27 1 21

19 i

WKB ii

i 1 1 1.1................................. 1 1.2......................................... 1 1.2.1.................................. 1 1.2.2.................................. 3 1.3........................................ 4 1.3.1.................................... 4 1.3.2........................... 5 1.3.3............ 7 1.3.4................................ 8 1.4...................................... 9 1.5.................................. 10 1.5.1.................................... 10 1.5.2............................... 11 1.5.3................................... 12 1.5.4................................... 12 1.5.5................................... 13 1.5.6.................................... 14 1.6........................................ 15 2 17 2.1.................................. 18 2.1.1............................. 18 2.1.2................................ 21 2.1.3................................... 23 2.1.4................................ 23 2.2................................... 28 2.2.1........................... 28 iii

2.2.2................ 29 2.2.3.................................. 30 2.2.4.................................. 30 2.3................................... 31 2.3.1........................... 31 2.3.2.................................. 32 2.3.3......................... 34 2.3.4..................... 35 2.3.5.................................. 36 2.3.6................................... 37 2.4............................ 37 2.4.1........................... 37 2.4.2................................ 39 2.4.3............ 40 2.4.4.................................. 40 2.4.5.................................. 41 2.4.6........................ 44 2.5........................................ 45 3 53 3.1....................................... 54 3.1.1................................... 55 3.1.2................................... 56 3.2........................................ 61 3.2.1.................................. 61 3.2.2............................... 63 3.2.3................................ 65 3.3........................ 65 3.3.1.................................. 67 3.3.2.......................... 69 3.4........................ 69 3.4.1.................................. 70 3.4.2........................ 72 3.5.............................. 73 3.5.1 1.............................. 73 3.5.2 2.............................. 75 3.6.................................... 76 iv

3.6.1............................. 76 3.6.2........................... 77 3.6.3............................. 78 3.6.4................................ 79 3.7................................. 80 3.8........................................ 81 4 85 4.1...................................... 85 4.2..................... 87 4.2.1................................... 87 4.2.2............................. 88 4.3................................. 90 4.3.1............................. 90 4.3.2.............................. 92 4.4...................................... 92 4.5................................. 93 4.5.1 a......................... 93 4.5.2............................. 94 4.5.3................. 95 4.6................................. 97 4.6.1.................................. 97 4.6.2............................. 97 4.7...................................... 98 4.7.1............................. 98 4.7.2.......................... 100 4.8........................................ 100 5 105 5.1...................................... 107 5.1.1................................... 108 5.1.2............................. 109 5.1.3......................... 112 5.1.4............ 113 5.2....................................... 114 5.2.1........................... 114 5.2.2............................ 115 5.3................................... 115 v

5.3.1................................. 116 5.3.2.................................. 117 5.3.3................................ 119 5.3.4............................... 120 5.4......................... 120 5.5............................ 123 5.5.1...................... 123 5.5.2........................... 124 5.6...................................... 126 5.6.1.................................. 126 5.6.2........................... 127 5.6.3........................ 132 5.6.4 Legendre........................... 135 5.6.5........................ 135 5.7.................................. 136 5.8.................................. 137 5.8.1........................... 137 5.8.2 l 1 + l 2......................... 138 5.8.3 l 1 + l 2................... 140 5.8.4.................................. 141 5.8.5...................... 141 5.9............................. 143 5.9.1 3............................ 143 5.9.2................................ 144 5.10........................................ 144 6 149 6.1...................................... 149 6.2............................ 151 6.2.1.............................. 151 6.2.2........................ 153 6.3.............. 153 6.3.1............................ 156 6.4............................ 160 6.4.1.............................. 161 6.4.2.............................. 161 6.4.3 n........................... 161 vi

6.4.4................................... 161 6.5.................................. 162 6.6............................. 162 6.7....................................... 167 6.7.1.................................. 167 6.7.2.................................. 168 6.8........................................ 169 6.8.1..................................... 171 7 179 7.1............................ 179 7.1.1................. 180 7.1.2............................. 181 7.2.................... 181 7.3................................... 183 7.4............................. 184 7.5................................... 185 7.6............................. 186 7.7........................................ 187 8 191 8.1....................................... 191 8.2....................................... 191 8.3 2 2................................ 192 8.3.1............................... 192 8.3.2................................ 195 8.3.3................................ 195 8.4............. 197 8.5.............................. 198 8.5.1 ϵ 0...................................... 198 8.5.2 ϵ 1............................... 198 8.5.3 ϵ 2............................... 200 8.6................................... 201 8.7.............. 201 8.7.1............................... 203 8.8............................. 204 8.9.............................. 206 8.9.1........................ 206 vii

8.9.2.............................. 207 8.9.3........................... 208 8.10.............................. 209 8.10.1................................ 212 8.10.2..................... 213 8.10.3................................ 216 8.11...................................... 217 8.12........................................ 219 9 WKB 223 9.1.............................. 224 9.2...................................... 225 9.3......................... 226 9.3.1........................... 226 9.3.2.................................. 226 9.3.3.................................... 227 9.4............................. 228 9.5................ 229 9.5.1................................ 230 9.6............................... 230 9.7........................................ 231 10 233 10.1........................................ 233 10.2........................................ 235 10.3........................................ 235 11 237 11.1............................ 237 11.2........................ 239 11.2.1...................... 239 11.3................................. 240 11.3.1................................ 240 11.3.2................................... 241 11.4..................... 242 11.4.1................................ 243 11.4.2...................... 244 11.4.3........................... 245 viii

11.5............................... 250 11.5.1 WKB................................. 250 11.5.2 Born................................. 251 11.6.......................... 252 11.7........................ 254 11.7.1...................... 254 11.7.2............ 255 11.8............................... 259 11.9................................. 261 12 263 12.1.................................... 263 12.1.1.......................... 264 12.1.2.................................. 264 12.1.3................................ 265 12.2................................... 266 12.2.1..................................... 266 12.2.2.................................... 270 12.2.3................................ 270 12.3 mass singlarity:infrared divergence........................ 271 12.3.1........................... 273 12.3.2.............................. 273 12.4.................................... 273 12.5........................ 274 12.5.1........................... 275 12.5.2........................... 277 12.5.3........................ 279 12.5.4............... 279 12.6................... 279 12.6.1 EPR................... 280 12.7........................................ 280 13 283 13.0.1............................... 284 13.1............... 285 13.2...................................... 286 13.3.................................... 287 13.3.1...................... 287 ix

13.3.2 = 0............................ 288 13.4................................. 289 13.5............................ 289 13.6............................. 290 13.7................................... 291 13.8............................ 292 13.9...................................... 293 13.10................................. 294 13.11........................ 296 13.12.................................... 297 13.13........................................ 297 13.14........................................ 300 14 303 14.1 B.................. 303 14.1.1.............................. 303 14.1.2............................... 303 14.1.3.......................... 305 14.1.4................................... 309 14.1.5.................................. 313 14.1.6................................ 314 14.1.7........................... 316 14.1.8............................... 317 14.2 C............ 317 14.2.1............................. 317 14.2.2............................. 319 14.2.3...................... 319 14.3 D.................................... 320 14.3.1................................ 320 14.3.2 Legendre Legendre............... 321 14.3.3 Hermite Hermite................ 322 14.3.4............................... 326 14.3.5 Airy................................. 327 14.3.6................................ 327 14.3.7............................. 328 14.3.8 Laguerre.............................. 328 14.4 E......................... 328 x

14.5 F.................................. 329 14.6 G................................... 329 14.7.................................. 330 14.8...................................... 331 14.9....................................... 331 14.9.1........................ 332 14.9.2.......................... 333 14.9.3 ( )...................... 335 xi

1 1.1 1.2 1.2.1 T T 1

Wien P (T, ν) = ( 8παν3 )e βν c 3 T (1.1) Rayleigh-Jeans P (T, ν) = ( 8πν2 c 3 )kt (1.2) T E P (T, E) = e E kt (1.3) Wien Rayleigh-Jeans P (T, ν) = ( 8πhν3 1 ) c 3 e hν kt 1 (1.4) (1.4) h h (1.4) T ν e hν kt 1 e hν kt (1.5) (1.4) P (T, ν) = ( 8πhν3 c 3 )e hν kt (1.6) Wien (1.4) T ν e hν kt 1 hν kt (1.7) (1.4) P (T, ν) = ( 8πhν3 ) kt c 3 hν (1.8) Rayleigh-Jeans Wien Rayleigh-Jeans 2

(1.4) n= x n=0 x n = x 1 x (1.9) x = e hν kt (1.10) x 1 x = 1 e hν kt 1 (1.11) P = e E n kt (1.12) E n = hνn, n = (1.13) n 1.2.2 ν hν 3

E = hν (1.14) E = E E 0 (1.15) E 0 E 0 N hνn h 1.3 1.3.1 4

x(t) x(t) = v 0 t + x 0 (1.16) t = 0 x 0-2 1.3.2 3 4 5

4 5 p p = hk, k = 2π λ (1.17) 6

h = 6.62660693(11) 10 34 JS (1.18) 1.3.3 ν E = hν (1.19) δe ν δe = hν (1.20) 6 δe = c( 1 n 1 ), n, m (1.21) 2 m2 7

l λ λ l = 2πN (1.22) 1.3.4 x(t) δp δq = h (1.23) [q, p] = i h (1.24) i u 1, u 2 v v = z 1 u 1 + z 2 u 2 (1.25) 8

z 1, z 2 1.4 ψ(x, t) m U(x) ψ(x, t) i h ψ(x, t) = Hψ(x, t) (1.26) t H = p2 + U(x) (1.27) 2m H p x ψ(x, t) = e iωt ψ(x) (1.28) E = hν, ν = ω 2π (1.29) x p p = i h (1.30) (2.109) k p ψ(x, t) = e ikx ψ(t) (1.31) pψ(x, t) = hkψ(x, t) (1.32) p H = p2 2m + 1 2 kx2 (1.33) 9

ψ(x, t) = e i Ē h t ψ(x), E = hω(n + 1 2 ) (1.34) ψ(x, t) = ψ 1 (x, t) + ψ 2 (x, t) (1.35) ψ(x, t) 2 = ψ 1 (x, t) 2 + ψ 2 (x, t) 2 + 2Re(ψ 1 (x, t) ψ 2 (x, t)) (1.36) (1.37) E = c 1, n (1.38) n2 (1.21) Schroedinger 1.5 1.5.1 19 20 10

α α 1.5.2 11

1.5.3 1.5.4 10 15 cm 12

1 1.5.5 3 13

u-,d-,c-, s-,t-,b- 6 0.5MeV/c 2 100MeV/c 2 1.3GeV/c 2 1.5.6 14

1.6 m k L L = m 2 (ẋ(t))2 k 2 x(t)2 (1.39) k E = m 2 ẋ2 + k 2 x2 I = dxp, p = E k 2 x2 (1.40) I L = m 2 ( d dt x(t))2 α 1 r (1.41) L = m 2 (ṙ2 + r 2 θ2 + r 2 sin θ 2 ϕ2 ) α 1 r (1.42) 15

r θ ϕ p r = L = mṙ, (1.43) ṙ p θ = θ L = mr2 θ, (1.44) p ϕ = ϕ L = mr2 θ 2 ϕ (1.45) ν hν/c hν ν v m mc2 (1 v 2 /c 2 ) 1/2 (1 v 2 /c 2 ) 1/2 (1) (2) (3) 5 ν hν ν 2000 mc2 (1 v 2 /c 2 ) 1/2 6 p λ = h/p (1) (2) 2 10 8 M/S (3) 300K 3/2kT 16

2 Schroedinger Schroedinger Schroedinger Schroedinger Schroedinger 17

2.1 2.1.1 P V V v v 1, v 2 c 1, c 2 v 1, v 2 V c 1 v 1 + c 2 v 2 V (2.1) r 1, r 2 x 1 r 1 + x 2 r 2 = r (2.2) 18

v 1, v 2, v 3, v 4,, v n V 1, v 2, c 1, c 2, c i (i = 1, n) c 1 v 1 + c 2 v 2 + c n v n = 0 (2.3) c 1 = c 2 = = c n = 0 (2.4) v 1, v 2, v 3, v 4,, v n c 1, c 2 c 1 v 1 + c 2 v 2 + c n v n = 0 (2.5) c 1 0, c 2 0 (2.6) v 1 = 1 c 1 (c 2 v 2 + c n v n ) (2.7) v 1 n n c 1 v 1 + c 2 v 2 + c n v n = 0 (2.8) c 1 = c 2 = = c n = 0 (2.9) c 1 v 1 + c 2 v 2 + c n+1 v n+1 = 0 (2.10) c 1 = c 2 = = c n = c n+1 = 0 (2.11) 19

c n+1 c n+1 0 (2.12) v n+1 = 1 c n+1 (c 1 v 1 + c 2 v 2 + c n v n ) (2.13) v n+1 u v u v (u, v) = i ū i v i (2.14) (u, c 1 v 1 + c 2 v 2 ) = i (c 1 u 1 + c 2 u 2, v) = i c i (u, v i ) (2.15) c 1 (u i, v) (2.16) (u, v) = 0 (2.17) u, v (u, u) 0 (2.18) (u, u) = 0 (2.19) u = 0 (2.20) 20

n e i, i = 1, n (e i, e j ) = 0, i j (2.21) (e i, e i ) = 1, i = j (2.22) (e i, e j ) = δ i,j (2.23) u n u = c i e i (2.24) i=1 (e j, u) = = n c i (e j, e i ) (2.25) i=1 n c i δ i,j = c j (2.26) i=1 2.1.2 ψ ψ ψ 1 ψ 2 dxψ 1(x)ψ 2 (x) (2.27) 21

ψ (2.28) ψ (2.29) ψ 1 ψ 2 ( ) ψ 1 ψ 2 (2.30) ψ 1 ψ 2 ψ 1 ψ 2 ψ 1 ψ 2 { u n }, n = 1, 2, 3,, (2.31) u n u m = δ nm (2.32) ψ ψ = n c n u n (2.33) u m u m ψ = n c n u m u n = c m (2.34) 22

ψ = n c n u n = n u n u n ψ (2.35) ψ u n u n = 1 (2.36) n u v O = u v (2.37) ψ ψ O ψ = u c (2.38) c = v ψ (2.39) ψ O = d v (2.40) d = ψ u (2.41) 2.1.3 Hermite 2.1.4 H u i, i = 1, 2 H u i = v i (i = 1, 2) (2.42) H(c 1 u 1 + c 2 u 2 ) = c 1 v 1 + c 2 v 2 (2.43) 23

H β, α H 1 H 2 u u β Hu α = (H β,α ) (2.44) H 1 + H 2 (2.45) H 1 H 2 (2.46) (H 1 + H 2 ) u = H 1 u + H 2 u (2.47) H 1 H 2 u = H 1 (H 2 u (2.48) (H 1 + H 2 ) u = (H 2 + H 1 ) u (2.49) H 1 (H 2 u ) H 2 (H 1 u ) (2.50) H 1 + H 2 H 2 + H 1 (2.51) H 1 H 2 H 2 H 1 (2.52) Hermite H Hermite H [H 1, H 2 ] = H 1 H 2 H 2 H 1 (2.53) H β,α = H α,β (2.54) H α,β = H β,α (2.55) 24

Hermite Hermite Hermite H 11 H 22 H 33 3 3 H 11 H 12 H 13 H = H12 H 22 H 23 H13 H23 H 33 H 1, H 2 Hermite Hermite (H 2 + H 1 ) = H 2 + H 1 (2.56) (H 2 H 1 ) = H 1 H 2 H 2 H 1 (2.57) Hermite Hermite α u α Hu α = αu α (2.58) ᾱ = α (2.59) u α, u β = 0, α β (2.60) Hermite Hermite x x Hermite x x x = x x (2.61) x p p Hermite p p p = p p (2.62) 25

p x x x p x i p j (2.63) ( x i p j ) = p j x i (2.64) p j x i Hermite (ϵ ijk x j p k ) = ϵ ijk p k x j (2.65) j k p k x j Hermite O Õ Õ x i p j O Õ Hermite Hermite Hermite O O O (2.66) Hermite Hermite H K O O = H + ik (2.67) H = H, K = K (2.68) O Hermite O Hermite H = O + O 2 K = O O 2i (2.69) (2.70) OO (2.71) O O (2.72) (OO ) = (O ) O = OO (2.73) (O O) = O (O ) = O O (2.74) 26

Hermite U U = 1 (2.75) v 1 v 2 (Uv 1, Uv 2 ) = l Ūv 1l (Uv 2 ) l (2.76) = Ūlm 1 v 1m1 U lm2 v 2l = v 1m1 (U U) m1 m 2 v 2m2 = (v 1, v 2 ) (2.77) (U 1 + U 2 ) = U 1 + U 2 = U 1 1 + U 1 2 (2.78) (U 1 + U 2 )(U 1 1 + U 1 2 ) = U 1 U 1 1 + U 1 U 1 2 + U 2 U 1 1 + U 2 U 1 2 (2.79) = 1 + U 1 U 1 2 + U 2 U 1 1 + 1 1 (U 1 U 2 ) = U 2U 1 = U 1 2 U 1 1 (2.80) U 1 2 U 1 1 U 1 U 2 = 1 Hermite i t Hermite U = e ith, H = H (2.81) U = e ith (2.82) U U = e ith e ith = 1 (2.83) t U(t) t 27

2.2 2.2.1 α, β u α, u β f = u α, u β (2.84) α β P = f 2 (2.85) α β P α,β 1 (1)P α,β 0 (2.86) (2) β P α,β = 1 (2.87) α β 1 β 2 P α,β1 + P α,β2 (2.88) 1 A A x 1 + x 2 + x 3 (2.89) x 4 + x 4 + x 6 (2.90) x 7 + x 8 + x 9 (2.91) (2.92) 28

x 1, x 2, 2.2.2 ψ x ψ(x) = x ψ (2.93) x x ψ dx ψ(x) 2 = 1 (2.94) 1 ρ(x) j(x) v ρ(x) = ψ(x) 2 (2.95) j(x) = ψ(x) vψ(x) (2.96) ρ(x, t) + j(x) = 0 (2.97) t dxx ψ(x) 2 = x (2.98) x Hermite H ψ H ψ (2.99) H H H h = h h (2.100) 29

ψ H ψ (2.101) = ψ h h H h h ψ = ψ h 2 h (2.102) h P (h) = ψ h 2 (2.103) (2.102) Hermite h H h = h h h = h δ(h h ) h = (2.104) δ(h h ) 2.2.3 t = u β t = + u+ α S α,β = u + α u β (2.105) t = β t = + α t = ± t t = ± 2.2.4 ψ ρ = ψ ψ (2.106) O T r(ρo) = ψ O ψ (2.107) 30

2.3 q p L p = L q (2.108) q q p Hermite 2.3.1 [q, p] = i h, h = h 2π (2.109) 1 h 2π h u q u c u (2.110) p u = c u [q, p] u = i h u h (1.18) q + q (2.109) q q q 1 = q 1 q 1 (2.111) q 2 q q 1 = q 1 q 2 q 1 (2.112) q 2 [q, p] q 1 = i h q 2 q 1 (2.113) 31

q i q 2 = q 1 q l q l 1 l q l [q, p] q l = q l (qp pq) q l q l l (2.114) = q l q q m q m p q l l p q m q m q q l = 0 l,m l,m q (2.115) i h q l q l = i h l = i hl 0 (2.116) q l L q i p q 2 (qp pq) q 1 = (q 2 q 1 ) q 2 p q 1 (2.117) (q 2 q 1 ) q 2 p q 1 = i h q 2 q 1 (2.118) 2.3.2 A, B [A, B] = AB BA (2.119) A, B A, B, C [A, B] = [B, A] (2.120) [A, B + C] = [A, B] + [A, C] (2.121) [A, BC] = [A, B]C + B[A, C] (2.122) 32

3 [A, BC] = ABC BCA (2.123) [A, B]C + B[A, C] = ABC BAC + BAC BCA (2.124) = ABC BCA (2.125) A, B, C [[A, B]C] + [[B, C]A] + [[C, A]B] = 0 (2.126) [[A, B]C] = (AB BA)C C(AB BA) (2.127) [[B, C]A] = (BC CB)A A(BC CB) (2.128) [[C, A]B] = (CA AC)B B(CA AC) (2.129) [[A, B]C] = A(BC) + B( AC) + C( AB + BA) (2.130) [[B, C]A] = A( BC + CB) + B(CA) + C( BA) (2.131) [[C, A]B] = A( CB) + B( CA + AC) + C(AB) (2.132) [[A, B]C] + [[B, C]A] + [[C, A]B] = A(BC BC + CB CB) + B( AC + CA CA + AC) (2.133) +C( AB + BA BA + AB) = 0 q p [q, p 2 ] = 2i hp, [q, p l ] = i hlp l 1 (2.134) 33

p q [q, F (p)] = i hf (p) (2.135) [q 2, p] = 2i hq, [q l, p] = i hlq l 1 (2.136) [G(q), p] = i hg (q) (2.137) {q, p} P B = 1, {q, q} P B = 0, {p, p} P B = 0 (2.138) {q, F (p)} P B = F (p), {G(q), p} P B = G (p) (2.139) (2.135), (2.135) (2.139) 1 i h [q, F (p)] {q, F (p)} P B (2.140) 1 i h [G(q), p] {G(q), p} P B (2.141) 2.3.3 q 1 q 2 (2.118) (q 2 q 1 ) q 2 p q 1 = i h q 2 q 1 = 0, q 1 q 2 (2.142) 34

q 1 f f(q) = q f f = 1 q 1 q 1 = (2.143) q 1 p q 1 = (2.144) q 2 q 2 q 2 = 1 (2.145) q 2 q 1 q 2 q 2 f = q 1 f (2.146) q 2 q 1 q 2 f(q 2 ) = f(q) (2.147) q 2 q 1 q 2 = 1 (2.148) q 2 (q 2 q 1 ) q 2 p q 1 = i h (2.149) singular δ(q 1 q 2 ) q 1 q 2 = δ(q 1 q 2 ) (2.150) q 2 p q 1 = i h d δ(q 1 q 2 ) dq 1 (2.151) δ(q 1 q 2 ) 2.3.4 (2.151) p = i h q (2.152) f(q) [q, i h q ]f(q) = q( i h q )f(q) ( i h )qf(q) (2.153) q = i h(qf (q) f(q) qf (q)) = i hf(q) (2.154) 35

2.3.5 0 θ π (2.155) 0 ϕ 2π (2.156) singular [θ, p θ ] = i h (2.157) [ϕ, p ϕ ] = i h (2.158) p ϕ m = m m (2.159) m = 1 e imϕ 2π (2.160) h m 1 [ϕ, p ϕ ] m 2 = i h m 1 m 2 (2.161) h(m 2 m 1 ) m 1 ϕ m 2 = i h m 1 m 2 (2.162) m ϕ m (2.163) m 2 = m 1 [sin θ, p θ ] = i h cos θ (2.164) [cos θ, p θ ] = i h sin θ (2.165) [e iϕ, p ϕ ] = i 2 he iϕ (2.166) 36

m m 1 [e iϕ, p ϕ ] m 2 = i 2 h m 1 e iϕ m 2 (2.167) h(m 2 m 1 ) m 1 e iϕ m 2 = i 2 h m 1 e iϕ m 2 (m 2 m 1 + 1) m 1 e iϕ m 2 = 0 (2.168) m 1 e iϕ m 2 = 0 (m 2 m 1 + 1) = 0 (2.169) 2.3.6 q i, i = 1, N p i = L(q i, q j ) q i (2.170) [q i, p j ] = i hδ ij (2.171) [q i, q j ] = [p i, p j ] = 0 (2.172) 2.4 ψ(x, t) 2.4.1 m U(x) ψ(x, t) i h ψ(x, t) = Hψ(x, t) (2.173) t H = p2 + U(x), p = i h 2m x 37 (2.174)

H L(x i, ẋ i ) H = p i ẋ i L(x i, ẋ i ) (2.175) U(x) m L = mẋ2 U(x) (2.176) 2 p i = L = mẋ i ẋ i H = p i ẋ i L = p2 i 2m + U(x) q i p j H(q i, p j ) ψ(q i, p j, t) U(t) i h t ψ(q i, p j, t) = H(q i, p j )ψ(q i, p j, t) (2.177) t U (t) U(t) = exp( H t) (2.178) i h i h t U(t) = H(q i, p j )U(t) (2.179) ψ(q i, p j, t) = U(t)ψ(q i, p j, 0) (2.180) U (t) = exp( H t) (2.181) i h U (t)u(t) = U(t)U (t) = 1 (2.182) U(t) U (t) [U(t), H] = [U (t), H] = 0 (2.183) 38

U(t) U(0) = 1 (2.184) U(t 1 )U(t 2 ) = U(t 1 + t 2 ) (2.185) 2.4.2 ψ 1 ψ 2 A ψ 1 (t) A ψ 2 (t) (2.186) ψ 1 (t) ψ 2 (t) (2.180) (2.186) ψ 1 (t) A ψ 2 (t) = ψ 1 (0) U (t)au(t) ψ 2 (0) (2.187) q i (t) = ψ(0) U (t)q i U(t) ψ(0) (2.188) p i (t) = ψ(0) U (t)p i U(t) ψ(0) d dt q i(t) = 1 i h ψ(0) U (t)[q i, H]U(t) ψ(0) (2.189) d dt p i(t) = 1 i h ψ(0) U (t)[p i, H]U(t) ψ(0) [q i, H] = i h H = i h p i (2.190) p i m [p i, H] = i h H = i h U q i q i d dt q i(t) = p i(t) m (2.191) d dt p i(t) = U (t) q i 39

2.4.3 (2.186) ψ 1 (0) A H (t) ψ 2 (0) (2.192) A H (t) = U (t)au(t) (2.193) i h t A H(t) (2.194) = i h t U (t)au(t) + U (t)ai h t U(t) = U (t)[a, H]U(t) = [A H (t), H] d dt q ih(t) = p ih(t) m d dt p ih(t) = U(q ih) q i H(t) (2.195) 2.4.4 Hψ E (q i, p j ) = Eψ E (q i, p j ) (2.196) 40

E ψ E1 ψ E2 = 0, E 1 E 2 (2.197) (2.174) t i h t ψ E(x, t) = Eψ E (x, t) (2.198) ψ E (x, t) = exp( E i h t)ψ E(x, 0) (2.199) exp( E t) i h E ν ν = E h2π = E h (2.200) 2.4.5 A H A H A H = 1 i h [H, A H] (2.201) [H, A H ] = 0 (2.202) [H, A] = 0 (2.203) 41

A B [H, B] = 0 (2.204) A B [A, B] H A B [H, [A, B]] = [A, [B, H]] [B, [H, A]] = 0 (2.205) A B H [A, B] H [A, B] (1) [A, B] = 0 (2.206) [A, B] 0 (2) [A, B] 0 (2.207) [A, B] A B C d 1, d 2 [A, B] = C + d 1 A + d 2 B (2.208) N Q i, i = 1, N N Q i Q i [Q i, Q j ] (2.209) [Q i, Q j ] = k f k ijq k (2.210) f k ij 42

[Q i, Q j ] = k fij k Q k + Z (2.211) N Q i Z [H, [Q i, Q j ]] = [H, k C k ijq k + Z] = [H, k C k ijq k ] + [H, Z] = [H, Z] (2.212) Z N Q i, i = 1, N [Q i, Q j ] = k f k ijq k (2.213) fij k N fij k ϵ ijk, i, j, k = 1, 3 H = 1 2m p2 (2.214) [p i, H] = 0 [x i, p j ] = i hδ ij (2.215) [p i, p j ] = 0 (2.216) a p i U(a) = e h (2.217) U(a)ψ(x)U(a) 1 (2.218) = ψ(x) + a i i h [p i, ψ(x) + a ia j [ 2! i h, p j i h, ψ(x)]] + 2 = [1 + a i + a i a j + ]ψ(x) x i x i x j = ψ(x + a) x a 43 p i

2.4.6 Q α = i,j C α ijp j q j (2.219) L(q i, q j ) U ij δ ij q i (t) U ij q j (t) (2.220) q i (t) δ ij q j (t) = q i (t) (2.221) 1 1 U ij A ϵ α, α = 1, A C ij U ij = δ ij + ϵ α C α ij (2.222) q i (t) q i (t) + δq i (t), (2.223) δq i (t) = ϵ α C α ijq j (t) (2.224) L(q i + δq i (t), d dt (q i + δq i (t))) = L(q i (t), q i (t)) + δl (2.225) δl = L δq i + L δ q i q i q i = d L δq i + p i δ q i dt q i = d dt (p ic α ijq j )ϵ α ϵ α δl = 0 (2.226) 44

d dt Qα = 0, (2.227) Q α = p i Cijq α j (2.228) Q α Q α Q α [Q α, Q β ] [Q α, Q β ] = γ f αβγ Q γ (2.229) Q α Q α 1 i h [Qα, q i ] = Cijq α j (2.230) 1 h [Qα, p i ] = Cijp α j (2.231) (2.224) 2.5 1 u 2 = (u, u) (2.232) u + v u + v (2.233) ((u, v) + (v, u)) 2 4(u, u)(v, v) 0 (2.234) t u + tv 2 = u 2 + t 2 v 2 + t((u, v) + (v, u)) 0 (2.235) 45

2 P 1 P 2 P i P i 0, P iorj = P i + P j, i P i = 1 (2.236) 3 [p, q] = i h (2.237) ξ q ξ q ξ q = ξ ξ, q η = η η, (2.238) ξ [p, q] η = (η ξ) ξ p η (2.239) ξ [p, q] η = i h ξ η (2.240) (η ξ) ξ p η = i h ξ η (2.241) η ξ 0, ξ p η = 0, (2.242) dξ(η ξ) ξ p η = i h (2.243) 46

θ π π π dθ f(θ) 2 = 1 (2.244) θ L z θ π [L z, θ] = i h (2.245) θ θ i = θ i θ i (2.246) θ 1 [L z, θ] θ 2 = i h θ 1 L z θ 2 (2.247) θ 1 [L z, θ] θ 2 = (θ 2 θ 1 ) θ 1 L z θ 2 (2.248) 4 ψ(t, x) 2 = ψ(0, x) 2, (2.249) d xψ (t, x)o(x, x )ψ(t, x) = 0 (2.250) t ψ(t, x) = e iα(t) ψ(0, x) (2.251) ψ(0, x) h α(t)ψ(0, x) = Hψ(0, x) (2.252) h α(t) = Hψ(0, x) ψ(0, x) 47 (2.253)

x t t, x E h α(t) = E = E Hψ(0, x) ψ(0, x) E (2.254) α(t) = Ē t, h (2.255) Hψ(0, x) = Eψ(0, x) (2.256) ψ(t, x) = l a l e ie lt ψ El (t, x) (2.257) ψ(t, x) = a 1 ψ E1 (t, x) + a 2 ψ E2 (t, x) (2.258) ψ(t, x) 2 = a 1 2 ψ E1 (0, x) 2 + a 2 2 ψ E2 (0, x) 2 (2.259) +a 1a 2 e i(e 1 E 2 )t ψ 1 (0, x) ψ 2 (0, x) + a 1 a 2e +i(e 1 E 2 )t ψ 1 (0, x)ψ 2 (0, x) 5 N Q i (i = 1, N) Q i Q j (?) [Q i, Q j ] Q i, i = 1, N (2.260) t Q i = 1 i h [H, Q i] = 0 (2.261) [H, Q i Q j ] = [H, Q i ]Q j + Q i [H, Q j ] = 0 (2.262) [H, [Q i, Q j ]] + [Q i, [Q j, H]] + [Q j [H, Q i ]] = 0 (2.263) 48

6 H = p2 2M + V ( x) (2.264) V M ṗ i = {H, p i } P B = H q i q i = {H, q i } P B = H p i = = V q i (2.265) p i 2M (2.266) {q i, q j } P B = {p i, p j } P B = 0, {q i, p j } P B = δ ij (2.267) ṗ i = 1 i h [H, p i] = H q i q i = 1 i h [H, q i] = H p i = = V q i (2.268) p i 2M (2.269) [q i, q j ] = [p i, p j ] = 0, [q i, p j ] = i hδ ij (2.270) 7 8 49

9 σ 3 = ( 1 0 ) 0 1 σ 2 = ( ) 0 1 1 0 σ 2 = ( ) 0 i i 0 10 [A, BC] = [A, B]C + B[A, C] (2.271) [A, BC] = {A, B}C B{A, C} (2.272) [A, B] = AB BA (2.273) {A, B} = AB + BA (2.274) a i a j b i b j [a i, a j ] = [a i, a j] = 0, [a i, a j] = δ ij (2.275) {b i, b j } = {b i, b j} = 0, {b i, b j} = δ ij (2.276) [a ia ij a j, a k B kla l ] = a i[a, B] ij a j (2.277) [b ia ij b j, b k B klb l ] = b i[a, B] ij b j (2.278) 50

11 l x l f(x) f( l) = f(l) f (x) f(x) = a 0 /2 + (a n cos nπ l x + b n sin nπ x) (2.279) l n=1 (1) (2) a m = 1 l b m = 1 l l l l l 1 l dx f(x) 2 = a 2 l o/2 + l l (3)f(x) = x 2 a 0, a l, b l f(x) cos mπ xdx (2.280) l f(x) sin mπ xdx (2.281) l (a 2 l + b 2 l ) (2.282) 12 l x l f(x) f( l) = f(l) f (x) f(x) = n= n= (1) (2) (3)l c m = 1 2l l l 1 l dx f(x) 2 = 2l l l f(x) = 1 (2π) 1/2 a(k) = 1 (2π) 1/2 (c n e inπx l ) (2.283) f(x)e imπx l dx (2.284) 51 c 2 l (2.285) a(k)e ikx dk (2.286) f(x)e ikx dx

3 m x t ψ(x, t) i h ψ(x, t) = Hψ(x, t) (3.1) t H = p2 + U(x), p = i h 2m x (3.2) H p2 U(x) H 2m (3.1) dxρ(x, t) (3.3) ρ(x, t) j(x, t) ρ(x, t) = ψ (x, t)ψ(x, t) (3.4) j(x, t) = ψ (x, t) i h i h ψ(x, t) 2m x 2m x ψ (x, t)ψ(x, t) (3.5) (3.1) t ρ = t ψ (x, t)ψ(x, t) + ψ (x, t) ψ(x, t) (3.6) t 53

= 1 (( i h i h 2m x )2 + U(x))ψ (x, t)ψ(x, t) + 1 i h ψ (x, t)(( i h 2m x )2 + U(x)) = 1 (( i h i h 2m x )2 ψ (x, t)ψ(x, t) + ψ (x, t)( i h 2m x )2 ψ(x, t)) x j = 1 (( i h i h 2m x )2 ψ (x, t)ψ(x, t) + ψ (x, t)( i h 2m x )2 ψ(x, t)) (3.7) ρ(x, t) + j(x, t) = 0 (3.8) t x dxρ(x, t) = dx t x ρ(x, t) = [ρ(x)] x= = 0 (3.9) ψ = 0 3.1 U(x) = 0 x i h p2 ψ(x, t) = ψ(x, t) (3.10) t 2m i h h2 ψ(x, t) = t 2m ( x )2 ψ(x, t) (3.11) ψ 1 ψ 2 ψ = c 1 ψ 1 + c 2 ψ 2 (3.12) ψ(x, t) = exp( Et )ψ(x, 0) (3.13) i h ψ(x, 0) E h2 2m ( x )2 ψ(x, 0) = Eψ(x, 0) (3.14) E 54

3.1.1 (3.11) ψ(x, t) = e i( Ē h t kx) (3.15) (3.15) (3.11) E = p2, p = hk (3.16) 2m E t kx = C (3.17) h x t x t ν λ ν = λ = 2π k E 2π h = E h (3.18) (3.19) (1.18) K E = 3 kt 2 λ = h p p = 2m 3 2 kt (3.20) λ = = 3mkT h (3.21) 3mkT x t = ; (3.22) = ; (3.23) ρ(x, t) = 1 (3.24) j(x, t) = v, v = p m 55 (3.25)

H p He i( Ē h t kx) = p2 E 2m e i( h t kx) (3.26) pe i( Ē h t kx) = hke i( Ē h t kx) (3.27) (3.16) +p p (3.15) h2 k 2 2m ψ(x, t) = e i( Ē h t+kx) (3.28) x t ρ(x, t) = 1 (3.29) j(x, t) = p m (3.30) 3.1.2 ψ(x, t) = A + e i( Ē h t+kx) + A e i( Ē h t kx) (3.31) A + = A = A A + (e i( Ē h t+kx) + e i( Ē h t kx) ) = 2Ae i Ē h t cos kx (3.32) A + = A = A A + (e i( Ē h t+kx) e i( Ē h t kx) ) = 2iAe i Ē h t sin kx (3.33) 56

(3.32) (3.33) x = (n + 1 2 )π k x = nπ k (3.34) (3.35) ρ(x, t) = 4 A 2 1 (1 ± cos 2kx) (3.36) 2 j(x, t) = 4 A 2 1 sin 2kx (3.37) 2 ρ(x, t) = 4 A 2 1 2 (3.38) j(x, t) = 0 (3.39) + a 57

ρ(x, t) x= j(x, t) x= ρ(x, t) x= ψ(0) = ψ(a) = 0 (3.40) (3.31) k A + + A = 0 (3.41) A + e ika + A e ika = A + (2i sin ka) = 0 (3.42) sin ka = 0 (3.43) ka = nπ, n = (3.44) E = p2 n 2m, p n = hπ a n (3.45) u n (x) = 2 a sin k nx, k n = nπ a (3.46) (u n, u m ) = δ n,m (3.47) 58

j(x, t) x= ψ (0) = ψ (a) = 0 (3.48) (3.31) k A + A = 0 (3.49) A + e ika A e ika = A + (2i) sin ka = 0 (3.50) sin ka = 0 (3.51) ka = nπ, n = (3.52) E = p2 n 2m, p n = hπ a n (3.53) 59

u n = 2 a cos k nx, k n = nπ a (3.54) (3.47) x = 0 x = a ψ(x + a) = ψ(x) (3.55) (3.31) k e ika = 1 (3.56) ka = 2nπ, n = (3.57) E = p2 n 2m, p n = hπ a u n = 1 a e±iknx k n = nπ 2a 2n (3.58) (3.59) (3.47) pu n = p n u n, p n = hk (3.60) 60

3.2 (3.11) a(k) ψ(x, t) = E(k) i( dka(k)e h t kx), E = h2 k 2 2m (3.61) (3.11) = (i h t + h2 2m x dka(k)(e(k) h2 k 2 2 )ψ(x, t) (3.62) E(k) 2m )e i( h t kx) = 0 (3.61) a(k) 3.2.1 a(k) k 0 1 α a(k) = Ne α(k k 0) 2 (3.63) N = ( π α ) 1/4 (3.64) k = k 0 k k 0 k k 0 ± x (3.61) k (3.63) ψ(x, t) = dkne α(k k 0) 2 i( hk2 2m t kx) (3.65) 61

t = 0 = Ñ exp( 1 4(α i ht )(x v 0t) 2 i( hk2 0 2m t k 0x)) 2m v 0 = p 0 m, Ñ = N( π α i ht ) 1/2 (3.66) 2m ψ(x, 0) = N( π α )1/2 exp( 1 4α x2 ik 0 x) (3.67) t α ht 2m (3.68) ψ(x, t) = Ñ exp( 1 4α (x v 0t) 2 i( hk2 0 2m t k 0x)) (3.69) v 0 = p 0 m, Ñ = N(π α )1/2 (3.70) v 0 α α v 0 62

t ψ(x, t) = Ñ exp( 1 ht 2m α (3.71) 4 i ht 2m v 0 = p 0 m, Ñ = N( π i ht 2m (x v 0 t) 2 i( hk2 0 2m t k 0x)) (3.72) ) 1/2 (3.73) x 2 ht m (3.74) 3.2.2 t = 0 (3.69) δx δp ( x) 2 = ψ (x x ) 2 ψ (3.75) ( p) 2 = ψ (p p ) 2 ψ (3.76) O = ψ O ψ (3.77) x p = h 2 (3.78) (3.79) α x 0 (3.80) p (3.81) 63

p 0 (3.82) x (3.83) x p x 0 p 0 x p ψ x x 0 ψ = 0, (3.84) ψ p p 0 ψ = 0, (3.85) ψ ψ = 1 (3.86) ψ s (x x 0 + is(p p 0 )) ψ Ψ f(s) f(s) = Ψ Ψ (3.87) Ψ = ψ (x x 0 + is(p p 0 )), Ψ = (x x 0 is(p p 0 )) ψ (3.88) s f(s) x = x x 0, p = p p 0 (3.89) f(s) = ψ x 2 ψ + s 2 ψ p 2 ψ + s ψ i[ x, p] ψ (3.90) = ψ x 2 ψ + s 2 ψ p 2 ψ + s h = ψ p 2 h ψ (s + 2 ψ p 2 ψ )2 + ψ x 2 h 2 ψ 4 ψ p 2 ψ (3.91) f(s) ψ x 2 ψ h 2 4 ψ p 2 ψ 0 (3.92) 64

ψ p 2 ψ ψ x 2 ψ h 2 (3.93) s Ψ s Ψ = ( x is p) ψ = 0 (3.94) (x isp) ψ = (x 0 isp 0 ) ψ (3.95) (x + s x )ψ(x) = (x 0 ip 0 )ψ(x) (3.96) ψ(x) = Ne ip 0x 1 2s (x x 0) 2 (3.97) (3.69) 3.2.3 a(k) a(k) = Ne α(k k 0) 2 H n ( α(k k 0 )) (3.98) N = ( π α ) 1/4 C (3.99) 3.3 U(x) = V 0 (3.100) 65

H V 0 (3.101) U F F = U(x) (3.102) x V 0 U(x) = 0, x 0 (3.103) = V 0, 0 x a = 0, a x x = 0 x = a V 0 > 0 F F = V 0 (δ(x) δ(x a)) (3.104) x = 0 x = a x = 0 x = a 66

x = 0 x = a x x 3.3.1 ψ(x) 0, x (3.105) dx ψ(x) 2 (3.1) ( h2 2m ( d dx )2 + U)ψ(x) = Eψ(x) (3.106) ( d 2m(E U) dx )2 ψ(x) = ( h 2 )ψ(x) (3.107) ψ(x) = e ik x (3.108) k = ± 1 h 2m(E U), E U > 0 (3.109) k = ±i 1 h ( )2m(E U), E U < 0 (3.110) V 0 < E < 0 ψ(x) = B e 1 h 2mEx, x < 0 (3.111) 67

= A + e ī 2m(E+V h 0 )x + A e ī 2m(E+V h 0 )x, 0 < x < a (3.112) = B + e 1 h 2mEx, a < x (3.113) 0 < x a < x B A + A B + x = 0 ψ(0 ϵ) = B (3.114) ψ(0 + ϵ) = A + + A (3.115) ψ (0 ϵ) = B 1 h ψ (0 + ϵ) = (A + A ) ī h x = a ψ(a ϵ) = A + e ī h 2mE (3.116) 2m(E+V 0 )a + A e ī h 2m(E + V 0 ) (3.117) 2m(E+V 0 )a (3.118) ψ(a + ϵ) = B + e 1 h 2mEa, (3.119) ψ (a ϵ) = ī h 2m(E + V 0 )(A + e ī 2m(E+V h 0 )a A e ī 2m(E+V h 0 )a ) (3.120) ψ (a + ϵ) = 1 h 2mEB+ e 1 h 2mEa (3.121) ψ(0 ϵ) = ψ(0 + ϵ) (3.122) ψ (0 ϵ) = ψ (0 + ϵ) (3.123) ψ(a ϵ) = ψ(a + ϵ) (3.124) ψ (a ϵ) = ψ (a + ϵ) (3.125) B +, B, A +, A B = A + + A 1 B 2mE = (A+ A ) h ī h 2m(E+V 0 )a + A e ī h A + e ī h i h 2m(E + V 0 )(A + e ī h (3.126) 2m(E + V 0 ) (3.127) 2m(E+V 0 )a = B + e 1 h 2m(E+V 0 )a A e ī h 68 2mEa (3.128) 2m(E+V 0 )a ) = 1 h 2mEB+ e 1 h 2mEa (3.129)

E ( E)(E + V 0 ) 2E + V 0 = tan 2m(E + V 0 )a (3.130) (3.130) E (3.130) E (3.106) E ψ = 0 (3.106) E 3.3.2 (3.130) E V 0 a 3.4 x ψ(x) = 0, x (3.131) 69

dx ψ(x) 2 E > 0 x(t) ± E > 0 3.4.1 0 < E ψ(x) x ± = 0 B +, B, A +, A, C +, C ψ(x) = B + e i 1 h 2mEx + B e i 1 h 2mEx, x < 0 (3.132) = A + e ī 2m(E+V h 0 )x + A e ī 2m(E+V h 0 )x, 0 < x < a (3.133) = C + e i 1 h 2mEx + C e i 1 h 2mEx, a < x (3.134) ρ = B + 2 + B 2 + B + B e 2i 1 h 2mEx + B B+e 2i 1 h 2mEx, x < 0 (3.135) = A + 2 + A 2 + A + A e 2 ī 2m(E+V h 0 )x + A A +e 2 ī 2m(E+V h 0 )x, 0 < x < a(3.136) = C + 2 + C 2 + C + C e 2i 1 h 2mEx + C C +e 2i 1 h 2mEx, a < x (3.137) 3 4 x x 1 ρ = B + 2 + B 2, x < 0 (3.138) = A + 2 + A 2, 0 < x < a (3.139) = C + 2 + C 2, a < x (3.140) 2E 2E j = B + 2 ( m ) + B 2 ( ), x < 0 (3.141) m 2(E + V = A + 2 0 ) 2(E + V ( ) + A 2 0 ) ( ), 0 < x < a (3.142) m m 2E 2E = C + 2 ( m ) + C 2 ( ), a < x (3.143) m 70

B +, B, A +, A, C +, C x < 0 1 x > 0 B = 1, +C = 0 ψ(x) = e i 1 h 2mEx + B e i 1 h 2mEx, x < 0 (3.144) = A + e ī 2m(E+V h 0 )x + A e ī 2m(E+V h 0 )x, 0 < x < a (3.145) = C + e i 1 h 2mEx, a < x (3.146) + e i 1 h 2mEx B e i 1 h 2mEx C + e i 1 h 2mEx B C + 1 1 x = 0 B 2 + C + 2 = 1 (3.147) ψ(0 ϵ) = 1 + B (3.148) ψ(0 + ϵ) = A + + A (3.149) ψ (0 ϵ) = (1 B )i 1 h 2mE (3.150) ψ (0 + ϵ) = (A + A ) ī h 2m(E + V 0 ) (3.151) 71

x = a ψ (a ϵ) = ( ī h ψ(a ϵ) = A + e ī h 2m(E+V 0 )a + A e ī 2m(E+V h 0 )a (3.152) ψ(a + ϵ) = C + e i 1 h 2mEa (3.153) 2m(E + V 0 ))A + e ī 2m(E+V h 0 )a A e ī 2m(E+V h 0 )a ) (3.154) ψ (a + ϵ) = (i 1 h 2mE)C+ e i 1 h 2mEa (3.155) 1 + B = A + + A (3.156) (1 B )i 1 h 2mE = (A+ A ) h ī 2m(E + V 0 ) (3.157) A + e ī 2m(E+V h 0 )a + A e ī 2m(E+V h 0 )a = C + e i 1 h 2mEa (3.158) ( ī 2m(E + V 0 ))(A + e ī 2m(E+V h 0 )a A e ī 2m(E+V h 0 )a ) = (i 1 h 2mE)C+ e i 1 h 2mEa h (3.159) 4 B, A +, A, C + (3.129) E(> 0) E > 0 E E 3.4.2 (3.156) (3.159) A + A A +, A B C + A + (1 + 1 + V 0 E (A +e ī h 1 + V 0 E ) + A (1 2m(E+V 0 )a A e ī h 1 + V 0 E ) = 2 (3.160) 2m(E+V 0 )a ) = A + e ī 2m(E+V h 0 )a + A e ī 2m(E+V h 0 )a (3.161) B = A + + A 1 (3.162) C + = e ī h 2mEa (A + e ī 2m(E+V h 0 )a + A e ī 2m(E+V h 0 )a ) (3.163) 72

(3.161)(3.162) A +, A B, C + 1 + 1 + V 0 /E A + = 2 (1 + 1 + V 0 /E) 2 (1 1 + V 0 /E) 2 e 2 ī h ( 1 + 1 + V 0 /E)e 2 ī h A = 2 (1 + 1 + V 0 /E) 2 (1 1 + V 0 /E) 2 e 2 ī h B = (1 + C + = 4 (1 + V 0 E + ( 1 + 1 + V 0 /E) 2 (1 1 + V 0 /E) 2 (1 (1 + V 0 ī E ))e2 h 2m(E+V 0 )a 2m(E+V 0 )a 1 + V 0 /E) 2 e 2 ī h 1 + V 0 /E 1 + V 0 /E) 2 e 2 ī h 2m(E+V 0 )a 2m(E+V 0 )a 2m(E+V 0 )a 2m(E+V 0 )a (3.164) (3.165) (3.166) (3.167) B 2 = (3.168) C + 2 = (3.169) 3.5 3.5.1 1 U = U 0 / cosh 2 αx (3.170) d 2 dx ψ + 2m 2 h 2 (E + U 0 cosh 2 )ψ = 0 (3.171) αx 73

U 0 α E < 0 ξ = tanh αx (3.172) d dξ [(1 ξ2 ) d ψ] + [s(s + 1) ϵ2 ]ψ = 0 (3.173) dξ 1 ξ2 2mE ϵ =, 2mU 0 2 = s(s + 1) (3.174) hα α2 h s = 1 2 ( 1 + 1 + 8mU 0 2 ) (3.175) α2 h ψ = (1 ξ 2 ) ϵ 2 ω(ξ) (3.176) u(1 u)ω ( ) + (ϵ + 1)(1 2u)ω ( ) (ϵ s)(ϵ + s + 1)ω = 0, (3.177) u = 1 + ξ 2 ψ = (1 ξ 2 ) ϵ 1 ξ 2 F [ϵ s, ϵ + s + 1, ϵ + 1, 2 ] (3.178) E = h2 α 2 [ (1 + 2n) + 1 + 8mU 0 2 ]2 8m α2 h 74

n n s (3.179) [s] U 0 1 α m E 0 E E 0 = U 0 + E (3.180) E = h m (3.181) 3.5.2 2 V 0 V (x) = (3.182) 1 + e ax x ± V (x) 0, x = (3.183) V (x) V 0, x =, (3.184) x a x χ ψ = constant e ik 1x ; x +. (3.185) χ = e αx, (3.186) ψ = χ ik 0 α (C1 ( χ) i(k 0 k 1 ) α + C 2 ( χ) i(k 0+k 1 ) ) (3.187) Γ( 2ik 1 α C 1 = 2 α Γ( i(k 1 k 2 ) )Γ(1 + i(k 1 k 2 ) ) α α (3.188) Γ( 2ik 1 α C 2 = 2 α Γ( i(k 1 k 2 ) )Γ(1 + i(k 1 k 2 (3.189) ) ) α α hk 2 = 2m(E U 0 ), hk 1 = 2mE (3.190) 75

3.6 δp δq δp δq 3.6.1 a V 0 (3.103) x > 0 x < 0 ψ wp = dkn(k)(e ikx + e ikx B (k))e iet, x << 0 (3.191) ψ wp = dkn(k)(e ikx C + (k))e iet, x >> 0, (3.192) N(k) = N 1 e (k k 0 )2 2σ (3.193) N 1 =. (3.194) 76

3.6.2 V 0 U(x) = V 0, x 0 (3.195) = 0, a x. ψ(x) = e ikx + B e ikx, x < 0 (3.196) = C + e ik x, a < x (3.197), x < 0 B x > 0 C + k, k E = p2 2m + V 0 = p 2 2m, (3.198) p = hk, p = hk (3.199) B = k k k + k (3.200) C + = 2k k + k (3.201) ψ wp = dkn(k)(e ikx + k k k + k e ikx ), x < 0 (3.202) 2k = dkn(k) k + k eik x, a < x N(k) = N 1 e (k k 0 )2 2σ (3.203) N 1 = (3.204) 77

3.6.3 Schroedinger d 2 dx 2 ψ + 2m h U = U 0 cosh 2 αx. ϵ s (3.205) 2 (E U(x))ψ = 0 (3.206) ϵ = ( 2mE)1/2 hα (3.207) s = 1 2 ( 1 + (1 + 8mU 1/2 0 α 2 h 2 ) ) (3.208) 2mE k =, χ = tanh αx (3.209) h ψ = (1 χ 2 ) ϵ 2 ω(χ) (3.210) ω = F (ϵ s, ϵ + s + 1, ϵ + 1, 1 (1 χ)) (3.211) 2 F (a, b, c, x). ψ = N(e ikx A 1 (k) + e ikx A 2 (k)), x << 0 (3.212) ψ = Ne ikx B 1 (k), x >> 0 (3.213) A 1 (k) = Γ( ik)γ(1 ik) α α Γ( s)γ(1 + s) (3.214) A 2 (k) = Γ( ik)γ(1 ik) α α Γ( ik s)γ( ik + 1 + s) α α (3.215) B 1 (k) = 1. (3.216) e iet ψ ψ wp = dkn(k)(e ikx A 1 (k) + e ikx A 2 (k))e iet, x << 0 (3.217) ψ wp = dkn(k)(e ikx B 1 (k))e iet, x >> 0, (3.218) 78

ψ wp. N(k) = N 1 e (k k 0 )2 2σ (3.219) N 1 = (3.220) 3.6.4 (3.182) ψ wp = dkn(k)χ ik 0 α (C1 ( χ) i(k 0 k 1 ) α + C 2 ( χ) i(k 0+k 1 ) ) (3.221) N(k) = N 1 e (k k 0 )2 2σ (3.222) N 1 = (3.223) 79

3.7 U(x) = U 0, 0 x, x a (3.224) U(x) = 0, 0 < x < a ψ = C e ±κx (3.225) κ = ( 2m h 2 )(U 0 E) E = (3.226) U = U 0 / cosh 2 αx (3.227) n ξ = tanh αx (3.228) ψ = (1 ξ 2 ) ϵ 1 ξ 2 F [ϵ s, ϵ + s + 1, ϵ + 1, 2 ] (3.229) E = h2 α 2 [ (1 + 2n) + 1 + 8mU 0 2 ]2 8m α2 h E n 0 (3.230) 80

U = A(e 2ax 2e ax ) (3.231) ξ = 2 2mA e ax (3.232) a h s = 2mE a h (3.233) ψ = e ξ/2 ξ s F ( n, 2s + 1, ξ) (3.234) E = A[1 a h (n + 1/2)] 2 2mA n E n 0 (3.235) 3.8 1 V (x) = V 1 ; x 0, x 1 x (3.236) = 0 x x 1 V 1 81

2 V (x) = V 0, Nb < x < Nb + a, N (3.237) = 0, Nb + a < x < (N + 1)b M 1 3 4 2 5 6 λ = h p = 2π hc pc (3.238) hc = 200MeV fm, fm = 10 15 M (3.239) m e c 2 = 0.5MeV v 10 5 (I) v=0.5 c 82

λ = 2π 200MeV fm m e (0.5)cc = 2π 200MeV fm (0.5)0.5M ev = 4.8 10 3 10 15 M = 4.8 10 12 M = 6 200 fm (3.240) 0.25 (II) v = 10 5 c = 3 10 3 M/sec λ = 2π 200MeV fm m e (10 5 )cc = 2π 200MeV fm (10 5 )0.5MeV = 2.4 10 8 10 15 M = 2.4 10 7 M = 6 200 fm (3.241) 0.5 10 5 7 V (x) = V 0 /cosh 2 (αx) (3.242) 83

4 V (x) x 0 V (x) x=x0 = 0 (4.1) V (x) = V 0 + 1 2 V (x 0 )(x x 0 ) 2 (4.2) 4.1 k 2 V = k 2 x2 (4.3) k E = p2 2m + k 2 x2 (4.4) 85

E x x 0, x 0 = ( 2E k )1/2 (4.5) x(t) = 0 (4.6) p(t) = 0 (4.7) E = 0 (4.8) E e Et i h ψ(x) ψ(x) 2 ( h2 2m x 2 + k 2 x2 )ψ = Eψ (4.9) x (E k 2 x2 ) < 0 (4.10) 2 x 2 ψ ψ > 0 (4.11) x x x e ipx E dx ψ(x) 2 < (4.12) E 86

4.2 4.2.1 h2 2m 2 h2 2m x 2 e αx = ( hα)2 2m e αx (4.13) = h2 σ m e σx2 (2 hσ)2 2m x2 e σx2 (4.14) 2 x 2 e σx2 e αx (4.9) (4.14) (4.9) k = (2 hσ)2 m E = h2 σ m (4.9) mk e σx2, σ = h (4.9) (4.15) (4.16) (4.17) k E = h m (4.18) e σx2 x f(x) (4.9) ψ = f(x)e σx2 (4.19) (f (x) 4σxf (x) 2σf(x) + 2mE h 2 f(x))e σx2 = 0 (4.20) f(x) (f (x) 4σxf (x) + (β 2σ)f(x) = 0 (4.21) β = 2mE h 2 (4.22) 87

4.2.2 (4.21) p f(x) = a n x n (4.23) n=0 a n (4.21) f(x)e σx2 f(x) e σx2 x p f (x) = na n x n 1 (4.24) n=1 p f (x) = n(n 1)a n x n 2 (4.25) n=2 ((n + 2)(n + 1)a n+2 + (β 2σ 4σn)a n )x n = 0 (4.26) n (n + 2)(n + 1)a n+2 + (β 2σ 4σn)a n = 0 (4.27) β + 2σ + 4σn a n+2 = (n + 2)(n + 1) a n (4.28) (i) (ii) (i) n 0 n β + 2σ + 4σn 0 = 0 (4.29) a n 0, n n 0 a n = 0, n n 0 + 2 f(x) β β = 4σ(n 0 + 1/2) (4.30) E = h2 2m 4σ(n 0 + 1/2) (4.31) 88

(4.1) (ii) β + 2σ + 4σn 0 (4.32) a n f(x) n a n n a n+2 = β + 2σ + 4σn (n + 2)(n + 1) a n 4σ n a n (4.33) a 2n+2 = 2σ n a 2n = (2σ)n a 0 (4.34) n! f(x) f(x) = n (2σ) n a 0 x 2n = a 0 e 2σx2 (4.35) n! ψ(x) ψ(x) = f(x)e σx2 = a 0 e σx2 (4.36) x ± (4.1) (4.1) E (5.129) 89

4.3 4.3.1 (4.9) x y y = αx (4.37) α = (mk)1/4 h 1/2 (4.38) [ 2 y 2 + y2 ]ψ = βψ (4.39) m E β = 2 (4.40) k h [( y + y)( y + y)]ψ = (β 1)ψ (4.41) [a, a ] = 1 (4.42) a = 1 ( + y) (4.43) 2 y a = 1 2 ( y + y) a a a a βψ = [2a a + 1]ψ (4.44) [a, a a] = a, [a, a a] = a (4.45) (4.42) β [2a a + 1]ψ β0 = β 0 ψ β0 (4.46) 90

a a [2a a + 1]aψ β0 = (β 0 2)aψ β0 (4.47) [2a a + 1]a ψ β0 = (β 0 + 2)a ψ β0 a a 2 a a ψ β ψ β0 ψ β0 = 1 β = ψ β [2a a + 1]ψ β (4.48) = 2 a ψ 2 + 1 > 0 β β 0 β 1 Hermite n aψ β0 = 0 (4.49) [2a a + 1]ψ β0 = ψ β0 (4.50) n = a a (4.51) n n = n n (4.52) a 0 = 0 (4.53) n 0 = 0 (4.54) a n 1 N N a n = N n + 1 (4.55) n aa n = N 2 n + 1 n + 1 = N 2 (4.56) 91

[a, a ] = 1 n aa n = n a a + 1 n = n + 1 (4.57) N 2 = (n + 1) (4.58) n = N (a ) n 0 (4.59) 1 N = n! (4.49) y ( y + y) ψ β 0 = 0 (4.60) ψ β0 = Ne y2 /2 (4.61) (4.60) (4.9) 4.3.2 4.4 a a n a a n a n = n n 1, a 0 = 0 (4.62) a n = n + 1 n + 1 (4.63) 92

a a m a n = nδ m n+1 (4.64) m a n = n + 1δ m n 1 (4.65) a a y y m y n = 1 2 m a + a n (4.66) = 1 2 ( nδ m n+1 + n + 1δ m n 1 ) m y n = 1 2 m a a n (4.67) = 1 2 ( nδ m n+1 n + 1δ m n 1 ) 4.5 4.5.1 a a a z = z z (4.68) 93

a a a z a a z = Ne za 0 (4.69) a z = N[a, e za ] 0 (4.70) = Nze za 0 = z z (4.69) a a a a l z = z l z (4.71) 4.5.2 A B [[A, B], A] = [[A, B], B] = 0 (4.72) e A e B = e (A+B+ 1 2 [A,B]) (4.73) t f(t) = e ta e tb (4.74) f(t) f(t) t d dt f(t) = AetA e tb + e ta Be tb (4.75) = Ae ta e tb + e ta Be ta e ta e tb = (A + e ta Be ta )f(t) 94 (4.76)

A B t e ta Be ta = B + t[a, B] + t2 [A, [A, B]] + (4.77) 2 (4.72) e ta Be ta = B + t[a, B] (4.78) f(t) f(0) = 1 d f(t) = (B + t[a, B])f(t) (4.79) dt f(t) = e (tb+t2 /2[A,B]) f(0) (4.80) f(t) = e (tb+t2 /2[A,B]) (4.81) t = 1 A B e A e B = e (A+B+1/2[A,B]) (4.82) e B e A = e (A+B 1/2[A,B]) (4.83) e A e B = e B e A e +[A,B] (4.84) A B 4.5.3 (4.69) z z = N 2 0 e za e za 0 (4.85) 95

(4.84) a 0 = 0 U(z) U(z) z z = 1 (4.86) N 2 e zz[a,a ] 0 e za e za 0 = N 2 e zz (4.87) N 2 e zz = 1 (4.88) N = e 1 2 zz (4.89) z = e 1 2 zz e za 0 (4.90) z = U(z) 0 (4.91) U(z) = e za za (4.92) U(z 1 )U(z 2 ) = e z 1a z 1 a e z 2a z 2 a = e z 1a z 1 a+z 2 a z 2 a 1 2 (z 2 z 1 z 1 z 2 ) = U(z 1 + z 2 )e 1 2 (z 2 z 1 z 1 z 2 ) = U(z 2 )U(z 1 )e (z 2 z 1 z 1 z 2 ) U(z) z 1 z 2 = e z 1z 2 1 2 ( z 1z 1 + z 2 z 2 ) (4.93) (4.94) 0 e ikx 0 (4.95) 0 e ikx 0 = 0 e ik/ 2(a+a ) 0 (4.96) = 0 e ik/ 2(a ) e ik/ 2(a) e k2 /4[a,a ] 0 = e k2 /4 96

4.6 4.6.1 V (r) = 1 k r2 = 1 2 k(x2 + y 2 + z 2 ) (4.97) x y z H = i H (i) (4.98) H i = ( p2 i 2m + k 2 x2 i ) ψ(x) = X(x)Y (y)z(z) (4.99) H i X(x i ) = E i X(x i ) (4.100) H i X(x i ) = ( i E i ) i X(x i ) (4.101) i E i 4.6.2 n 1 n 2 n 3 x y z ψ = n 1 1 n 2 2 n 1 3 (4.102) E = hω(n + 3/2) (4.103) N = n 1 + n 2 + n 3 (4.104) 97

N N N = 1 3 N = 2 (n 1, n 2, n 3 ) = (1, 0, 0), (0, 1, 0), (0, 0, 1) (4.105) (n 1, n 2, n 3 ) = (2, 0, 0), (0, 2, 0), (0, 0, 2) (4.106) = (1, 1, 0), (1, 0, 1), (0, 1, 1) 6 N 4.7 4.7.1 h 2 2m H = p2 2m + F x (4.107) d 2 ψ(x) + F xψ(x) = Eψ(x) (4.108) dx2 98

x x 2mF x = (x E h F ) (4.109) ψ xψ = 0 (4.110) Appendix D ψ = constant dte xt t3 /3 (4.111) Appendix D P Q C P = t 2, Q = 1, Z = e t3 /3, V = e xt t3 /3 (4.112) t 3 t t 0 t ( xt t3 3 ) t=t 0 = 0 (4.113) x t 2 = 0, t 0 = ± x (4.114) ( t )2 t3 3 t=t 0 = 2t 0 e 2iβ (4.115) β x > 0 x < 0 t 0 = x, β = π ; x > 0 (4.116) 2 t 0 = ±i x, β = ± π ; x < 0 (4.117) 4 ψ = 1 2 x 1/4 e 2/3 x3/2 ; x > 0 (4.118) ψ = x 1/4 sin( 2 3 x 3/2 + π ); x < 0 (4.119) 4 x > 0 x < 0 x < 0 x π 4 99

4.7.2 4.8 M k H = p2 2M + k 2 x2 (4.120) ( h2 2M ( x )2 + k 2 x2 )ψ = Eψ (4.121) ψ(x) = H(x)e ( x 2 R 0 2 ) (4.122) H(x) = 1 1 H(x) = 2x H(x) = 4x 2 2 e t2 +2tξ = (1) n=0 H n (ξ) t n (4.123) n! H n+1 (ξ) 2ξH n (ξ) + 2nH n 1 (ξ) = 0 (4.124) d dξ H n(ξ) = 2nH n 1 (ξ) (4.125) H n+1 (ξ) = 2ξH n (ξ) d dξ H n(ξ) (4.126) 100

(2) H 0 (ξ) = 1 (4.127) H 1 (ξ) = 2ξ (4.128) H 2 (ξ) = 4ξ 2 2 (4.129) H 3 (ξ) = 8ξ 3 12ξ (4.130) (3) H n (ξ) = ( 1) n e ξ2 dn dξ n e ξ2 (4.131) (4) (5) H n (ξ) dξh n (ξ)h m (ξ)e ξ2 = 2 n n!(π) 1/2 δ nm (4.132) n m x p n x m = (4.133) n p m = (4.134) x = a + a 2 p = a a 2i (4.135) (4.136) 101

x = a H = p2 2M + k 2 (x a)2 (4.137) ( h2 2M ( x )2 + k 2 (x a)2 )ψ = Eψ (4.138) ψ(x a) = H(x a)e ( (x a) 2 R 0 2 ) (4.139) 6 2 H = p2 x + p 2 y 2m + k 2 (x2 + y 2 ) (4.140) ψ = X(x)Y (y) (4.141) 7 2 H = p2 x + p 2 y 2m + k 2 (x2 + y 2 ) (4.142) ψ = R(r)Θ(θ) (4.143) 102

8 i, j = 1, 2 [a i, a j ] = 0, [a i, a j] = δ ij, [a i, a j] = 0 (4.144) 0 e c a d a e 0 (4.145) 103

5 m V (x) x i p j H = p2 + V (x) (5.1) 2m [x i, p j ] = i hδ ij (5.2) p = i h (5.3) 1 i h ψ(x, t) = Hψ(x, t) (5.4) t H i h d xψ (x, t)ψ(x, t) (5.5) t = d x( t ψ (x, t)ψ(x, t) + ψ (x, t) ψ(x, t)) t = d x(ψ ( H )(x, t)ψ(x, t) + ψ (x, t)hψ(x, t)) = 0 105

H H = H (5.6) ρ(x, t) j(x, t) ρ(x, t) = ψ (x, t)ψ(x, t) (5.7) j(x, t) = ψ (x, t) p p ψ(x, t) ( 2m 2m ψ )ψ(x, t) (5.8) ρ(x, t) + j(x, t) (5.9) t = t ψ (x, t)ψ(x, t) + ψ (x, t) ψ(x, t) t + ψ (x, t) p 2m ψ(x, t) + ψ (x, t) p p ψ(x, t) 2m 2m ψ (x, t)ψ(x, t) p 2m ψ (x, t) ψ(x, t) i h2m ψ (x, t)ψ(x, t) + ψ (x, t) p2 i h2m ψ(x, t)ψ (x, t) p p ψ(x, t) 2m 2m ψ (x, t)ψ(x, t) = p2 = = 0 p 2m ψ (x, t)ψ(x, t) ψ (x, t) ρ(x, t) p 2m ψ(x, t) + ψ (x, t) p p ψ(x, t) 2m 2m ψ (x, t)ψ(x, t) P = dxρ(x, t) (5.10) t P = d x ρ( x, t) (5.11) = d x j = ds j (5.12) t P = 0 (5.13) P = dxρ(x, t) (5.14) 106

ρ(x, t) ψ x ρ(x) P P ψ(x, t) = e iet h ψ(x) (5.15) ψ(x, t + a) = e iea h ψ(x, t) ρ(x, t) j(x, t) ρ(x, t + a) = ρ(x, t) (5.16) j(x, t + a) = j(x, t) (5.17) Hψ(x) = Eψ(x) (5.18) (5.4) 5.1 i h h2 ψ(x, t) = t 2m 2 ψ(x, t) (5.19) 2 = 2 x 2 + 2 y 2 + 2 z 2 (5.20) 107

5.1.1 (5.19) ψ(x, t) = e i( Ē h t k x) (5.21) E = p2, p = hk (5.22) 2m ψ(x, t) = ψ 1 (x, t)ψ 2 (y, t)ψ 3 (z, t) (5.23) E(kx) i( t k ψ 1 (x, t) = e h x x) ψ 2 (y, t) = e i( E(k y) h t k y y) E(kz) i( t k ψ 3 (z, t) = e h z z) (5.24) (5.25) (5.26) E(k) = E(k x ) + E(k y ) + E(k z ) (5.27) E(k x ) = hk x 2 2m, E(k y) = hk 2 y 2m, E(k z) = hk x 2m 2 (5.28) (5.21) He i( Ē h t k x) = E(k)e i( Ē h t k x) (5.29) pe i( Ē h t k x) = hke i( Ē h t k x) (5.30) 108

p H phe i( Ē h t k x) = hke(k)e i( Ē h t k x) (5.31) Hpe i( Ē h t k x) = E(k) hke i( Ē h t k x) (5.32) [p, H]e i( Ē h t k x) (5.33) = ( hke(k) E(k) hk)e i( Ē h t k x) = 0 (5.34) [H, p] = 0 (5.35) 5.1.2 ψ(x, t) = dkdea(k, E)e i( Ē h t k x) (5.36) a(k, E) k E ψ(x, t) (5.36) (5.19) dedk(e p2 E )a(k, E)e i( h t k x) = 0, p = hk (5.37) 2m (5.22) E a(k, E) = δ(e h2 k 2 )a(k) (5.38) 2m ψ(x, t) E(k) i( ψ(x, t) = dka(k)e h t k x) (5.39) a(k) k t = 0 ψ(x, 0) = dka(k)e ik x (5.40) 109

a(k) = ( 1 2π )3 dxψ(0, x)e ik x (5.41) a(k) t ψ(t, x) t = 0 ψ(0, x) G(t, x) ψ(x, t) = dξψ(ξ, 0)G(t, ξ x) (5.42) G(t, ξ x) = ( 1 E(k) 2π )3 i( dke h t k (ξ x)) (5.43) G(t, x) [i h t h2 2m 2 ]G(t, x) = 0 (5.44) G(0, x) = δ(x) (5.45) (5.43) G(t, ξ x) t = 0 ξ = x (5.42) t = 0 ξ ψ(ξ, 0) G(t, ξ x) G(t, ξ x) dxg(t, ξ 1 x)g (t, ξ 2 x) (5.46) 1 = dk (2π) 6 1 dk 2 dxe i( E(k 1 ) t k h 1 (ξ 1 x)) e i( E(k 2 ) t k h 2 (ξ 2 x)) 1 = (2π) 6 1 = dk (2π) 3 1 e ik 1 (ξ 1 ξ 2 ) = δ(ξ 1 ξ 2 ) dk 1 dk 2 (2π) 3 δ(k 1 k 2 )e i( E(k 1 ) h t k 1 (ξ 1 )) e i( E(k 2 ) h t k 2 (ξ 2 )) G(t, x) (5.58) (5.46) G(t, x) (5.46) dx ψ(x, t) 2 (5.47) = dxdξ 1 ψ (ξ 1, 0)G (t, x ξ 1 )dξ 2 G(t, x ξ 2 )ψ(ξ 2, 0) 110

= = dξ 1 dξ 2 ψ (ξ 1, 0)δ(ξ 1 ξ 2 )ψ(ξ 2, 0) dξ 1 ψ(ξ 1, 0) 2 θ(t) G R (t, x) = θ(t)g(t, x) (5.48) [i h t h2 2m 2 ]G R (t, x) = i hδ(t)δ(x) (5.49) G R (t, x) (i h t h2 2m 2 ) α a(k) = Ne α(k k 0) 2 (5.50) N = ( π α ) 3/4 (5.39) ψ(x, t) = t = 0 dkne α(k k 0) 2 i( hk2 2m t k x) (5.51) = Ñ exp( 1 4(α + i ht )(x v 0t) 2 i( hk2 0 2m t k 0 x)) 2m v 0 = p 0 m, Ñ = N( π α + i ht ) 3/2 (5.52) 2m ψ(x, 0) = N( π α )3/2 exp( 1 4α x2 ik 0 x) (5.53) t α ht 2m (5.54) 111

ψ(x, t) = Ñ exp( 1 4α (x v 0t) 2 i( hk2 0 2m t k 0 x)) (5.55) v 0 = p 0 m, Ñ = N(π α )3/2 v 0 α α v 0 t ht 2m α (5.56) ψ(x, t) = Ñ exp( 1 4 i ht (x v 0 t) 2 i( hk2 0 2m t k 0 x)) (5.57) 2m v 0 = p 0 m, Ñ = N( π i ht ) 3/2 2m x 2 ht m (5.58) 5.1.3 ψ 1 ψ 2 ψ 1 t = 0 x 1 ψ 1 (x, t) = dkne α(k k 0) 2 i( hk2 2m t k (x x 1)) = Ñ exp( 1 4(α + i ht )(x x 1 v 0 t) 2 i( hk2 0 2m t k 0 (x x 1 ))) 2m ψ 2 t = 0 x 2 ψ 2 (x, t) = dkne α(k k 0) 2 i( hk2 2m t k (x x 2)) = Ñ exp( 1 4(α + i ht )(x x 2 v 0 t) 2 i( hk2 0 2m t k 0 (x x 2 ))) 2m (5.59) (5.60) 112

ψ 1 ψ 2 ψ = ψ 1 + ψ 2 (5.61) 5.1.4 113

5.2 r, θ, ϕ 2 Ψ(x) = d θ,ϕ 2 Ψ(x) = 1 r r(r 2 2 r Ψ(x)) + 1 r 2 d2 θ,ϕψ(x) (5.62) 1 sinθ θ(sinθ θ Ψ(x)) + 1 sinθ 2 φ 2 Ψ(x) (5.63) r, θ, ϕ L L 2 L 2 = h 2 d θ,ϕ 2 (5.64) r θ, ϕ r θ, ϕ 5.2.1 h2 2m ( 1 r 2 r(r 2 r Ψ(x)) + 1 r 2 d θ,ϕ 2 Ψ(x)) + V (r)ψ(x) = EΨ(x) (5.65) Ψ(x) r θ, ϕ Ψ(x) = R(r)Y (θ, ϕ) (5.66) h2 2m Y (θ, ϕ)( 1 r 2 r(r 2 r R(r) + 1 r 2 R(r)d θ,ϕ 2 Y (θ, ϕ)) + V (r)r(r)y (θ, ϕ) = ER(r)Y (θ, ϕ) (5.67) R(r)Y (θ, ϕ) h2 1 2m R(r) ( r(r 2 r R(r)+) + (V (r) E)r 2 = 1 Y (θ, ϕ) d θ,ϕ 2 Y (θ, ϕ)) (5.68) 114

r θ ϕ r r, θ, ϕ R(r) r h2 2m (( 1 r r(r 2 2 r R(r))) 1 l(l + 1))R(r) + V (r)r(r) = ER(r) (5.69) r2 Y (θ, ϕ) d θ,ϕ 2 Y (θ, ϕ) = l(l + 1)Y (θ, ϕ) (5.70) Y (θ, ϕ) l(l + 1) (5.70) l(l + 1) 5.2.2 Ψ 1 (x) = R(r) 1 Y 1 (θ, ϕ) (5.71) Ψ 2 (x) = R(r) 2 Y 2 (θ, ϕ) (5.72) (Ψ 1 (x), Ψ 2 (x)) = r 2 dr sin θdθdϕ(r(r) 1 Y 1 (θ, ϕ), R(r) 2 Y 2 (θ, ϕ) (5.73) = 0 π r 2 drr(r) 1R(r) 2 sin θdθ 0 2π 0 dϕy 1 (θ, ϕ) Y 2 (θ, ϕ) (5.74) r θ, ϕ r r 2 θ sin θ 5.3 r h2 2m ( 1 r r(r 2 2 r R(r)) + ( h2 2m 1 l(l + 1) + V (r))r(r) = ER(r) (5.75) r2 115

(1)r (2) V (r) l h2 1 l(l + 1) 2m r 2 (3) (5.74) r 2 (1) R(r) r u(r) R(r) = u(r) r (5.76) r R(r) = u (r)r u(r) r 2 (5.77) r (r 2 r R(r)) = r (u (r)r u(r)) (5.78) = u (r)r h2 (u (r) + ( h2 1 l(l + 1) + V (r))u(r) = E u(r) 2m r 2m r2 r r h2 2m u (r) + ( h2 1 l(l + 1) + V (r))u(r) = Eu(r) (5.79) 2m r2 (2) h2 1 l(l + 1) 2m r 2 (3) u(r) 0 r 2 drr(r) 1R(r) 2 = r 0 dru 1(r)u(r) 2 (5.80) r 0 (5.81) 5.3.1 V (r) R(r) = r γ (5.82) 116

γ r γ (γ(γ + 1) l(l + 1))r γ 2 = 0 (5.83) γ = l, l 1 (5.84) γ = l 1 γ = l R(r) = r l (5.85) l 1 l(l + 1) 5.3.2 V (r) u(r) h2 2m r 2 u(r) = Eu(r) (5.86) E 0 u(r) = Ne ±ikr, k = 2mE h (5.87) u(r) = N cos kr, N sin kr (5.88) N 117

R(r) ρ(r) = N 2 1 r 2 (5.89) j(r) = ± N 2 e r v 1 r 2 + O( 1 r 3 ) (5.90) v = hk m r r 2 v ρ(r) R r r 2 R drr 2 sin θdθdϕρ(r) = 4π N 2 R (5.91) 0 R j j ds = ±4π N 2 v (5.92) r=r v = hk m E 0 r R(r) = e kr r, k = 2mE h (5.93) ρ(r) = e 2pr r 2 (5.94) j(r) = 0 (5.95) r ρ(r) R = drr 2 4πρ(r) = 4π N 2 1 (5.96) 0 2p R j ds = 0 (5.97) r=r E 0 118

5.3.3 h2 2m ( 1 r r(r 2 2 r R(r)) + ( h2 2m 1 l(l + 1))R(r) = ER(r) (5.98) r2 2 r R(r) + 2 r rr(r) 1 2mE l(l + 1)R(r) + r2 h 2 R(r) = 0 (5.99) r = cz 2 z R(z) + 2 z zr(z) 1 2mE l(l + 1)R(z) + c2 z2 h 2 R(z) = 0 (5.100) c 2 = ( 2mE h 2 ) 1, c = h p = 1 k z 2 R(z) + 2 z zr(z) + (1 (5.101) l(l + 1) z 2 )R(z) = 0 (5.102) R(z) = j l (z), n l (z) (5.103) j l r l n l r l 1 j l l j 0 (ρ) = sinρ ρ, (5.104) j 1 (ρ) = sinρ ρ 2 cosρ ρ, (5.105) j 2 (ρ) = ( 3 ρ 1 3 ρ )sinρ 3 cos ρ, (5.106) ρ2 n l l n 0 (ρ) = cosρ ρ, (5.107) n 1 (ρ) = cosρ ρ 2 sinρ ρ, (5.108) n 2 (ρ) = ( 3 ρ 1 3 ρ )cosρ 3 sin ρ, (5.109) ρ2 119

R + (r) = j l (kr) + in l (kr) (5.110) R (r) = j l (kr) in l (kr) (5.111) R + (r) l = 0 l = 1 R (r) l = 0; R + (r) = i eikr kr (5.112) l = 1; R + (r) = e ikr 1 kr (1 + i 1 kr ) (5.113) l = 0; R (r) = i e ikr kr (5.114) l = 1; R (r) = e ikr 1 kr (1 + i 1 kr ) (5.115) l 0 r l r l = 0 r Y lm (θ, φ) l m j l n l (5.113) 5.3.4 (5.75) 1 r r2 r 2 5.4 V = V 0, r R (5.116) = 0, r R (5.117) 120

r R R < r h2 2m ( 1 r r(r 2 2 r R(r)) + ( h2 2m 1 p2 l(l + 1))R(r) = R(r) (5.118) r2 2m r = R p 2 2m = (E V 0), r R, (5.119) = E, R < r (5.120) V 0 < 0 V 0 < E < 0 p in = ± 2m(E V 0 ), r R, (5.121) p out = 2mE, R < r (5.122) k in = p in h k out = p out h (5.123) (5.124) r < R R < r N l (k in )j l (k in r), r R, (5.125) M l (k out )(j l (i k out r) + in l (i k out r)), R < r (5.126) r = R 121

N l (k in )j l (k in R) = M l (k out )(j l (i k out R) + in l (i k out R)) (5.127) N l (k in ) r j l(k in r) r=r = M l (k out ) r (j l(i k out r) + in l (i k out r)) r=r (5.128) N l, M l r j l(k in r) r=r j l (k in R) = r (j l(i k out r) + in l (i k out r)) r=r j l (i k out R) + in l (i k out R) (5.129) l = 0 j 0 (kr) = D (5.129) C e kr r k in R cos k in R sin k in R R sin k in R = k outr + 1 R sin kr r j 0 (ikr) + in 0 (ikr) = (5.130) k in R cos k in R sin k in R = k out R + 1 sin k in R (5.131) k in R(tan k in R) 1 1 = k out R + 1 (5.132) tan(k in R) k in R = 1 k out R + 2 (5.133) E E < 0 V 0 tan 2m(E V 0 )R = (5.134) 122