ( ) i = ψ m ψ + ( ) i m ψ ψ = i m [ ψ ( ψ) + ( ψ )ψ] (15) j = i m [ ψ ( ψ) + ( ψ )ψ] (16) 6

7 ρ = j (17) t M d ρdv = jdv = dt j ds (18) M M M M M M j M ρ 1: 1.3 SI H(x, p; t) = 1 m (p qa(x, t)) + qϕ(x, t) (19) (ϕ, A) E = ϕ + A, B = A. (0) t p p qa, H H qϕ (1) 7

8 1.1 (19) E B ˆp = i p A(x) p x A x p x ˆp ˆx () ( ˆp ˆr) = (ˆr ˆp) ( ˆp ˆr) (3) ( ˆp qa(ˆx)) = ˆp q( ˆp A(ˆx) + A(ˆx) ˆp) + q A(ˆx) (4) [ ] 1 m ( ˆp qa) + qϕ ψ = i t ψ (5) A A = A + λ, ϕ ϕ = ϕ t λ, λ (x, t) ψ ψ = e iqλ/ ψ. (6) j 1.4 Ĥ(q, ˆp)ψ(q) = Eψ(q) (7) 8

9 n Ĥ(q, p) = Ĥ i (q i, p i ) (8) i=1 1 U(x) U(x) = U x (x) + U y (y) + U z (z) (9) Ĥ = Ĥx + Ĥy + Ĥz, Ĥ i = 1 m ˆp i + U i (x i ), (i = x, y, z). (30) ψ(x) = ψ x (x)ψ y (y)ψ z (z) (31) (31) (7) (Ĥxψ x )ψ y ψ z + ψ x (Ĥyψ y )ψ z + ψ x ψ y (Ĥyψ z ) = Eψ x ψ y ψ z (3) ψ x ψ y ψ z Ĥ x ψ x ψ x + Ĥyψ y ψ y + Ĥzψ z ψ z = E (33) x, y, z 3 E x, E y, E z Ĥ x ψ x = E x ψ x, Ĥ y ψ y = E y ψ y, Ĥ z ψ z = E z ψ z (34) E = E x + E y + E z (35) 9

10 (31) ψ x (x) dx = ψ y (y) dy = ψ z (z) dz = 1 (36) ψ(x) d 3 V = 1 (37) (31) x x + δx x+δx x+δx dx dy dz ψ(x) = ψ x (x) dx ψ y (y) dy ψ z (z) dz x x = ψ x (x) δx (38) ψ x (x) x 1.4 U(x) = 1 kx (39) 1.5 L x, L y, L z U(x) = 0 0 x L x, 0 y L y, 0 z L z, = outside (40) 0 1 ψ(x) = 0 Ĥψ(x) = Eψ(x) (41) 10

11 Ĥ = Ĥx + Ĥy + Ĥz (4) 3 Ĥ i = + V i (x i ). (43) m x i V i (x i ) 0 x i L i 0 + (45) ψ(x) = ψ x (x)ψ y (y)ψ z (z) (44) Ĥ i ψ i (x i ) = E i ψ i (x i ), (i = x, y, z) (45) 0 ψ i,ni = sin n iπ x i, n i = 1,, 3,... (46) L i L i Ĥi ( ) E i,ni = ni π. (47) m L i ( E = E x,nx + E y,ny + E z,nz = π n x m n x, n y, n z L x + n y L y ) + n z, n L x, n y, n z = 1,, 3,.... z (48) x 1 x x 1, x ψ(x 1, x ) = x 1, x ψ (49) 11

12 H = 1 m 1 p m p + U(x 1 x ) (50) x 1 x p 1 ˆp 1 = i 1, p ˆp = i. (51) i (i = 1, ) ( ) i =,, x i y i z i (5) X = 1 M (m 1x 1 + m x ), x = x 1 x (53) M = m 1 + m µ = m 1m M P = MẊ p = µẋ H = 1 M P + 1 µ p + U(x) (54) P = MẊ = m 1ẋ 1 + m ẋ = p 1 + p, p = µẋ = m 1m M ẋ1 m 1m M ẋ = m M p 1 m 1 M p (55) ˆP = i X, ˆp = i x (56) 1.6 (53) X x 1 (55) Ĥ = M X µ x + U(x) (57) 1

13 X x ψ(x, x) = ψ CM (X)ψ rel (x) (58) M Xψ CM (X) = E CM ψ CM (X) (59) ] [ µ x + U(x) ψ rel (x) = E rel ψ rel (x) (60) E = E CM + E rel (61) X ψ CM (X) = exp(ik X) (6) E CM = M k (63) ψ rel (60) 13

14 .1 U(x) r = x (r, θ, ϕ) (x, y, z) x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (64) z θ r y φ x : 1 L = µ [ ṙ + r ( θ + sin θ ϕ ] ) U(r) (65) (r, θ, ϕ) p r = µṙ, p θ = µr θ, pϕ = µr sin θ ϕ. (66) H = i p i q i L = 1 µ ) (p r + L + U(r). (67) L L = (µv r) = (µr ) ( θ + sin θ ϕ ) = p θ + 1 sin θ p ϕ (68) 1 L L r 14

15 v L p θ p ϕ (68) 1 U eff (r) = L + U(r). (69) µr Ĥ = + U(r) (70) µ x r θ ϕ = sin θ cos ϕ x + sin θ sin ϕ y + cos θ z, = r cos θ cos ϕ x + r cos θ sin ϕ y r sin θ z, = r sin θ sin ϕ x + r sin θ cos ϕ y (71) x y z x = sin θ cos ϕ r + 1 r cos θ cos ϕ θ 1 sin ϕ r sin θ ϕ, = sin θ sin ϕ y r + 1 r cos θ sin ϕ θ + 1 cos ϕ r sin θ ϕ, = cos z r 1 r sin θ θ. (7) = 1 r r r r + 1 r S (73) S = 1 sin θ θ sin θ θ + 1 sin θ ϕ (74) 15

16 [ Ĥ = 1 m r r r r + 1 ] r S + U(r). (75) (75) i r p r, i θ p θ, i ϕ p ϕ (76) (67) S (68) ˆL = S (77).1 L S (75) Ĥ [ { 1 µ r r r r + 1 ] } r S + U(r) ψ = Eψ (78). (78) µr / [ r r r + µ ] r (U(r) E) S ψ = 0. (79) r (θ, ϕ) ψ(r, θ, ϕ) = R(r)Y (θ, ϕ) (80) S Y (θ, ϕ) = λy (θ, ϕ), (81) [ r r r + µ ] r (U(r) E) + λ R(r) = 0. (8) (81) (8) (77) λ L = λ (83) 16

17 f(r) = rr(r) R(r) [ d + U(r) + µ dr µ λ r ] f(r) = Ef(r). (84) (69) (81) Y (θ, ϕ) λ λ (8) (84) R(r) f(r) E.3 (76) p r i r, p θ i θ, p ϕ i ϕ (85) (67) (75) [ Ĥ = µ r + 1 ( r θ + 1 )] sin + U(r) (86) θ ϕ Ĥ ψ 1 ψ ψ 1 Ĥψ = Ĥψ 1 ψ (87) 17

18 Ĥ ψ i Ĥψ i ψ 1 ψ = ψ1ψ dx = ψ1ψ r sin θdrdθdϕ (88) Ĥ ψ 1(Ĥψ )r sin θdrdθdϕ =? (Ĥψ 1) ψ r sin θdrdθdϕ (89) ψ 1 ψ r θ ϕ (89) ( ) d ψ1 ( ) dr ψ r dr =? d dr ψ 1 ψ r dr. (90) r r Ĥ. (86) 18

19 3 3.1 S Y (θ, ϕ) = λy (θ, ϕ) (91) Y (θ, ϕ) λ L = x p (9) L x = yp z zp y, L y = zp x xp z, L z = xp y yp x. (93) ˆL z = xˆp y yˆp x = i (x y y x ) (94) ˆL x, ˆL y ˆL = (ˆL z, ˆL y, ˆL z ) ˆL = i (x ) (95) ˆL ˆL [ˆL x, ˆL y ] = i ˆL z, [ˆL y, ˆL z ] = i ˆL x, [ˆL z, ˆL x ] = i ˆL y (96) ˆL x, ˆL y, ˆL z 3.1 (96) 3. (96) ˆL ˆL = i ˆL ˆL ˆL = ˆL ˆL = ˆL x + ˆL y + ˆL z (97) 19

20 ˆL ˆL [ˆL z, ˆL ] = 0. (98) ˆL ˆL z ˆL z = i ϕ, ( ˆL x = i sin ϕ cot θ cos ϕ θ ϕ ( ˆL y = i cos ϕ cot θ sin ϕ θ ϕ ), ). (99) (77) ˆL = S 3.3 ˆL z (94) ˆL x ˆL y (7) (99) 3. r µ r µ L = µ v = µ r ( θ + sin θ ϕ ) (100) v L = µrv H = 1 µr L. (101) 1 µr ˆL ψ(θ, ϕ) = Eψ(θ, ϕ) (10) (91) 0

21 3.3 m = 0 ˆl = 1 ˆL. (103) (91) ˆl Y (θ, ϕ) = λy (θ, ϕ) (104) ˆl ˆl z (104) ˆlz Y (θ, ϕ) = my (θ, ϕ). (105) m λ Y (θ, ϕ) π 0 sin θdθ π 0 dϕ Y (θ, ϕ) = 1. (106) i Y (θ, ϕ) = my (θ, ϕ), (107) ϕ ( 1 sin θ θ sin θ ) θ + m sin Y (θ, ϕ) = λy (θ, ϕ). (108) θ ϕ θ Y (θ, ϕ) Y (θ, ϕ) = Θ(θ)Φ(ϕ). (109) i d Φ(ϕ) = mφ(ϕ), (110) dϕ ( 1 d sin θ dθ sin θ d ) dθ + m sin Θ(θ) = λθ(θ). (111) θ π 0 sin θdθ Θ(θ) = 1, π 0 dϕ Φ(ϕ) = 1. (11) (110) Φ(ϕ) = 1 π e imϕ. (113) 1

22 S 1 Φ(ϕ) Φ(ϕ + π) = Φ(ϕ) (114) m Θ(θ) z = cos θ Θ(θ) = f(z) (115) d dθ = sin θ d dz (116) (111) ( d dz (1 z ) d ) dz + λ m f(z) = 0 (117) 1 z z 1 z 1 (118) f(z) Θ(θ) (11) 1 1 f(z) dz = 1. (119) m (111) m = 0 ( d dz (1 z ) d ) dz + λ f(z) = 0. (10) f(z) f(z) = a n z n (11) n=0 (10) 0 = n = n a n [{λ n(n + 1)}z n + n(n 1)z n ] [a n {λ n(n + 1)} + a n+ (n + )(n + 1)]z n (1)

23 0 a n+ = n(n + 1) λ (n + )(n + 1) a n (13) 0 a 0 a 1 n a n (13) n a n+ n + 1 n + 3 a n (14) (n + 1)a n n a even A n, a odd B n (15) n n + 1 n f(z) f(z) = n:even = A + B = A + B A n zn + n:odd n=1 B n zn + z n n + A B ( z) n + n n=1 log(1 z) + A B log(1 + z) + (16) A B 0 z = ±1 A = B = 0 0 a n 0 f(z) n = l l = 0, 1,,... (13) 0 λ λ = l(l + 1), l = 0, 1,,.... (17) l a 0 0, a 1 = 0 l a 0 = 0, a 1 0 λ = l(l + 1) (10) l P l (1) = 1 Legendre polynomials P l (z) l = 0, 1,, 3 P l (z) 3

24 P 0 (z) = 1, P 1 (z) = z, P (z) = 1 ( 1 + 3z ), P 3 (z) = 1 ( 3z + 5z3 ). (18) P 0 z P 1 z P z P3 z z : l = 0, 1,, 3 l = 50 (10) ĥf(z) = λf(z), ĥ = d dz (1 z ) d dz. (19) 1 z 1 f f ĥ (f, g) = 1 1 (f(z)) g(z)dz (130) ĥ 1 P l (z)p l (z)dz = C l δ ll (131) 1 P l (1) = 1. (13) 4

25 P l (z) z l 3.4 (18) 4 (131) (131) C l = /(l + 1) 1 P l (ζ) dζ = 1 l + 1 (133) (119) f(z) l f(z) f l (z) l + 1 f l (z) = P l (z). (134) l m Y lm (θ, ϕ) (113) (134) m = 0 Y l0 = l + 1 4π P l(cos θ). (135) P 50 (z) l P l (z) l P l (z) 3.4 ( d dz (1 z ) d ) + l(l + 1) P l (z) = 0. (136) dz ψ ( 1 ψ(r, θ, ψ) = r r r r + 1 ) r S ψ(r, θ, ϕ) = 0 (137) r = 0 ψ ψ(r, θ, ϕ) = r n f n (θ, ϕ) (138) (137) f n [ S + n(n + 1)]f n (θ, ϕ) = 0 (139) 5

26 r n f n (θ, ϕ) l = n f l (θ, ϕ) ϕ f l (θ) Y l0 (θ, ϕ) P l (cos θ) (140) ϕ (137) (139) ϕ P l (cos θ) ψ(r, θ) P n (cos θ) ψ(r, θ) ϕ (137) G(r, cos θ) = 1 x + y + (z 1) = 1 r r cos θ + 1 = 1 r rξ + 1 (141) ξ = cos θ (x, y, z) = (0, 0, 1) r < 1 G(r, ξ) = c n r n P n (ξ) (14) n=0 c n ξ = 1 (13) P n (1) = 1 G(r, 1) = 1 1 r = c n r n (143) n=0 c n = 1 G(r, ξ) = 1 r rξ + 1 = r n P n (ξ). (144) n=0 (144) P n (ξ) G(r, ξ) r 1 (k)! = 1 x ( k k!) xk (145) G(r, ξ) = k=0 k=0 (k)! ( k k!) rk (ξ r) k (146) r n k n r k (ξ r) k r n k k n k k n 6

27 (n/) k n k r n P n (ξ) = (k)! ( k k!) k C n k (ξ) k n ( 1) n k. k = 1 ( 1) n k (k)! n k!(n k)! (k n)! ξk n. (147) k k P n (ξ) = 1 ( 1) n k d n n k!(n k)! dξ n ξk k 1 d n n! = n n! dξ n k!(n k)! ξk ( 1) n k. (148) k k P n (ξ) = 1 d n n n! dξ n (ξ 1) n. (149) P l 1 1 P l (ζ)p l (ζ)dζ = C l δ ll (150) C l (150) 1 G(r, ζ) dζ = r l C l (151) 1 1 [ dζ 1 r rζ + 1 = 1 ] 1 r log(1 rζ + r ) = 1 1 r log 1 + r 1 r = l + 1 rl l=0 (15) l=1 C l = l + 1 (133) (153) 7

28 3.5 m 0 (r, ϕ) R = 1 r r r r + 1 r ϕ. (154) µ ( R + k )ψ(r, ϕ) = 0, (155) k = µe/ ϕ ψ(r, ϕ) = R km (r)e imϕ (156) (155) ( 1 d r dr r d ) dr m r + k R km (r) = 0. (157) z = kr k R km (r) = R m (z) k ( 1 d z dz z d ) dz m z + 1 R m (z) = 0. (158) m m m R m (z) m 0 m = 0 ( 1 d z dz z d ) dz + 1 R 0 (z) = 0 (159) R 0 (z) m R m (z) R m (z) = z m χ m (z) (160) χ m (z) (157) χ m + n + 1 χ m + χ m = 0. (161) z 8

29 z z ( ) 1 z χ m + m + 3 ( ) ( ) 1 1 z z χ m + z χ m = 0. (16) (161) (16) χ m/z χ m+1 χ m+1 (z) = 1 z χ m(z). (163) m χ m (z) χ 0 (z) = R 0 (z) ( ) m 1 R m (z) = z m d R 0 (z). (164) z dz R 0 J m (z) = ( z) m ( 1 z d dz ) m J 0 (z), Y m (z) = ( z) m ( 1 z d dz ) m Y 0 (z). (165) J m (z) Y m (z) m = 0, 1,, J 0 z J 1 z J z J 3 z Y 0 z Y 1 z Y z Y 3 z z z : H m (1) H m () H m (1) = J m (z) + iy m (z), H m () = J m (z) iy m (z). (166) 9

30 3.6 m 0 m 0 ϕ Y lm (θ, ϕ) = eimϕ π Θ lm (θ) = eimϕ π f lm (ξ), ξ = cos θ. (167) Θ lm (θ) f lm (ξ) ( 1 d sin θ dθ sin θ d ) dθ m sin + l(l + 1) Θ lm (θ) = 0. (168) θ m m 0 π 0 dθ sin θ Θ lm (θ) = 1 (169) (157) θ (168) (157) (157) (157) (168) (157) (168) d r sin θ, dr d dθ, k l(l + 1). (170) (160) g m Θ lm (θ) (sin θ) m g m (θ) (171) g g + (m + 1) c s g + [l(l + 1) m(m + 1)]g = 0. (17) s = sin θ c = cos θ (161) θ sin θ ( ) 1 s g + (m + 3) c ( ) ( ) 1 1 s s g + [l(l + 1) (m + 1)(m + )] s g = 0. (173) g m/ sin θ g m+1 g m+1 = 1 d sin θ dθ g m. (174) 30

31 g m g 0 (164) ( ) m 1 Θ lm (θ) sin m d θ P l (cos θ) (175) sin θ dθ ξ = cos θ P m l (ξ) = (1 ξ ) m ( d dξ ) m P l (ξ) (176) associated Legendre functions m 0 m < 0 Pl m (ξ) = P m l (ξ). (177) 5 5 P 0 3 z z P 1 3 z 5 P 3 z 10 P 3 3 z 15 5: Pl m (z) l = 3 Mathematica LegendreP[l,m,z] Y l,m (θ, ϕ) Pl m (cos θ)e imϕ Pl m P m l (ξ) dξ = 1 1 [ ] d dξ(1 ξ ) m m dξ P l(ξ) (178) m (149) l + m 1 Pl m (ξ) 1 1 [ ] d dξ = dξ(1 ξ ) m l+m l l! dξ (1 l+m ξ ) l = ( 1)l+m l l! dξ(1 ξ ) l dl+m dξ (1 l+m ξ ) m dl+m dξ (1 l+m ξ )(179) l. 31

32 m 0 l + m l + m ξ l + m 0 ξ 1 1 P m l (ξ) dξ = = = = 1 1 l l! 1 l l! dξ(1 ξ ) l dl+m dξ dξ dl+m ξm ξl l+m l+m dξ(1 ξ l (l + m)!(l)! ) (l m)! 1 l+1 l! (l + m)!(l)! l l! (l + 1)!! (l m)! (l + m)! (l + 1) (l m)! (180) m = 0 (153) f lm (ξ) l + 1 (l m)! f lm (ξ) = (l + m)! P l m (ξ). (181) ξ = cos θ l m : l = 10 m = 100 flm (z) 3.6 l m flm (θ) Y lm = ( 1) (m+ m )/ l + 1 (l m )! 4π (l + m )! P l m (cos θ)e imϕ. (18) 3

33 m ( 1) (m+ m )/ l: l = 0, 1,,... m: m = l, l 1,..., l 1, l l = 0, 1,, 3,... s, p, d, f,... l = 1 p π 0 π dθ dϕ sin θylmy l,m = δ ll δ mm. (183) 0 Y lm = ( 1) m Y l m. (184) Y lm F (θ, ϕ) = l,m c lm Y lm (θ, ϕ). (185) l = 0, 1, (θ, ϕ) n x = sin θ cos ϕ, n y = sin θ sin ϕ, n z = cos θ (186) l = 0 4πY0,0 = 1. (187) l = πY1,+1 = sin θeiϕ = (n x + in y ), (188) 4πY1,0 = 3 cos θ = 3n z, (189) 4πY1, 1 = 3 3 sin θe iϕ = (n x in y ). (190) 33

34 l = πY,+ = 8 sin θe iϕ = 8 (n x + in y ), (191) πY,+1 = sin θ cos θeiϕ = n z(n x + in y ), (19) 5 5 4πY,0 = 4 (3 cos θ 1) = 4 (n3 z n x n y), (193) πY, 1 = sin θ cos θe iϕ = n z(n x in y ), (194) πY, = 8 sin θe iϕ = 8 (n x in y ). (195) 7 7: Y lm 34

35 4 4.1 ) ( + U(r) ψ(x) = Eψ(x) (196) m e m e U(r) - U(r) = e 1 4πϵ 0 r (197) m e - µ = m e m p /(m p + m e ) m p m e 000 m e µ m e = Js, m e = kg, s = e 4πϵ 0 = Jm (198) E 0 = m es = J, a 0 = m e s = m (199) a 0 E 0 E h (196) m e c E 0 E 0 m e c = s c = α. (00) α α = 1 e = 1 c 4πϵ (01) E 0 /(m e c )

36 ψ(x) = R(r)Y lm (θ, ϕ) (0) ] [ d m e dr + U eff(r) f(r) = Ef(r) (03) U eff (r) U eff (r) = e 1 4πϵ 0 r + l(l + 1) (04) m r U eff r r 8: r 0 r 0 = l(l + 1)a 0, U eff (r 0 ) = E 0 l(l + 1) (05) ρ = r a 0, κ = E E 0. (06) E < 0 κ κ (03) ( d l(l + 1) + ) dρ ρ ρ κ f(ρ) = 0. (07) ρ ρ 36

37 ρ 0 f(ρ) ρ (07) ( d l(l + 1) dρ ρ ) f(ρ) = 0. (08) f(ρ) = c 1 ρ l+1 + c ρ l. (09) c 1 c c = 0 ρ 0 f(ρ) ρ l+1 ρ (07) ( ) d dρ κ f(ρ) = 0 (10) f(ρ) c 1e κρ + c e κρ (11) c = 0 f(ρ) (07) g + f(ρ) ρ l+1 e κρ g(ρ) (1) (l + 1) g κg [κ(l + 1) 1] g = 0 (13) ρ ρ g(ρ) g(ρ) = c k ρ k (14) k=0 0 (k + 1)[k + l + ]c k+1 = [κ(k + l + 1) 1]c k (15) c k c 0 ρ e +κρ ψ(x) ρ 0 c k = 0 c k 0 k k max κ = 1 k max + l (16)

38 k max = 0, 1,,... k max n = k max + l + 1 (17) n k max = 0, 1,,... n n l + 1 l n g(ρ) g(ρ) g nl (ρ) g nl (ρ) ρ n l 1 3 n l m n l m m l < n. (18) 3 ψ nlm (x) ρ l e κρ g nl (ρ)y lm (θ, ϕ) (19) g nl (ρ) κ = 1/n E = E 0, (n = 1,,...) (0) n n l m m l n l m l = 0, 1,..., n 1, m = l, l + 1,..., +l. (1) n n 1 (l + 1) = n () l=0 4. (19) g nl (ρ) (13) κ = 1/n g nl(ρ) (l + 1) + g ρ nl(ρ) n g nl(ρ) [ ] 1 (l + 1) 1 g nl (ρ) = 0 (3) ρ n 38

39 0 l=0(s) l=1(p) l=(d) l=3(f) n=4(n) n=3(m) n=(l) 4s 3s s 4p 3p p 4d 3d 4f -13.6eV n=1(k) 1s 9: ρ = nz/ g(ρ) = h(z) z(h nl h nl) + (l + 1)h nl + (n l 1)h nl = 0 (4) h nl = dh nl/dz a k a a k = z d dz k, a = 1 d dz. (5) a a k a a a k+1 = 1. (6) a k k (4) [ a a l 1 (n + l) ] h(z) = 0. (7) n + l h nl ψ h nl (a ) n+l ψ (8) ψ 0 k (6) a 1 a n l 1 ψ(z) = 0. (9) 39

40 ψ = z n l 1 h nl h nl (a ) n+l z n l 1 = ( 1 d dz ) n+l z n l 1. (30) L n (z) z(l n L n) + L n + nl n = 0. (31) L n (z) = 1 dn ez n! dz n (e z z n ) = 1 ( ) n d n! dz 1 z n. (3) (a a 0 n)l n = 0 L m k (z) = ( 1) m dm dz m L k+m(z). (33) h nl (z) h n,l (z) L l+1 n l 1 (z) (34) ( ) l ( ) R nl (r) = N 1/ r nl e r na r 0 L l+1 n l 1 na 0 na 0 (35) N nl 0 N nl = n4 4 r R nl(r)dr = 1 (36) (n + l)! (n l 1)! a3 0 (37) 40

41 R 10 (ρ) = a 3 0 e ρ, 1 R 0 (ρ) = e ρ/ ( ρ), a R 1 (ρ) = e ρ/ ρ, 6a 3 0 R 30 (ρ) = 81 e ρ/3 (7 18ρ + ρ ), 3a 3 0 R 31 (ρ) = 1 ( e ρ/3 4ρ ) 7 3a ρ, R 3 (ρ) = e ρ/3 ρ. (38) 81 15a 3 0 R 10 Ρ R 0 Ρ R 30 Ρ R 1 Ρ R 31 Ρ R 3 Ρ : R nl (ρ) 41

42 Mathematica L n (z) = LaguerreL[n,z], L m n (z) = LaguerreL[n,m,z] (39) X nl = 1 = ρ nl (l + 1)n, 3 1 = 1 ρ n, nl 1 nl = 1, 0 ρ nl = 3n l(l + 1), Xr R nl(r)dr (40) ρ nl = 5n4 (3l + 3l 1)n. (41) 1 = 1 (36) H = H 0 + H (4) H 0 H H 0 n, l, m E n n, l, m n, l, m n, l, m n 1, l 1, m 1 n, l, m E = m e 4 (4πϵ 0 ) 4 ( 1 n 1 1 n ) (43)

43 1 = E ( 1 λ (n1,n ) hc = R 1 ) n 1 n (44) R = me4 8h 3 cϵ 0 = 1 E 0 hc m 1 = 1 91nm. (45) E 0 / = 13.6eV 91nm (43) n 1 n n 1 < n λ (n1,n ) n 1 = 1,, 3, 4 L α : λ (1,) = 1nm, L β : λ (1,3) = 103nm, L γ : λ (1,4) = 97nm,... (46) H α : λ (,3) = 656nm, H β : λ (,4) = 486nm, H γ : λ (,5) = 434nm,... (47) P α : λ (3,4) = 1875nm, P β : λ (3,5) = 18nm, P γ : λ (3,6) = 1094nm,... (48) 656nm 486nm 434nm 410nm 11: e Ze e 4 Z e 4 Z 43

44 5 5.1 n, l, m Z Ze Z n, l, m n l n =, l = 1 p l = 0, 1,, 3,... s, p, d, f, s s p s p m = 0, ± s, 3p, 3d 11-3s 3p 3d E 3s < E 3p < E 3d. (49) 0 44

45 3s Na : (1s) (s) (p) 6 (3s) 1 (50) 3p 3s 590nm D 1 D 1 D 1 : nm, D : nm. (51) D D 1 1: D D D 1 D D 1 D 1 45

46 5. H int H int B ext C F = qẋ B ext (5) M = x F = qx (ẋ B ext ) (53) 3 M = q x (ẋ B ext )dt = q x (dx B ext ) = q (x dx) B ext (54) T T T C T B µ M = µ B (55) (54) (55) µ = q (x dx) = q ds (56) T T C ds L = 1 mx ẋdt = m x dx = m ds (57) T C T C T µ = q m L (58) 3 C x (dx B ext) 1 C (x dx) B ext = C [(x B ext)dx (x dx)b ext 1 (x B ext)dx+ 1 (dx B ext)x] = C d[ 1 (x B ext)x 1 (x x)b ext] = C

47 H int = µ B ext = q m B ext L. (59) (19) B ext A ext A ext = 1 B ext x. (60) (19) 0 p /(m) 1 m (p qa ext) = 1 m p q m A ext p + O(A ext) (61) A ext H int = q m A ext p = q m (B ext x) p = q m B ext L (6) (59) e m e µ = µ B l, µ B = e m e = ev/t (63) µ B +e m p µ p = µ N l, µ N = e m p = ev/t (64) µ N 1 : 000 g g µ e = gµ B l, µ p = gµ N l (65) 47

48 g 5.6 D g D 3p 3s z Ĥ int = µ B B extˆlz (66) 3s ˆl z m = 0 (Ĥ0 + Ĥint) 3, 0, 0 = E 3s 3, 0, 0 (67) Ĥ0 3p m = 0, ±1 (Ĥ0 + Ĥint) 3, 1, +1 = (E 3s + E) 3, 1, +1, (Ĥ0 + Ĥint) 3, 1, 0 = E 3s 3, 1, 0, (Ĥ0 + Ĥint) 3, 1, 1 = (E 3s E) 3, 1, 1. (68) 3 E = µ B B ext (69) l l + 1 3s 3p 3 3 D 1 D 1 D 1 D E 48

49 N S N +q m S q m S N d µ = q m d (70) N S z db F z = q m B(z N ) q m B(z S ) = q m d z dz = µ db z dz (71) z N z S N S z µ z z l z m = +l m = l l + 1 µ z l + 1 l Ag : (1s) (s) (p) 6 (3s) (3p) 6 (3d) 10 (4s) (4p) 6 (4d) 10 (5s) 1 (7) 5s l l + 1 l = 1/

50 / ˆL = ˆx ˆp Ŝ = (Ŝx, Ŝy, Ŝz) [Ŝx, Ŝy] = i Ŝz, [Ŝy, Ŝz] = i Ŝx, [Ŝz, Ŝx] = i Ŝy. (73) Ĵ 1 Ĵ Ĵ [Ĵx, Ĵy] = iĵz, [Ĵy, Ĵz] = iĵx, [Ĵz, Ĵx] = iĵy. (74) Ĵ x, Ĵ y, Ĵ z (Ĵx) = Ĵx, (Ĵy) = Ĵy, (Ĵz) = Ĵz. (75) J ± Ĵ ± = Ĵx ± iĵy. (76) (Ĵ+) = Ĵ, (Ĵ ) = Ĵ+. (77) (74) [Ĵ, Ĵz] = 0, (78) [Ĵ, Ĵ±] = 0, (79) [Ĵz, Ĵ±] = ±Ĵ±, (80) Ĵ Ĵ ± = Ĵ Ĵz(Ĵz ± 1). (81) 50

51 (78) Ĵ Ĵz λ, m Ĵ λ = λ λ, m, Ĵ z λ = m λ, m. (8) (80) Ĵ± λ, m 0 Ĵ Ĵz λ m ± 1 0 λ, m (81) λ, m ± 1 Ĵ± λ, m. (83) λ, m Ĵ Ĵ± λ, m = (λ m(m ± 1)) λ, m λ, m (84) Ĵ± λ, m λ m(m ± 1) 0. (85) Ĵ± λ, m = 0 (85) m λ, m Ĵ+ Ĵ 0 m m max m min Ĵ+ λ, m max = Ĵ λ, m min = 0 (85) λ = m max (m max + 1) = m min (m min 1). (86) m min m max m min = m max. (87) m min = m max +1 (86) m min m max m max j m max = j, m max = j, λ = j(j + 1). (88) λ, m min λ, m max λ, m J ± m m min m max j = m max m min j j m j = 0, 1, 1, 3,..., m = j, j + 1,..., j 1, j. (89) 51

52 ψ(r, θ, ϕ) ϕ ψ(r, θ, ϕ + π) = ψ(r, θ, ϕ) (90) m j Ĵ± m Ĵ Ĵz j, m Ĵ j, m = j(j + 1) j, m, Ĵ z j, m = m j, m. (91) 6.1 j m j m Ĵ+ Ĵz m J + j, m = c j,m j, m + 1 (9) c j,m j, m j, m = 1 (93) c j,m c m,j (9) c j,m j, m + 1 j, m + 1 = j, m Ĵ Ĵ+ j, m = (j m)(j + m + 1) j, m j, m (94) (81) (93) c j,m c j,m = (j m)(j + m + 1). (95) Ĵ + j, m = (j m)(j + m + 1) j, m + 1 (96) 5

53 j, m + 1 j, m Ĵ j, m + 1 = (j m)(j + m + 1) (97) Ĵ j, m + 1 = (j m)(j + m + 1) j, m. (98) Ĵ z j, m = m j, m, Ĵ + j, m = (j m)(j + m + 1) j, m + 1, Ĵ j, m = (j m + 1)(j + m) j, m 1. (99) 0 j, m Ĵz j, m = m, j, m + 1 Ĵ+ j, m = (j m)(j + m + 1), j, m 1 Ĵ j, m = (j + m)(j m + 1). (300) j 0 j (j + 1) (j + 1) j Ĵ z = j 1... j (301) Ĵ + = Ĵ = 0 1 j 0 (j 1) 0 1 j 0 (j 1) j 1 0 j 1 0 (30) (303) 53

54 0 Ĵx Ĵy Ĵ x = 1 (Ĵ+ + Ĵ ) 0 1 j = 1 j (j 1) 0 (j 1) 0 Ĵ y = 1 i (Ĵ+ Ĵ ) 0 i 1 j i 1 j 0 i = (j 1) i 0 (j 1)... j 1 j 1 0,... i j 1 i j 1 0 (304) 6.1 Ĵi (74) Ĵ (74) (99) ˆl l m l, m Y l,m (θ, ϕ) (99) ˆl± = ˆl x ± iˆl y ( ˆl± = e ±iϕ ± θ + i cot θ ) ϕ (305) m ˆl ± m 0 Y lm ˆl + (18) m 0 Y lm (θ, ϕ) = ( 1) m l + 1 (l m)! 4π (l + m)! P l m (cos θ)e imϕ. (306) 54

55 (305) ˆl + ( ) ˆl+ Y lm (θ, ϕ) = ( 1) m l + 1 (l m)! d 4π (l + m)! dθ m cot θ Pl m (cos θ)e i(m+1)ϕ. (307) (176) ( ) d dθ m cot θ Pl m (cos θ) = sin m θ ( 1 sin θ Pl m (cos θ) = sin m θ d ( 1 dθ sin m θ P m ( = sin m+1 1 d sin θ dθ ) m d P l (cos θ) (308) dθ ) l (cos θ) ) m+1 P l (cos θ) = P m+1 l (cos θ) (309) ˆl+ Y lm (θ, ϕ) = ( 1) m+1 l + 1 (l m)! 4π (l + m)! P m+1 l (cos θ)e i(m+1)ϕ = (l m)(l + m + 1)Y l,m+1 (θ, ϕ) (310) (99) (18) ( 1) (m+ m )/ l, m (9) c j,m 6.3 ϵ ( ψ ψ = ) i ϵ ˆL ψ (311) ψ = x x ˆx x = x x ( ψ = ) i ϵ ˆL x (31) 55

56 ( ˆx x = ˆx ) i ϵ ˆL x ( = ) [ ( i ϵ ˆL ˆx x + ˆx, )] i ϵ ˆL x = (x + ϵ x) x (313) [ˆx, ϵ ˆL] = i ϵ ˆx (314) (313) ψ x = x + ϵ x (315) ϵ n ϵ ϵ ϵ = ϵn n x ϵ x 13 (311) n x ε x 13: x ψ = dx x ψ(x) (316) ψ = (1 + δ) ψ = dx x + ϵ x ψ(x) (317) x = x ϵ x x x ψ = dx x ψ(x ϵ x) (318) ψ (x) = ψ(x ϵ x). (319) 56

57 (31) ( ψ (x) = ) i ϵ ˆL ψ(x) = ψ(x) ϵ (x )ψ(x) = ψ(x) (ϵ x) ψ(x) = ψ(x ϵ x) (30) Ô δô = 1 i [Ô, ϵ ˆL] (31) Ô = ˆx (314) δ ˆx = 1 i [ˆx, ϵ ˆL] = ϵ ˆx (3) (315) δ ˆp = 1 i [ ˆp, ϵ ˆL] = ϵ ˆx, δ ˆL = 1 i [ ˆL, ϵ ˆL] = ϵ ˆL (33) ϵ O(ϵ) θ θ N = θ/ϵ n θ Û(n, θ) ( Û(n, θ) = lim 1 + θ ) N ( ) 1 N N i n ˆL θ = exp i n ˆL (34) exp  = 1 +  + 1! + 1 3!Â3 + (35)   = diag{a i} exp  = diag{ea i } ψ = Û(n, θ) ψ, (36) 57

58 Ô = Û 1 (n, θ)ôû(n, θ) (37) 6. U(x) Ĥ (31) δĥ = 0 s Ŝ s(s + 1) s x z s z x, s z, s z = s, s + 1,..., s. (38) ψ(x, s z ) = x, s z ψ (39) x s z ψ sz (x) s z s + 1 ψ(x) = ψ s (x) ψ s 1 (x). ψ s (x) (330) ˆL i s z ˆL i ψ s (x) ˆL i ψ s 1 (x) ˆL i ψ(x) = (331). ˆL i ψ s (x) s z (Ŝi) s,s (Ŝi) s,s 1 (Ŝi) s, s (Ŝi) s 1,s (Ŝi) s 1,s 1 (Ŝi) s 1, s Ŝ i ψ(x) = (Ŝi) s,s (Ŝi) s,s 1 (Ŝi) s, s ψ s (x) ψ s 1 (x). ψ s (x) (33) x s z [ˆL i, Ŝj] = 0. (333) 58

59 ˆL i Ŝi Ĵ = ˆL + Ŝ. (334) (34) ˆL Ĵ ( ) θ Û(n, θ) = exp i n Ĵ = Ûorb(n, θ)ûspin(n, θ) (335) ( ) ( ) θ Û orb (n, θ) = exp i n ˆL θ, Û spin (n, θ) = exp i n Ŝ (336) ( ψ (x) = ) i ϵ Ĵ ψ(x) = (1 + ϵ R (s) )ψ(x ϵ x) (337) s (330) R (s) = (R x (s), R y (s), R z (s) ) Ŝ/i s R x (s) R y (s) R z (s) (s + 1) (s + 1) s = 1 i R (1) x = i 0 i 0 i 0, R (1) y = , R (1) z = i i. (338) 6.4 s = 1 ψ(x) 3 E(x) B(x) A(x) 1 V (x) = V x (x) V y (x) V z (x) 59 (339)

60 (319) x ϵ x ϵ V V (x) = V (x ϵ x) + ϵ V (x) = (1 + ϵ R (vec) )V (x ϵ x) (340) R (vec) 3 (R (vec) k ) ij = ϵ ikj R (vec) x = , R (vec) y = , R (vec) z = (341) (340) 1 (337) R (1) R (vec) U U =. R (vec) = U 1 R (1) U (34) 1 i i 0 (343) UV (x) 1 ψ (1) (x) U 1 ψ (1) (x) 1 1 = 1 (344)

61 0 0 = 0 (345) = (346) 6.5 1/ 1/ x z s z x, s z ψ ψ sz (x) = x, s z ψ (347) s z +1/ 1/ ψ(x) = ( ψ (x) ψ (x) ) (348) Ĵ i = σ i (349) σ i ( ) ( ) ( i 1 0 σ x =, σ y =, σ z = 1 0 i ). (350) n θ ( ) θ Û(n, θ) = exp i n Ĵ = exp ( iθ ) n σ (351) 61

62 (n σ) = 1 Û(n, θ) = k=0 ( 1 iθ ) k k! n σ = 1 cos θ in σ sin θ. (35) θ ( ) cos θ i sin θ Û(e x, θ) = i sin θ cos θ, ( ) cos θ sin θ Û(e y, θ) = sin θ cos θ, ( ) e iθ 0 Û(e z, θ) =. (353) 0 e iθ 6.3 ψ x ψ Ŝz ( ) Ŝ z ψ = + ψ, ψ = 1. (354) 0 y π/ Ŝx 6.4 ψ π Û(n, π)ψ = ψ (355) 6

63 n = 1,, 3,... l = 0, 1,..., n 1 m = l, l + 1,..., l s z = 1, 1 n p (mc ) + (cp) = mc + 1 m p 1 8m 3 c (p ) + (356) E = 1 8m 3 c (p ) (357) n, j, m ( 1 4m ( ˆp ) = ˆK 1 = (Ĥ Û) = E0 4n n ( 1 = E0 4n 1 ) 4 n + 4 n 3 (l + 1) ( = E0 1 n n 4 l ) 4 63 ) ρ ρ (358)

64 E 0 (199) ρ = r/a 0 (41) E ( E = α n n 4 l ) E 0. (359) 8 l l α α 7. ±q m q m +q m d µ = q m µ 0 d (360) E = ϵ 1 0 D v B q v f = qv B q m f = q m v D (361) ±q m M = d f = d (q m v D) = µ 0 µ (v D) (36) D = e/(4πr 3 )x M = µ 0e µ (v x) = µ 0e µ l 4πr3 4πr 3 e (363) m e l e = m e x v (363) µ B = µ 0e 4πr 3 m e l e = µ 0µ B πr 3 l e (364) 64

65 µ B µ B = e m e (365) (361) (361) 14 B + 14: r ω l e ω = 1 m e r l e (366) I = e ω π = e πm e r l e (367) B = µ 0 I/(r) (364) (364) s e µ e = g e µ B s e, (368) g e g g e =.003 g e = (364) (368) E = B µ e = g e µ 0 µ B πr 3 l e s e (369) LS 65

66 LS LS (369) c E = cl e s e (370) 3s 3p 3s 589nm D 3p 3s.1eV LS (370) 3s 3p (370) l e s e 1/ s e 1/ l e l 3s l = 0 l e = 0 (370) 0 3p l = 1 l e 1 l e s e E = +c/ E = c/ 15 3p 3p LS coupling 3p c.1ev 3s D 3s D1 15: LS 3s D 3p 0.00eV (369) c = 10 4 ev 66

67 7.3 1/ µ p = g p µ N s p (371) µ N µ N = e m p (37) m p /m e = 1836 µ N = µ B /1836 g g p = (373) g e = 1s 1s l = s p e 1s p e 1cm 16: 1s 1cm 1cm s, s z s = 1/ s z s z = ±1/ = 1, 1, = 1, 1. (374) 67

68 - x z s (p) z z s (e) z x, s (p) z p, s (e) z e (375) x p e (376) x, (377) 4 ψ (x) x, ψ ψ (x) ψ(x) = ψ (x) = x, ψ x, ψ ψ (x) x, ψ (378) Ĥ 0 = m e 1 4πϵ 0 r (379) 1s Ĥ 0 ψ s (p) z s (e) z (x) = E 1sψ s (p) z s (e) z (x) (380) 4 1s ψ 1s (x) = 1 πa 3 0 e r a 0 (381) 4 u u ψ(x) = ψ 1s (x)u, u = u u (38) x ψ 1s (x) u 68

69 (38) (38) 1s s z (p) s (e) z = 1s, s (p) z s (e) z (383) 1s s (p) z s (e) z (384) ĤSS µ p B = µ 0 4πr 3 [3(n µ p)n µ p ] + µ 0 3 µ pδ(x) (385) n = x/ x B = 0 x 0 0 µ e H SS = µ e B = µ 0 4πr 3 [3(n µ p)(n µ e ) (µ p µ e )] µ 0 3 (µ p µ e )δ(x) (386) [ 1 Ĥ SS = µ 0 g p µ N g e µ B 4πr [3(n ŝ p)(n s 3 e ) (s p s e )] + ] 3 (ŝ p ŝ e )δ(x) (387) (38) ψ 1 (x) = ψ 1s (x)u 1, ψ (x) = ψ 1s (x)u (388) ψ 1 ĤSS ψ = ψ 1(x)ĤSSψ (x)d 3 x (389) (387) r 3 n 0 ψ 1s (x) x δ(x) ψ 1 ĤSS ψ = ζu 1(ŝ p ŝ e )u (390) 69

70 ζ ζ = µ 0 g p µ N g e µ B ψ 0 (0) µ 0 3 = g pg e 3 = g pg e 3 m e 4µ 0 µ B m p 4πa 3 0 m e α E 0 m p = E 0 (391) (389) 1s u Ĥ SS = ζ(ŝ p ŝ e ) (39) ŝ p ŝ e 1/ H SS = ζ/4 H SS = ζ/4 Ĥ SS = ζ 4, Ĥ SS = ζ 4, Ĥ SS = ζ 4, Ĥ SS = ζ 4. (393) ĤSS 1/ ( ) ( ) ( ) i 1 0 ˆσ x = 1, ˆσ y = i, ˆσ z = (394) ŝ x = 1, ŝ y = i, ŝ z = 1, ŝ x = 1, ŝ y = i, ŝ z = 1. (395) ŝ (p) ŝ (e) ŝ (e) x x, = ŝ (e) x ( x p e ) = x p (ŝ (e) x e ) = x p ( 1 e) = 1 x, (396) 70

71 ĤSS ŝ (p) x ŝ (e) x = 1 4, ŝ(p) y ŝ (e) y = 1 4, ŝ(p) z ŝ (e) z = 1 (397) 4 Ĥ SS = ζ (398) 4 Ĥ SS = ζ 4 + ζ, Ĥ SS = ζ 4 + ζ Ĥ SS = ζ. (399) 4 ĤSS + Ĥ SS = ζ +, 4 Ĥ SS = 3ζ 4. (400) 4 1s H SS = 3ζ/4 H SS = ζ/4 1s 0 1s 1 ζ 1s 1 1s 0 λ = hc ζ 1cm = 1cm. (401) 7.4 ŝ (p) z ŝ (e) z Ĥ = Ĥ0 + ĤSS (40) ĤSS [ŝ (p) i, ĤSS] = iζ(ŝ (p) ŝ (e) ) i, [ŝ (e) i, ĤSS] = iζ(ŝ (p) ŝ (e) ) i (403) 71

72 0 ŝ = ŝ (p) + ŝ (e) (404) LS LS Ĵ = ˆl + ŝ (p) + s (e) Ĥ SS = ζ(s (p) s (e) ) = ζ (s s (p) s (e) ) = ζ ( s 3 ) (405) ĤSS 4 ĤSS 1s 1,1 = 1s,, 1s 1,0 = 1s 1, 1 = 1s,, 1s 0,0 = 1s, + 1s,, 1s, 1s,. (406) ŝ ŝ z ŝ 1s 1,1 = 1s 1,1, ŝ z 1s 1,1 = 1s 1,1, ŝ 1s 1,0 = 1s 1,0, ŝ z 1s 1,0 = 0, ŝ 1s 1, 1 = 1s 1, 1, ŝ z 1s 1, 1 = 1s 1, 1, ŝ 1s 0,0 = 0, ŝ z 1s 0,0 = 0. (407) 1s 1,m 1 1, m m = 0, ±1 1s 0,0 0 0, 0 s (p) s (e) s j j + 1 [j] 1/ [ 1] [ 1 ] = [1] + [0] (408) 7

73 [j] j + 1 j 1 j [j 1 ] [j ] = [?] + [?] + (409) [j 1 ] [j ] Ĵ (1) Ĵ () Ĵ = Ĵ (1) + Ĵ () [j 1 ] [j ] (j 1 + 1)(j + 1) (409) Ĵz m = m 1 + m j 1 = 3 j = 35 m 1 -m 17 m = m 1 + m m m= 1 m= m= 3 m=5 m=4 m=3 m1 m= 4 m= m= 5 m=0 m=1 17: j 1 = 3 j = (j 1 + 1)(j + 1) = 35 m m 18 [j] (409) [1] [] [3] [4] [5] m=m1+m 18: 35 [j] [j] m = j m = +j j + 1 [3] [] [3] [] = [5] + [4] + [3] + [] + [1] (410) 73

74 [j 1 ] [j ] = j 1 +j j= j 1 j 7.1 (411) j [j]. (411) j 1 j j j 1 + j. (41) j 1, j, j (41) j j j1 19: 7.5 (411) j, m = m 1,m j 1, m 1 j, m C j1 m 1 j m jm (413) C j1 m 1 j m jm 7. (406) CG 0 z j 1 j j j 1 + j. (414) m 1 + m = m. (415) 74

75 (413) (413) j, m = j 1, m 1 ; j, m = j 1, m 1 j, m (416) m 1,m j 1, m 1 ; j, m C j1 m 1 j m jm. (417) (416) j 1, m 1 ; j, m CG C j1 m 1 j m jm = j 1, m 1 ; j, m j, m. (418) CG CG [1] [ 1 ] (411) 0 [1] [ 1 ] = [ 3 ] + [ 1 ] (419) 0 (1) 6 c a m e f b d A m1 (1) () C E = J + B D [1/] F [3/] m=m1+m : [1] [ 1 ] = [ 3 ] + [ 1 ] a : 1, 1 1, 1, b : 1, 0 1, 1, c : 1, 1 1, 1, d : 1, 1 1, 1, e : 1, 0 1, 1, f : 1, 1 1, 1. (40) 75

76 () A : 3, 3 B : 3, 1 C : 1, 1 D : 3, 1 E : 1, 1 F : 3, 3. (41) (41) (40) A A m = m 1 + m 3/ (40) A : 3, 3 = 1, 1 1, 1. (4) m = 1/ (1) () b c B C j, m (96) j, m + 1 = Ĵ + (j m)(j + m + 1) j, m (43) A (4) B B : 3, 1 = 1 3 Ĵ + 3, 3 = 1 3 (Ĵ (1) + + Ĵ () + ) 1, 1 1, 1 = 1 [ ] (Ĵ (1) + 1, 1 ) 1, 1 () + 1, 1 (Ĵ 3 + 1, 1 ) = 1 [ ] 1, 0 1, 1 + 1, 1 1, 1 3 = 3 1, 0 1, , 1 1, 1 (44) 3 76

77 Ĵ+ D F D : 3, 1 = 1 Ĵ+ 3, 1 = 1 3 1, 1 1, , 0 1, 1, F : 3, 3 = 1 Ĵ + 3, 1 = 1, 1 1, 1. (45) 3 (40) m = 1/ m = 1/ C E C C : 1, 1 = 1 3 1, 0 1, 1 3 1, 1 1, 1 (46) C E Ĵ+ E : 1, 1 = Ĵ+ 1, 1 = 3 1, 1 1, , 0 1, 1 (47) D 3, 1 [1] [ 1 ] = [ 3 ] + [ 1 ] (413) CG 7.6 D LS D 1 D D 3p 3s D λ D λ D = λ D = 589nm LS hc E 3p E 3s (48) E 3p E 3s =.10eV (49) Ĥ LS = ξ(ˆl ŝ) (430) ζ 3p 3s ξ 3p ξ 3s 77

78 3s l = 0 ˆl = 0 LS [0] [ 1 ] = [ 1 ] j = 1/ 3s 1/ E 3s1/ = E 3s. (431) 3p [1] [ 1] = [ 3] + [ 1] j = 3/ 3p 3/ 3p 1/ LS Ĥ LS = ξ 3p (Ĵ l ŝ ) = ξ ( 3p j(j + 1) 11 ) (43) 4 E 3p3/ = E 3p + ξ 3p, E 3p 1/ = E 3p ξ 3p. (433) λ D1 = hc E 3p E 3s + 1ξ, λ D = 3p E 3p E 3s ξ 3p ξ 3p hc E 3p E 3s ξ 3p. (434) λ D λ D1 = 3 (435) λ D E 3p E 3s (49) ξ 3p ξ 3p = eV (436) D 1 D Ĥ B = B (µ L + µ S ) = µ B B (ˆl + ŝ) (437) µ L µ S g g e = µ L = ˆµ Bˆl, µs = ˆµ B ŝ. (438) LS B ξ 3p µ B 4.5T. (439) 78

79 3s 1/ 3p 3/ 3p 1/ LS z Ĥ B = µ B B (ˆl z + ŝ z ) (440) 3p /3 3p 1/ (41) 6 (41) j, m 3p j,m 3p 3 4 3p 3, 3 = 1, 1 1, 1, 3p 3, 1 3p 3, 1 = = 1 1, 1 1, , 0 1, 1, 3 1, 0 1, , 1 1, 1, 3 3p 3, 3 = 1, 1 1, 1. (441) ĤB 4 4 = p 1 3p 1, 1 = 3p 3, 3 ĤB 3p 3, 3 = µ B B, 3p 3, 1 ĤB 3p 3, 1 = 3 µ BB, 3p 3, 1 ĤB 3p 3, 1 = 3 µ BB, 3p 3, 3 ĤB 3p 3, 3 = µ B B. (44) 3 1, 1 1, , 0 1, 1, 3p 1, 1 = 1 1, 0 1, , 1 1, 1. (443) 0 3p 1, 1 ĤB 3p 1, 1 = 1 3 µ BB, 3p 1, 1 ĤB 3p 1, 1 = 1 3 µ BB. (444) 79

80 3s 1/ l = 0 (440) 3s 1, 1 ĤB 3p 1, 1 = µ B B, 3s 1, 1 ĤB 3p 1, 1 = µ B B. (445) 3 3p 3/, 3p 1/, 3s 1/ 1 3p 3/ E 3p3/ +µ B B E 3p3/ +(3/)µ B B E 3p3/ (3/)µ B B E 3p3/ µ B B 3p 1/ E 3p1/ +(1/3)µ B B E 3p1/ (1/3)µ B B 3s 1/ E 3s1/ +µ B B E 3s1/ µ B B 1: 3p 3/ 3s 1/ D s 1, 1 3s 1, 1 3p 3, 3 3p 3, 3 6 E 3p3/ E 3s1/ + E, E = 5 3 µ BB, µ B B, 1 3 µ BB, 1 3 µ BB, µ B B, 5 3 µ BB. (446) D1 6 (/3)µ B B 3p 1/ 3s 1/ D 4 E 3p1/ E 3s1/ + E, E = 4 3 µ BB, 3 µ BB, 3 µ BB, 4 3 µ BB. (447) 80

81 D 4 (/3)µ B (4/3)µ B (/3)µ B E A A E B B E A E B E = E 0 sin(ωt) (448) ϕ(x) = E x = E 0 x sin(ωt) (449) Ĥ I = eϕ(x) = ee x = E d (450) d = qx (electric dipole moment) electric dipole transition M = B ĤI A (451) 0 A B 0 ĤI A B s z M = 0 81

82 s z 0 3p 3, 3 3s 1, 1 3p 3, 3 3s 1, 1 A B A = l A, m A, B = l B, m B (45) n x ϵ B ĤI A Yl B,m B (ϵ n)y la,m A dω (453) ( )dω ( )dω = π 0 π dθ dϕ sin θ( ). (454) 0 ϵ n θ ϕ ϵ n = ϵ z cos θ + 1 ϵ +e iϕ sin θ + 1 ϵ e iϕ sin θ [ 4π = ϵ 3 Y 10 1 ϵ + Y ] ϵ Y 11 3 (455) ϵ ± = ϵ 1 ± iϵ Y l B,m B Y 1m Y la,m A dω (456) m = 0, ±1 0 1, m l A, m A l B, m B 0 (414) (415) l A 1 l B l A + 1, m B = m A + m, (m = 0, ±1) (457) electric dipole transition l = 0, ±1, m = 0, ±1, s z = 0. (458) 8

83 b 0 π b r ϕ θ : φ = 0 (r, φ) E = m (ṙ + r φ ) + U(r), L = mr φ. (459) b v 0 L = mbv 0, E = m v 0. (460) φ L φ = L mr (461) φ E = m ) (ṙ + L + U(r) (46) m r 83

84 ṙ = ± (E U(r)) L (463) m m r 0 r r = r 0 (463) 0 (461) (463) dφ/dr r dφ dr = φ ṙ = ± m L mr = ± L (E U(r)) m r b. (464) r 1 U(r) b E r (460) r r 0 φ φ = π θ = r 0 dφ dr dr = r 0 bdr. (465) r 1 U(r) b E r π 8. (465) θ b 3: 84

85 r 1 r r 1 + r b < r 1 + r 0 r 1 + r 4 r 1 +r 4: r 1 + r σ σ = π(r 1 + r ) (466) n nσ nσ 1 nσ I 85

86 N N = Inσ (467) 3 I n n σ θ b (465) b b + db θ θ + dθ θ θ + dθ b b + db dσ = πb db (468) N(θ)dΩ = Indσ (469) dω θ θ + dθ dω = π sin θdθ (470) N(θ) θ dσ dω = b db sin θ dθ (471) dω dσ 5: 4 b = R cos θ (47) 86

87 R = r 1 + r 8.1 (465) (47) db = R sin θdθ (473) dσ dω = b db sin θ dθ = b R sin θ sin θ = R cos θ sin θ sin θ = R 4. (474) θ dσ R σ = dω = dω 4 4π = πr. (475) 0 0 θ = 0 θ U(r) = qq 1 4πϵ 0 r (476) k E = m k. (477) m a 0 = 4πϵ 0 m qq (478) 87

88 qq = e m = m e U(r) E = ± 1 a 0 k r (479) (465) r u = a 0 /r φ u = du (480) 0 ( a 0 b ) + 1 (u ± 1 ) (b k ) b k 6 AB = a 0 /b E D du D B dϕ/ ϕ/ a0/b (1/b k )+u 1/b k A C 6: BC = 1/(b k ) ABC ADC DC = 1/(b k ) + u AD D D CD du DAD dφ/ u = 0 AD 0 u φ / EAB ACB φ = tan 1 a 0 /b 1/(b k ) = tan 1 (a 0 bk ) (481) θ = π φ = tan 1 1 a 0 bk (48) 88

89 (48) dσ dω = b sin θ db dθ = b sin θ dθ a 0 b k cos θ tan θ = 1 a 0 bk (483) 1 cos θ = db a 0 b k (484) ( = 1/a 0k 4 1 sin θ cos θ tan θ ) 3 = 1 4a 0k 4 sin 4 θ (485) U(x) U(x) ) ( m + U(x) ψ(x) = Eψ(x) (486) k E = m k (487) U(x) ψ = e ik x (488) k k = k g(x) ψ(x) = e ik x + g(x) (489) g(x) (486) ( + m ) U(x) (e ik x + g(x)) = k (e ik x + g(x)) (490) 89

90 U(x) g(x) 1 U = g = 0 1 ( + k )g(x) = m U(x)eik x. (491) g(x) e ikr /r r ( ) 1 ( + k ) r eikr (49) = 1 r d dr r d dr (493) r > 0 ( ) 1 ( + k ) r eikr = 0 (494) r = 0 r r = 0 ( ) ( ) ( 1 1 ikr dv ( + k ) r eikr = ds r eikr = 4πr eikr 1r ) eikr (495) r 0 4π (49) δ 4π G(x) = 1 4πr eikr (496) ( + k )G(x) = δ(x). (497) δ f(x) ( + k )g(x) = f(x) (498) g(x) = dx G(x x )f(x ) (499) 90

91 ( + k ) dx G(x x )f(x ) = dx δ(x x )f(x ) = f(x) (500) (491) g(x) = dx G(x x ) m U(x )e ik x = m 4π dx eik x x x x U(x ) (501) r x U(x ) r x 0 x x x x = r r cos θ. (50) θ x x 7 (501) x x-x θ x 7: x x g(x) = m e ikr π r dx e ikr cos θ U(x )e ik x (503) x k k (503) g(x) = m e ikr π r U(x) Ũ(k) = dx e ik x U(x )e ik x = m e ikr π r dx U(x )e i(k k ) r (504) dxu(x)e ik r (505) 91

92 g(x) = m e ikr π r Ũ(k k) (506) dω r r + dr g(x) r dωdr (507) dσ dr dσdr dσ dω = r g(x) = ( m π ) Ũ(k k) (508) 1 ψ(x) = g 0 (x) + g 1 (x) + g (x) + g 3 (x) +. (509) g 0 (x) g 0 (x) = e ik x. (510) g 1 (x) U(x) 1 g(x) g n (x) n =, 3, 4,... n (509) (486) U(x) ( + k )g n+1 (x) = m U(x)g n(x) (511) n = 0 (491) n g n (x) g n+1 (x) (491) g n+1 (x) = m G(x x )U(x )g n (x )dx. (51) ( ) n m g n (x) = dx n dx 1 G(x x n )U(x n )G(x n x n 1 ) G(x x 1 )U(x 1 )e ik x (513) n 8 9

93 U(x ) U(x 1 ) U(x 3 ) 8: 9. U(x) = a r (514) U(x) = a r e br (515) b 0 (515) Ũ(k) = dx a r e br e ik x (516) k k ϕ r Ũ(k) = 0 = πa = πa r dr 0 π 0 π dr 0 π 0 sin θdθ π 0 dϕ a r e br ikr cos θ e (b+ik cos θ)r sin θdθre sin θdθ (b + ik cos θ) (517) b = 0 r b t = cos θ Ũ(k) = πa 1 1 dt (b + ikt) = 4πa (518) b + k 93

94 (515) ( b )U(x) = 4πaδ(x) (519) (497) k = ib dk U(x) = Ũ(k)eik x dk, δ(x) = eik x (50) (π) 3 (π) 3 = ( b dk ) Ũ(k)eik x dk = (π) 3 (π) 3 Ũ(k)( k b )e ik x, dk = 4πa eik x (51) (π) 3 (518) Ũ(k) b = 0 (506) (508) g(x) = m e ikr 4πa π r k k, dσ ( m ) dω = 4πa π k k (5) a a = q 1q 4πϵ 0 (53) k k θ k k = k sin θ (54) dσ dω = ( mq1 q 8π k ϵ 0 ) 1 sin 4 θ ( ) q1 q 1 = 8πmv ϵ 0 sin 4 θ (55) 94

95 9.3 k k x k = kn, x = rn. (56) ψ(rn ) = e ikrn n + 1 r f(n, n )e ikr (57) (508) dσ dω = f(n, n ). (58) dω n F in (n) n ψ(rn ) = dωf in (n)e ikrn n + eikr r dωf in (n)f(n, n ) (59) F in r n ±n n ±n θ < 1 kr (530) F in (n) k/r n = ±n dωf in (n)e ikrn n = F in (n ) = F in (n ) n n dωe ikrn n n n dωe ikrn n + F in ( n ) + F in ( n ) n n dωe ikrn n n n dωe ikrn n (531) n n n = n n 1 π sin θdθe ikr cos θ = π dze ikrz = π ikr eikr (53) 0 95

96 0 ψ(rn ) = π ikr eikr F in (n ) π ikr e ikr F in ( n ) + eikr dωf in (n)f(n, n ) (533) r ψ out n rn -n n 9: ψ in ψ in (rn ) = π ikr e ikr F in ( n ), [ F in (n ) + ik ψ out (rn ) = π ikr eikr ] dω 4π F in(n)f(n, n ) (534) ψ in (rn ) F in (n ) F in (n ) ψ out dω F out (n ) = F in (n ) + ik 4π F in(n)f(n, n ). (535) kn F in F out f(n, n ) n ˆf dω ˆfF in (n ) = 4π F in(n)f(n, n ) (536) (535) F out (n ) = ŜF in(n ) = (1 + ik ˆf)F in (n ). (537) Ŝ = 1 + ik ˆf Ŝ Ŝ Ŝ = 1. (538) 96

97 ˆf ˆf ˆf = ik ˆf ˆf (539) dω f(n, n ) f (n, n) = ik 4π f (n, n )f(n, n ) (540) n = n Im f(n, n) = k 4π dω f(n, n ) (541) (58) σ = dω f(n, n ) (54) Im f(n, n) = k 4π σ (543) f(n, n) Ŝ ψ ψ (533) ψ(rn ) = π ikr e ikr F in ( n ) + π ikr eikr ŜF in (n ) (544) ψ (rn ) = ψ (rn ) = π ikr e ikr (ŜF in(n )) + π ikr eikr F in( n ) (545) F in (n ) = Ŝ G (n ) (546) ψ (rn ) = π ikr e ikr G(n ) + π ikr eikr ˆP Ŝ T G(n ) (547) ˆP ˆP G(n ) = G( n ) (548) 97

98 G(n ) (533) (533) G(n ) = F in ( n ) = ˆP F in (n ) (549) ˆP ŜT ˆP = Ŝ (550) Ŝ = 1 + ik ˆf ˆf ˆP ˆf T ˆP = ˆf. (551) S(n, n ) = S( n, n), f(n, n ) = f( n, n) (55) n -n n -n 30: S 1 = S 1 f(n, n ) n n θ (489) (57) g(rn ) = eikr r f(n, n ) (553) (5) g(x) f(n, n ) = m 4πa π k n n. (554) Ŝ Ŝ 98

99 r = 0 0 ψ klm (r, θ, ϕ) = R kl (r)y lm (θ, ϕ) (555) U(r) = 0 k m E = k. (556) ] [ d l(l + 1) + R dr r kl (r) = k R kl (r) (557) z = kr k z l R kl (r) R l (z) R l (z) [ 1 d d ] l(l + 1) z dz z + 1 R dz z l (z) = 0. (558) χ l (z) χ l (z) + R l (z) = z l χ l (z). (559) (l + 1) χ z l(z) + χ l (z) = 0. (560) l = 0 0 = zχ 0(z) + χ 0(z) + zχ 0 (z) = (zχ 0 (z)) + zχ 0 (z) (561) χ 0 (z) e±iz z. (56) 99

100 l 1 χ 0 χ l (560) z z ( ) ( ) ( ) 1 (l + ) 1 1 z χ l(z) + z z χ l(z) + z χ l(z) = 0. (563) (560) (1/z)χ l (z) χ l+1(z) χ l+1 (z) 1 z χ l(z). (564) l χ l ( ) l ( ) l 1 R l (z) = z l χ l (z) z l d 1 χ 0 z l d e ±iz z dz z dz z. (565) R ± l (z) R ± l (z) = zl ( 1 z ) l d e ±iz dz z. (566) z (566) 1/z 1/z z z e ±iz R ± l z (z) 1 z ( ) l d e ±iz = (±i)l e ±iz. (567) dz z R + l (z) R l (z) 1/z z z 1/z R ± l (z) U(r) = 0 U(r) 0 z U(r) 0 z U(r) = 0 R ± l (z) z = 0 1/z e±iz z = 0 ( ) l R ± z 0 1 l (z) z l d 1 z dz z = 1 ( 1)l (l 1)!!. (568) zl+1 100

101 R ± l (z) z = 0 z = 0 R ± l (z) ( ) 1 ( l R + 1 l i (z) R l (z)) = z l d sin z z dz z. (569) (1/z) sin z (1/z)d/dz (1/z)d/dz z l z (1/z) sin z l ( ) 1 ( l R + l i (z) R l (z)) z 0 1 z l d ( 1) l z dz (l + 1)! zl = ( 1)l (l + 1)!! zl. (570) (570) R + l (z) + R l (z) j l (z) = ( 1)l i y l (z) = ( 1)l+1 ( ) ( l R + 1 l (z) R l (z)) = ( 1) l z l d sin z z dz z. (571) ( ) ( l R + 1 l (z) + R l (z)) = ( 1) l+1 z l d cos z z dz z. (57) j 0 z j 1 z j z j 3 z z y 0 z y1 z y z y3 z z 31: j l (z) y l (z) 101

102 z = 0, z = j l (z) z 0 1 (l + 1)!! zl, j l (z) z 1 ( z sin z lπ ). (573) y l (z) z 0 1 (l 1)!! z, l+1 y l(z) z 1 ( z cos z lπ ). (574) e ±iz ( ) l h (1) 1 l (z) = j l (z) + iy l (z) = i( 1) l z l d e iz z dz z. (575) ( ) l h () 1 l (z) = j l (z) iy l (z) = i( 1) l z l d e iz z dz z. (576) z = 0 z = h (1) l (z) z 0 1 (l 1)!! z, l+1 h(1) l (z) z i ( z exp iz lπi ). (577) h () l (z) z 0 1 (l 1)!! z, l+1 h() l (z) z i ( z exp iz + lπi ). (578) π π j l (z) = z J l+ 1 (z), y l (z) = z Y l+ 1 (z), h (1) π l (z) = z H(1) (z), h () π l+ 1 l (z) = z H() (z). (579) l+ 1 U(r) = 0 j l (z) U eff r l z l 0 R kl (r) = kj l (kr) z ( r sin kr lπ ) = i lπ r e i(kr ) i lπ r ei(kr ). (580) 10

103 j 100 z : j 100 (z) r R kl (r) 1 WKB U eff (r) = l(l + 1) mr (581) ae = m k (58) r k l(l + 1) k = ± k (583) r r = l + 1 k (584) l l(l + 1) l + 1 WKB ϕ(r 1, r ) = r r 1 kdr = k r r 1 dr r r r (585) r = t + r (586) ϕ(r 1, r ) = k t t 1 t dt t + r = [ kt (l + 1 ) tan 1 t r ] t t 1 (587) 103

104 r 1 = r t 1 = 0 r t r ϕ(r, r) r kr (l + 1 )π (588) r = r π/4 ψ(r) r 1 ( r sin (l + 1/)π kr + π ) = 1r ( 4 sin kr lπ ) (589) (580) 0 l δ l R kl (r) z r sin (kr lπ + δ l ) (590) 10. U(r) = 0 (r a), U(r) = + (r a). (591) m ψ = Eψ (59) ψ(x) = R kl (r)y lm (θ, ϕ) (593) E = k /(m) z = kr R kl (r) j l (kr). (594) r = a ψ(x) r=a = 0 j l (ka) = 0 (595) k 104

105 1: j l (z) n 0 z l,n l = 0 l = 1 l = l = 3 l = 4 l = 5 l = 6 n = n = n = n = n = l = 0 j 0 (z) = (1/z) sin z ka = nπ n = 1,, 3,... l j l (z) n 0 z l,n l n 1 4 j l (ka) = 0 n k = z l,n /a zl,n ma (596) zl,n / : (ma / )E zl,n / l = 0 l = 1 l = l = 3 l = 4 l = 5 l = 6 n = n = n = n = n = U(r) = 0 (r a), U(r) = U 0 (r a). (597) 0 < E < U 0 k b m k = E, m b = U 0 E (598) 4 Mathematica BesselJZero[l + 1/,n] 105

106 k + b = m U 0. (599) R kl j l (kr) (600) k ib 0 R kl i l h (1) l (ibr) (601) 1 r = a l = 0 ψ 1 r sin(kr), ψ 1 exp( br). (60) r ψ /ψ k cot(ka) 1 a = b 1 a (603) b = k cot(kr) (604) k-b (599) (604) k (ma / )U 0 = E = m k (605) ψ = e ikz (606) 106

107 3: (ma / )E (ma / )U 0 = 50 l = 0 l = 1 l = l = 3 l = 4 l = 5 l = 6 n = n = n = z ψ ϕ m = 0 e ikz = a l ψ kl0 = a l R kl (r)y l0 (θ) (607) l=0 a l r cos θ kz = kr cos θ (ik) n r n cos n θ (608) n! n=0 r n cos n θ R kl (r) = kj l (kr) r j l r = 0 (573) R kl (r) = l=0 kl+1 (l + 1)!! rl + O(r l+1 ) (609) l n r n Y l0 (θ) l + 1 l + 1 Y l0 = 4π P (l 1)!! l(cos θ) = cos l θ + (610) 4π l! cos θ l cos n θ l n r n cos n θ l = n (607) a n k n+1 r n cos n θ (611) π(n + 1) n! n=0 (608) (611) a n = in k π(n + 1) (61) 107

108 e ikz = l=0 i l π(l + 1)Rkl (r)y l0 (θ) = k l=0 l + 1 k il R kl (r)p l (cos θ) (613) n a l ψ klm 1 r r + dr dr v v (607) l = n v a n b v L = mbv l = n n (n + 1) mv n b (n + 1) (614) mv ( ) π n = πn mv p (615) v l v a n a n = πn k (616) n (61) ψ(x) = e ikz + f(θ) r eikr (617) 108

109 f(θ, ϕ) = (l + 1)f l P l (cos θ) (618) l=0 (613) ikz r e 1 kr (l + 1)i l P l (cos θ) sin l=0 ( kr lπ ) (619) r ψ r l + 1 [ ( i l r k sin kr lπ ) ] + f l e ikr P l (cos θ) (60) l=0 l f l (60) [ ] ) [ ] = il k sin = il ik ( kr lπ + f l e ikr ) (e ikr lπ i e ikr+ lπ i + f l e ikr = 1 + ikf l e ikr 1 ik ik ( 1)l e ikr (61) 1 + ikf l = 1 (6) 1 + ikf l = S l = e iδ l (63) S l δ l S l Ŝ F in = Y l,m, F out = Y l,m (64) dωf outŝf in = S l δ ll δ mm (65) f l ˆf 109

110 δ l (61) [ ] = eiδ l ik eikr 1 ik ( 1)l e ikr = il e iδl k sin ( kr lπ ) + δ l (66) δ l l σ tot = f l = eiδ l 1 ik dω f(θ) = π = eiδ l k sin δ l. (67) d cos θ (l + 1)f l P l (cos θ) l (68) d cos θp l (cos θ)p l (cos θ) = l + 1 δ ll (69) σ tot = 4π l=0 (l + 1) f l = 4π k l (l + 1) sin δ l. (630) l=0 σ l = 4π(l + 1) f l = 4π k (l + 1) sin δ l. (631) l l 4 δ l = π mod π 11. WKB (465) 0 f(θ) = l=0 e iδ l 1 (l + 1)P l (cos θ) (63) ik 110

111 e iδ l 1 1 θ = 0 1 f(θ) = 1 ik (l + 1)P l (cos θ)e iδ l (633) l=0 WKB P l (cos θ) l sin[(l + 1 )θ + π 4 ] πl sin θ (634) f(θ) = 1 ( { [ ( l exp i δ l l + 1 ) θ π ]} k π sin θ 4 l=0 { [ ( exp i δ l + l + 1 ) θ + π ]}) 4 (635) exp l l θ dδ l ± θ = 0. (636) dl δ l l WKB WKB π r m[e U(r)] (l + 1/) dr (637) r r 0 π/4 r 0 kr πl δ l = 1 r m[e U(r)] (l + 1/) dr + r ( l + 1 (638) ) π kr (639) δ l /δl l (l + ) = mbv 1 dδ l dl r = r 0 = r 0 (l+ 1 ) r dr + m[e U(r)] (l+1/) r b dr + r 1 U(r) b E r 111 π π (640)

112 r θ r 0 bdr = r 1 U(r) b E r π θ (641) (465) r 0 l 1 U(r) 0 l 1 0 l = 0 [ ] d m dr + U(r) ψ(r) = Eψ(r). (64) 0 0 ψ(r) = a(r r eff ). (643) a r eff r 0 a ψ(r) = lim k 0 k sin[k(r r eff)] (644) sin(kr + δ 0 ) r l = 0 δ 0 = kr eff. (645) σ = σ 0 = 4π k sin δ 0 = 4πr eff (646) 11

113 11.4 r 0 r, E, p / / ρ = r/r 0, E = E, k = p. (647) a 0 ρ r r 0 = m = = 1 l j l y l mr 0 R kl c 1 j l (kr) + c y l (kr). (648) c 1 c c 1 j l (kr 0 ) + c y l (kr 0 ) = 0 (649) c 1 c δ l c 1 j l (kr) + c y l (kr) r 1 [ ( c 1 sin kr lπ ) ( c cos kr lπ )] kr ( c = kr sin kr lπ ) + δ l (650) c = c 1 + c δ l c 1 c = cos δ l, c c = sin δ l (651) σ l = 4π k (l + 1) sin δ l (65) l = 50 k 33 exact WKB k l l(l + 1)/ (l + 1/) / E = k / l + 1/ > k δ l = 0 l + 1/ < k r = r 0 WKB 113

114 Σ WKB exact k 33: σ k l k l : δ l = (r 0, r) [ (r, r) + π ] 4 = (r, r 0 ) π 4 = ˆl ˆl cos 1 kr0 kr ˆl π 0 4 (653) ˆl = l + 1/ δ l 33 WKB ˆl = k δ l 35 l σ l = 0 r = r 0 114

115 Σ E 35: n l < n k 0 l l = 0 4π = k > l sin δ l sin δ = 1/ σ tot = l=0 k 4π k (l + 1)1 π (654) π e ikz e ikz 36: π 115

116 π 116