A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c 3 λ = c/ν Wien u(ν, T )dν = 8πhν3 c 3 e hν kt dν E/ν ν/t 3. λ = 6000Å 100W (a) J ev (b) 1
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = 5.67 10 8 W m 2 K 4 5. T λ ũ(λ, T ) Planck ũ(λ, T )dλ = 8πhc λ 5 1 e hc kt λ 1 dλ λ max λ max T = b Wien b hc 4.965k = 2.9 10 3 m K 4.965 (1 x 5 )ex = 1 2
6. Stefan-Boltzmann Wien λ max R = 7 10 8 m; d = 1.5 10 11 m; C = 1.4 10 3 J m 2 s 1 ; 7. 3K Penzias Wilson λ max (E = hν = hc/λ) B 8. 2.0eV 2000Å ev C 9. λ 0 E 0 p 0 λ = λ 0 + λ E p θ K p e ϕ (a) λ λ λ 0 = λ c (1 cos θ) 3
λ c h m e c = 2.43 10 12 m = 0.0243Å (b) cot θ 2 = (1 + hν 0 m e c ) tan ϕ, 2 ν 0 c λ 0 (c) 2hν K 0 m = e sin 2 θ c 2 2 E 0 1 + 2hν 0 m e sin 2 θ c 2 2 m e = 0.511 MeV/c 2 = 9.109 10 31 kg D 10. (?) de Broglie (a) 100m 10.00s 72kg (b) 155km/hour 145g (c) 10 6 m/s 11. K m de Broglie E 12. ρ e Thomson (a) (b) e = 1.6 10 19 C R = 1.0 10 10 m 4
13. k 0 = 1 4πϵ 0 = 9.0 10 9 N m 2 /C 2 1eV = 1.6 10 19 J m = 9.1 10 31 kg e = 1.6 10 19 C h = 6.6 10 34 J s c = 3.0 10 8 m/s (1) v r e 2 k 0 r = mv2 2 r L = mvr L L = n h 2π (n = 1, 2, ) n v n r n E n v n = 2πk 0e 2, r n = hn E n = 2π2 k0e 2 4 m h 2 n 2 h 2 4π 2 k 0 e 2 m n2, (2) 1 r 1 [m] E 1 [ev] r 1 r B r B r e k 0e 2 mc = 2.82 2 10 15 m α k 0e 2 hc = 1 137 5
(3) λ e [m] 2πr B λ e R 1/α λ e = h mc, R 2πr B λ e (4) v n, r n, E n α, m, c, r e, n (5) ( h h 2π ) mc, mc 2, h mc, h mc 2 (6) Rydberg R k2 0e 4 m 4π h 3 c = 1.097 107 m 1 E n Bohr ν = E i E f h = k2 0e 4 m 4π h 3 ( 1 n 2 f 1 ) n 2 i Rydberg 1/λ 6
14. f(x) = g(k)e ikx dk g(k) = { 0 (k < K, K < k) N ( K < k < K) N (a) f(x) (b) 15. f(x) 2 dx = 1 N (c) x k x k x k > 1 x = 2π/K f(x, t) = g(k)e i(kx ω(k)t) dk, g(k) = e α(k k 0) 2 k k = k 0 α β 1 ( d 2 ) ω k 2 dk 2 k=k 0 f(x, t) ( ) dω v g = dk k=k 0 7
16. f(x, t) = g(k)e i(kx ω(k)t) dk p E = p2 2m E = hω p = hk ω = p2 2m h, v g dω dk = p m ψ(x, t) = 1 2π h ϕ(p)e i(px Et)/ h dp ψ(x, t) i h h2 ψ(x, t) = t 2m 2 ψ(x, t) x2 17. 18. r p h H = p2 2m k e 2 0 r 19. x p h/2 H = p2 2m + 1 2 mω2 x 2 20. E t h m π r 0 ( c t) = 1.4 10 15 m MeV/c 2 8
21. Schrödinger i h ( ) t ψ( r, t) = h2 2m + V ( r) ψ( r, t) 2 x 2 + 2 y 2 + 2 z 2 (a) ρ t + J = 0 ρ J ρ ψ ψ = ψ 2 J i h 2m [ψ ( ψ) ( ψ) ψ] (b) t = 0 ψ( r, t) 22. (a) ψ( r, t) = a exp[ ī ( p r Et)] (b) h ψ(r, t) = a r exp[ ī (pr Et)] h 23. ψ( r, t) Schrödinger V ( r) Ehrenfest d r dt = p m d p dt = V A ψ ( r, t)aψ( r, t)d 3 x 9
( ) π 1 ( ) 4 24. ψ(x) = exp αx2 α 2 α (a) x, x 2, x x 2 x 2 (b) p, p 2, p p 2 p 2 (c) x p 2a 3 1 25. ψ(x) = π x 2 + a a 2 dx x 2 + a = π 2 a 2 (a) x, x 2, x (b) p, p 2, p (c) x p x 2 x 2 p 2 p 2 26. θ( π θ π) ψ(θ) ψ(π) = ψ( π) L = i h d dθ 27. ϕ(p) ψ(x) x = p = ϕ(p) = 1 2π h ψ (x)xψ(x)dx = ϕ (p)pϕ(p)dp = ψ(x)e ipx/ h dx ϕ (p)i h d dp ϕ(p)dp ψ (x) h i d dx ψ(x)dx 10
28. p x [x, p] = i h e A e A e ipa/ h xe ipa/ h = x + a n=0 A n n! 29. Schrödinger i h ( t ψ(x, t) = h2 2 ) 2m x + V (x) ψ(x, t) 2 ψ(x, t) ψ(x, t) = ( 2πi hϵ [ ( 1 i 1 m ) 2 exp h 2 m(x x ) 2 )] V (x)ϵ ψ(x, t 0 )dx ϵ ϵ t t 0 30. p E (E 2 = p 2 c 2 + m 2 c 4 ) ψ(x, t) = 1 2π h ϕ(p)e i(px Et)/ h dp ψ(x, t) ( 1 2 ) ( ) c 2 t 2 mc 2 ψ(x, t) + ψ(x, t) = 0 2 x 2 h ρ t + J x = 0 ρ J J i h ( ψ ψ ) 2m x ψ x ψ ρ 11
31. Schrödinger i h h2 ψ(x, t) = t 2m 2 ψ(x, t) + V (x)ψ(x, t) x2 ψ(x, t) = T (t)ϕ(x) ϕ(x) Schrödinger 32. a) O 1 ψ(x) = x 3 ψ(x) b) O 2 ψ(x) = x d dx ψ(x) c) O 3 ψ(x) = λψ (x) d) O 4 ψ(x) = exp(ψ(x)) e) O 5 ψ(x) = d x dx ψ(x) + a f) O 6ψ(x) = ψ(x )x dx 33. (a) O 6 ψ(x) = λψ(x) (b) (1) [O 1, O 2 ], (2) [O 2, O 6 ] 34. ( a < x < a) m V (x) = 0 ( x < a) = ( x > a) 12
a = 1 10 14 m [J] n (a) x, p (b) x x 2 x 2, p (c) x p p 2 p 2 35. V (x) = 0 ( x < a) = V 0 ( x > a) V 0 36. ψ(x) = 1 ( x < a) = 0 ( x > a) p p + dp 37. P ψ(x) = ψ( x) P P +1 1 13
38. Schrödinger d 2 h2 ψ(x) + V (x)ψ(x) = Eψ(x) 2m dx2 (a) (b) V (x) = V ( x) ψ(x) = ψ( x) ψ(x) = ψ( x) (c) V (x) x = x 1 dψ(x)/dx ψ(x) 39. V (x) = 0 (x < 0) = V 0 (x > 0) E m x R T (a) V 0 < E (b) 0 < E < V 0 40. V (x) = 0 ( x > a) = V 0 ( x < a) E m x R T E > 0 14
41. E m V (x) = 0 (x < 0, a < x) = V 0 (0 < x < a) x (a) V 0 < E R T (b) 0 < E < V 0 R T (c) V 0 E h 2 /2ma 2 T T V 0 E a 42. R T R + T = 1 43. x m d 2 h2 2m dx ψ(x) + 1 2 2 mω2 x 2 ψ(x) = Eψ(x) 44. n (a) x 2 p 2 (b) K = p2 2m V = 1 2 mω2 x 2 (c) x p h/2 45. K V lim T 1 T T 0 1 2 K dt = lim T T 15 T 0 x d dx V dt
Ô ψ (x, t)ôψ(x, t)dx K = p2 2m 2 K = x d dx V 46. U 0 δ(x) (U 0 ) m 47. V (x)( a V (x) = V (x + a)) (a) U exp(ipa/ h) Uψ(x) = ψ(x + a) (b) U H = p2 + V (x) 2m (c) ψ(x + a) = e iθ ψ(x) θ u(x) (u(x + a) = u(x)) ψ(x) ψ(x) = e ikx u(x) ka = θ 48. V (x) m V (x) = h2 γ ma n= δ(x + na) a γ 16
t ψ( x, t)  A ψ Âψd 3 x  A ψ Âψd 3 x = (Âψ) ψd 3 x 49. ˆp = i h Ĥ = h2 2m 2 + V ( x) ψ 0 50.  ψ 1 ψ 2 ψ 2Âψ 1d 3 x = (Âψ 2) ψ 1 d 3 x 51.  ˆB [Â, ˆB] = iĉ (a) Ĉ = Ĉ (b) A B C /2 ( A) 2 A 2 A 2 ( B) 2 B 2 B 2 17
(c) (Â A )ψ = iα(( ˆB B )ψ α α 52. Â (a) Â (b) 53. Â ˆB (a) A B Â ˆB (b) Â ˆB 54. (1) ψ(x), (2) ψ(x) = ϕ(x)e ikx (ϕ(x) ) (3) Ĥ = 1 2m p2 + V ( x) 55. Â d A = A dt t + ī [H, A] h 56. δ (1) f(x)δ(x)dx = f(0), f(x)δ(x a)dx = f(a) (2) δ( x) = δ(x) (3) xδ(x) = 0 (4) δ(ax) = δ(x), (a > 0) a (5) δ(a 2 x 2 ) = 1 (δ(x a) + δ(x + a)), (a > 0) 2a (6) ϕ(x) d dx δ(x)dx = d dx ϕ(0) 18
Ĥ Ĥ = ˆp2 2m + 1 2 mω2ˆx 2 ˆx ˆp [ˆx, ˆp] = i h 57 64 57. â â [â, â ] = 1 â mω 2 h ( ˆx + iˆp ) ( mω, â ˆx iˆp ) mω 2 h mω Ĥ Ĥ = hω( ˆN + 1 2 ) ˆN( â â) 58. Ĥ E n u n E n = hω(n + 1 2 ), u n = 1 n! (a ) n u 0, (n = 0, 1, 2,...) u 0 â u 0 = 0 â â â u n = n u n 1, â u n = n + 1 u n+1 u m u n = δ mn u n Ĥ E n (Ĥ u n = E n u n ) 19
59. ψ n (x) Ĥ u n ψ n (x) = x u n 57,58 ψ n (x) 43 Schrödinger d 2 h2 2m dx ψ(x) + 1 2 2 mω2 x 2 ψ(x) = Eψ(x) n 60. (a) u n ˆx u m, (b) u n ˆp u m, (c) u n [ˆx, ˆp] u m (d) x u n ˆx 2 u n u n ˆx u n 2 (e) p u n ˆp 2 u n u n ˆp u n 2 61. (1) e λâ f(â)e λâ = f(â λ), (2) e αâ+βâ = e βâ e αâ e αβ 2 λ α β 62. ˆx ˆp â â h ˆx = 2mω (â + â ), ˆp = 1 i m hω 2 (â â ) ( ) h e ikˆx = exp ik 2mω â h exp ik 2mω â exp hk2 4mω < u 0 e ikˆx u 0 > 63. Schrödinger Heisenberg Heisenberg â â 20
64. α â α â α = α α α α = 1 (a) α â α = α α α α = 1 α = exp ( 1 ) α n 2 α 2 u n n! = exp n=0 ( 1 2 α 2 ) exp(αâ ) u 0 = exp(αâ α â) u 0 (b) α N α ˆN α α n P n = u n α 2 P n N (P n Poisson (c) α ˆx ˆp ( x p = h/2) x α ˆx 2 α α ˆx α 2 p α ˆp 2 α α ˆp α 2 65. ˆQ ˆQ ( ˆQ σ + i d ) ( iw (x), dx ˆQ σ i d ) + iw (x) dx Ĥ σ + σ σ + = Ĥ 1 2 ( ˆQ ˆQ + ˆQ ˆQ) ( 0 1 0 0 ), σ = 21 ( 0 0 1 0 )
(a) ˆQ 2 ˆQ 2 ˆQ ˆQ (b) Ĥ (c) Ĥ ψ B(x) E(> 0) ψ F (x) = ˆQψ B (x) Ĥ ψ B (x) ( ) 0 ψ B (x) = ϕ(x) (d) Ĥ Ĥ (e) W (x) = λx λ (f) W (x) [x 0, x 1 ] W (x) W (x) x P (W (x P ) = 0) (W (x P ) 0) ψ B (x) ψ F (x) ±1 0 66. t 0 x t x G(x, t; x, t 0 ) = x exp( ī hĥ(t t 0)) x m G(x, t; x, t 0 ) G(x, t; x, t 0 ) Schrödinger Schrödinger ψ(x, t) 29 ψ(x, t) = G(x, t; x, t 0 )ψ(x, t 0 )dx 22
N 67. N V (x 1 x 2,..., x N 1 x N ) N 68. Schrödinger ( h2 2 h2 2 ) u(x 2m 1 x 2 1 2m 2 x 2 1, x 2 ) = Eu(x 1, x 2 ) 2 u(x 1, x 2 ) = ϕ 1 (x 1 )ϕ 2 (x 2 ) u(x 1, x 2 ) x X x = x 1 x 2, X = m 1x 1 + m 2 x 2 m 1 + m 2 69. Schrödinger ( h2 2 h2 2 2m 1 2m 2 x 2 1 x 2 2 ) u(x 1, x 2 ) + V (x 1 x 2 )u(x 1, x 2 ) = Eu(x 1, x 2 ) x X V (x) = 0 70. Ĥ = ˆp2 1 2m + ˆp2 2 2m + 1 2 mω2 x 2 1 + 1 2 mω2 x 2 2 23
E n = hω(n + 1 2 ) 71. Ĥ = ˆp2 1 2m + ˆp2 2 2m + V (x 1, x 2 ), V (x 2, x 1 ) = V (x 1, x 2 ) (a) ˆP 12 ˆP 12 (b) ˆP 12 72. 73. (0 x a) m 74. (0 x a) m N E n = h2 π 2 2ma 2 n2, (n = 1, 2, 3,...) Bose Fermi N Fermi E F E F 24
3 I 75. N ) N? V ( x 1,..., x N ) = N i=1 1 2 mω2 x 2 i 76. a ( m ) (a) (b) E F (c) N (d) E F n = N a 3 (e) E tot N n R (f) E F = h2 kf 2 2m k F = 2π λ F λ F d = n 1/3 77. 8.5 10 22 /cm 3 (a) ev (b) 25
3 II ( ) 78. H = h2 2 2m x + 2 2 y + 2 + V (r) r = (x 2 +y 2 + 2 z 2 z 2 ) 1/2 z θ { x = x cos θ y sin θ y = x sin θ + y cos θ Hψ(x, y, z) = Eψ(x, y, z) z 79. L = (L x, L y, L z ) ( L x = i h y z z ) (, L y = i h z y x x ) z ( L z = i h x y y ) x (a) [L x, L y ] = i hl z, [L y, L z ] = i hl x, [L z, L x ] = i hl y (b) L 2 L x L y L z ( ) (c) L z Ĥ = h2 2 2m x + 2 2 y + 2 + V (r) 2 z 2 26
80. L 2 + ( r p) 2 = r 2 p 2 + i h r p p 2 = 1 ( L r 2 h 2 1 r ) 2 h 2 1 2 r 2 r r r 81. ( ) ] [ h2 2 2µ x + 2 2 y + 2 + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z) 2 z 2 ( h2 2 2µ r + 2 2 r r + 1 r 2 sin θ ( sin θ ) + θ θ 1 2 ) r 2 sin 2 ψ(x, y, z) θ ϕ 2 + V (x, y, z)ψ(x, y, z) = Eψ(x, y, z) V = V (r) ψ = R nlm (r)y lm (θ, ϕ) R nlm (r) ( [ h2 d 2 2µ dr + 2 ) ] d l(l + 1) + V (r) R 2 r dr r 2 nlm (r) = ER nlm (r) Y lm (θ, ϕ) l(l +1) h,(l = 0, 1, 2,...) r r 2 V (r) 0 (a) u nlm (r) = rr nlm (r) u nlm (r) (b) r = 0 (c) r V (r) 0 27
82. V (r) = 0 (a) ρ = kr R nlm (r) (b) (c) ρ l (b) 83. (c) V (r) 0 84. V (r) l E (0) l 85. V (r) = 0 (r < a), V (r) = (r > a) l l = 0 86. V (r) = V 0 (r < a), V (r) = 0 (r > a) E (a) r < a (b) r > a r 0 (c) (a) (b) r = a (d) κa 1 ( E V 0 ) (c) E κ 2 = 2µ h 2 (V 0 + E) 28
87. (x, y, z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) L = (L x, L y, L z ) L ± L x ± il y 88. L z Φ m (ϕ) = m hφ m (ϕ) Φ m (ϕ) Φ m (ϕ) ϕ m 89. (a) I H = 1 2I L z 2 (b) N 2π N 90. L ± L x ± il y (a) L 2 = L + L + L z 2 hl z, L 2 = L L + + L z 2 + hl z (b) [L +, L ] = 2 hl z, [L z, L + ] = hl +, [L z, L ] = hl [ L 2, L + ] = 0, [ L 2, L ] = 0 29
91. L 2 L z (a) L 2 L z Y λm L 2 Y λm = λ h 2 Y λm L z Y λm = m hy λm λ m 2 (b) L ± Y λm L 2 L z (c) Y λmmax, Y λmmin L + Y λmmax = L Y λmmin = 0 (d) m max = l λ = l(l + 1) (l = 0, 1 2, 1, 3, ) 2 92. L 2 Y lm (θ, ϕ) L 2 Y lm = l(l + 1) h 2 Y lm, L z Y lm = m hy lm Y lm (θ, ϕ) = Θ(θ)e imϕ Θ(θ) Y lm (θ, ϕ) Y lm (θ, ϕ) = C m l e imϕ (1 u 2 ) m/2 d l m du l m (1 u2 ) l u = cos θ C m l L + Y ll = 0 30
93. m p e m e e V (r) = e 2 /r ( h2 d 2 2µ dr + 2 ) d l(l + 1) R 2 r dr r 2 nl (r) e2 r R nl(r) = ER nl (r) 8µ E µ(= m e m p /(m e + m p )) ρ = h 2 r, λ = e2 µ µ h 2 E = cα 2 E ˆR nl (ρ) = R nl (r) 94. ˆR nl (ρ) (a) ρ (b) ˆR nl (ρ) = e ρ/2 G(ρ) G(ρ) (c) G(ρ) = ρ l L(ρ) L(ρ) (d) L(ρ) = a k ρ k R nl (r) k=0 E n n(= λ) (e) ψ nlm = R nl (r)y lm (θ, ϕ) 31
95. R 10, R 20, R 21 (n = 1,l = 0) r r 96. 97. P nl (r) P nl (r) r 2 (R nl (r)) 2 P nn 1 (r) r 98. k + 1 n 2 r k (2k + 1)a 0 r k 1 + k 4 [(2l + 1)2 k 2 ]a 2 0 r k 2 = 0 a 0 = h2 Kramers µe2 99. Kramers r 1, r, r 2 100. 45 p 2 µ 32
101. ( m e) L (c ) L = 1 ( ) 2 d x 2 m eϕ( x, t) + e A( x, dt c t) d x dt (a) L m d2 x dt 2 = e ( E( x, t) + 1 ) d x c dt B( x, t) E = ϕ 1 c A t, B = A 102. (a) { ϕ( x, t) ϕ ( x, t) = ϕ( x, t) + 1 λ( x, t), c t A( x, t) A ( x, t) = A( x, t) λ( x, t) λ( x, t) 103. (a) H H = 1 ( p e A( x, 2m c 2 t)) + eϕ( x, t) H 33
104. ( m e) i h Ψ( x, t) = ĤΨ( x, t) t ( h 2 ( = ie A( x, 2m hc ) 2 t) + eϕ( x, t)) Ψ( x, t) (a) (b) (Ψ( x, t) = e iet/ h ψ( x)) ψ( x) (c) A = 1 2 x B, ϕ = e/r, B = (0, 0, B) (B ) ( h2 2m 2 eb 2mc L z + e2 B 2 ) 8mc 2 (x2 + y 2 ) e2 ψ( x) = Eψ( x) (b) r (d) (b) L z h, x 2 + y 2 a 2 0, r a 0 a 0 105. B (b) 106. ϕ( x) (b) (x = ρ cos φ, y = ρ sin φ) 107. ϕ( x) h 2 ( ie A( x) 2m hc ) 2 ψ( x) = Eψ( x) A = (0, Bx, 0) 34
108. u n = 1 n! (â ) n u 0 (a) â â ˆx ˆp Ĥ = hω(â â+ 1 2 ) (Ĥ) mn u m Ĥ u n = hω 1 0 0 2 0 3 0 2 0 0 5 2 (b) u 0 = (1, 0, 0,...) T u n u m u n = δ mn T 109. ψ = 1 2 (1, 1, 0, 0,...) T (a) Ĥ (b) ˆx, ˆx 2, ˆp, ˆp 2 (c) x p 35
L 2 l, m = h 2 l(l + 1) l, m L z l, m = hm l, m l, m L z l, m = hmδ m m l, m L ± l, m = h l(l + 1) m(m ± 1)δ m m±1 110. L z, L +, L (a) l = 1 2 (b) l = 1 (c) l = 3 2 (c) L x, L y, L z 111. n L n αk ( L n) k = 0 112. 1 2 ( S = h σ) 2 (a) S z (b) S x cos ϕ + S y sin ϕ S z h 2 113. n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) σ σ n σ n σ z 1 ψ (+) σ z 1 ψ ( ) 36
Ĥ Ĥ = M B 114 115 B M g = 2(1 + α 2π + ) = 2.0023192 S M = eg 2mc S 114. 1 2 ( S = h σ) 2 (a) ψ(t) = (ψ (+) (t), ψ ( ) (t)) T B = (0, 0, B) (b) ψ(0) = (ψ (+) 0, ψ ( ) 0 ) T (c) S x h 2 S x S y 115. S (a) B = (0, 0, B) (b) S z S x + is y 37
116. S 1 = h σ 1 /2 S 2 = h σ 2 /2 2 S 2 1 S 1z χ (1) +, χ (1) S 2 2 S 2z χ (2) +, χ (2) (a) S = S 1 + S 2 S 2 S z (b) S 1 S 2 117. L S = h σ/2 J = L + S ψ j,m+ 1 = αy l,m χ + + βy l,m+1 χ J 2 2 α β S + χ + = S χ = 0, S ± χ = hχ 118. L (a) S 2 S z S 1 (b) S z h, 0, h ξ +, ξ 0, ξ S + S (c) J 2 ψ j,m+1 = αy l,m ξ + + βy l,m+1 ξ 0 + γy l,m+2 ξ J J = L + S (d) J 2 ψ j,m = h 2 j(j + 1)ψ j,m α, β, γ 38
119. J 2 1 J 1z ψ (1) J 1,M 1 J 2 2 J 2z ψ (2) J 2,M 2 J 2 = ( J 1 + J 2 ) 2 J z = J 1z + J 2z ψ J,M ψ J,M ψ (1) J 1,M 1 ψ (2) J 2,M 2 ψ J,M = J 1 J 2 M 1 M 2 JM ψ (1) J 1,M 1 ψ (2) J 2,M 2 M 1,M 2 Clebsch-Gordan (a) Clebsch-Gordan JM J 1 J 2 M 1 M 2 J 1 J 2 M 1 M 2 J M = δ JJ δ MM M 1,M 2 J 1 J 2 M 1 M 2 JM JM J 1 J 2 M 1M 2 = δ M1 M 1 δ M 2 M 2 J (b) Clebsch-Gordan J(J + 1) M(M ± 1) J 1 J 2 M 1 M 2 JM ± 1 = J 1 (J 1 + 1) M 1 (M 1 1) J 1 J 2 M 1 1M 2 JM + J 2 (J 2 + 1) M 2 (M 2 1) J 1 J 2 M 1 M 2 1 JM 120. 1 J = L + 1, L, L 1 V (r) V (r) = V 1 (r) + ( S L) h 2 V 2 (r) + ( S L) 2 h 4 V 3 (r) 121. S 1 S 2 2 A,B,C H = A + B h 2 S 1 S 2 + C h (S 1z + S 2z ) (a) 1/2 (b) 1/2 1 39
122. H 0 {ϕ n } E 0 n H 0ϕ n = E 0 nϕ n λ H = H 0 + λh 1 ψ n E n λ λ H 0 123.? λ 124. H H = H 0 + λh ( 1 ) E 0 H 0 = 1 0 0 E2 0 ( ) α γ H 1 = γ β (a) λh 1 H 0 H λ (b) H λ λ (a) 125. H 0 H 0 = p2 2µ e2 r 40
ψ nlm = R nl (r)y lm (θ, ϕ) ψ 200,ψ 211,ψ 210, ψ 21 1 eez E H = H 0 + eez d 2 126. H 0 = h2 2m dx + mω2 2 2 x2 = hωa a λh 1 (a) λh 1 = λx 4 λ (b) λh 1 = λx 3 λ 127. ψ(x) H H H ψ (x)hψ(x)dx (a) ψ(x) H u n (x) ( E n ) ψ(x) = a n u n (x) H E n n a n (b) E 0 H E 0 41
(c)? ψ α () ψ 0 (x, α) α F (α) F (α) α α 0 ψ 0 (x, α 0 ) 128. H d 2 H = h2 2m dx + mω2 2 (a) 2 x2 ψ 0 (x, α) = Ae αx2 (α > 0, A ) ψ 0 (x, α 0 ) F (α 0 ) π e bx2 dx = b e bx2 x 2 dx = 1 2 π b 3 (b) (a) ψ 0 (x, α 0 ) H (c) ψ 1 (x, β) = (B + Cx)e βx2? (B, C β ) 42
1. 1 5 Get!! 2. 2 10 Get!! 3. 3 2 Get! 1 Get! 0 4. 10 Get!! 1. 2. 3. 4. 1 2 3 43
1. 2. 3. 4.!! a) Boltzmann b) E u(ν, T )dν = (ν ν + dν ) E (ν ν + dν ) = 8πν 2 dν/c 3!! Change! 44