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1 IA

2 Planck Bohr Schrodinger Schrodinger Schrodinger Schrodinger Heisenberg Schrodinger Heisenberg

3 A 41 B Bloch 41 C 4 D Hermite 43 E

4 1 1.1 Planck Newton 3 19 Maxwell Planck Planck Uν, T ) = 8πν n=0 c 3 nhν nhν e kt nhν n=0 e kt = 8πh c 3 ν 3 e hν kt 1, 1) 1 ν c k Boltzmann h h = π = J s, ) h h π ) Planck 1) ν hν E n = nhν, n = 0, 1,, 3, 3) 1905 Einstein 3) ν = ) E = hν Planck 1) T =.7 K Planck 1 1) 1 ω E = nω 4

5 1: b.jpg ) 1. Bohr 1885 Balmer Balmer 1890 Rydberg 1 ν = Rc m 1 ) n, m = 1,, 3,, n = m + 1, m +,. 4) c R = m 1 Rydberg Balmer m = 0 m = 1, 3, 4 Lyman Paschen Blackett Bohr Bohr m e v r = e 4πϵ 0 r, 5) : Balmer File:Emission_spectrum-H.png ) 5

6 m e e ϵ 0 v r Bohr m e vr = n, n = 1,,, 6) r v r = 4πϵ 0 m e e n, v = e 4πϵ 0 n, 7) E n = 1 m ev e 4πϵ 0 r = m ee 4 1 3π ϵ 0 n, 8) E n E m ν = E n E m h = m ee 4 64π 3 ϵ m 1 n ), 9) Rydberg 4) Rydberg R = m e e 4 /64π 3 ϵ 0 3 c Frank Hertz ) 8) 8) 196 Schrodinger Bohr Schrodinger Planck ν hν kt Rayleigh-Jeans ) ν hν kt Wien ) 1) 1). Boltzmann T E e E kt E n = nhν n = 0, 1,, ) 6

7 3. :390 nm :40 nm :450 nm :500 nm :590 nm :600 nm :700 nm hν 4. m e c = 0.5 MeV 5. ) 6. m r, θ) θ p θ Bohr p θ dθ = nh, n = 1,,, Bohr-Sommerfeld ) 7

8 Newton Huygens Huygens Young Maxwell c Maxwell Et, x) t c )E = 0, E = 0, 10) 3 10) e πik x+νt), 11) ν k λ k = 1/λ 10) ν = ±c k e πik x±c k t), 1) ±k c Et, x) = d 3 k Ak)e πik x c k t) + Ak) e πik x c k t)), 13) Ak) k = 0 { 1 )} E = E 0 cos π λ n x νt, n E 0 = 0, 14) ν λ = c/ν n = n 1, n, n 3 ) n λ E 0 n 3 = = i i = n x = n i x i = n 1 x 1 + n x + n 3 x 3 x i x i x i 8

9 . 0 Planck 3) E = nhν hν n Einstein 3: ν 0 ν 1905 Einstein 1 E p E = hν, p = h n, 15) λ 4 ν λ = c/ν n m ev = hν hν 0, 16) hν 0 h Einstein ν hν 4 m E p E = p c + m c 4 m = 0 E = p c 9

10 15) n E E = E 0 cos { 1 )} { 1 π λ n x νt ) } = E 0 cos p x Et, 17) 4: 193 Compton X X Compton X λ X λ hc λ + m ec = hc λ + p c + m ec 4, 18) h λ = h h cos θ + p cos ϕ, λ sin θ = p sin ϕ, 19) λ ϕ p θ λ λ = h 1 cos θ), 0) m e c.3 1. L 0 13). 10

11 Zeeman D ) ) Lorentz 1897 J. J. Thomson Zeeman Lorentz Rutherford proton) 1918 Rutherford neutron) 1935 Chadwick de Broglie J. J. Thomson 1909 Milikan e = C 1 fm = m de Broglie 15) V [V] p m e = ev, 1) 5 Lorentz

12 15) λ = h p = h me ev = m, ) V 100 V m m 197 Davisson Germer G. P. Thomson J. J. Thomson E e/m e 5:. 1 fm ) E m e c 1

13 4 1 Schrodinger 4.1 Schrodinger 1 t x E ν p λ de Broglie { x )} ψt, x) exp πi λ νt E = hν, p = h λ, 3) { i ) } exp px Et, 4) ψt, x) Eψ = i t ψ, pψ = i ψ, 5) x Ê = i t, ˆp = i x, 6) E = p m 5) i t ψ = ψ, 7) m x Schrodinger Schrodinger 7) { i ) ψt, x) exp px Ep)t }, Ep) = p m, 8) ψt, x) de Broglie Schrodinger Born dx ψt, x) = 1, 9) t x x + dx ψt, x) dx Born ψt, x) 13

14 7) ψ 8) ψ 1 ψt, x) = 1 π dp ψp) { i ) exp px Ep)t }, Ep) = p m. 30) ψp) 7) ψp) p = p 0 ψp) Gauss { ψp) = A exp 1 4σ p p 0) }, A = 1 π) 1/4, 31) σ1/ 30) p = p 0 + p ψt, x) = A π = A π { dp exp p { i dp exp Ep 0 ) + p 0 m p + p ) } t m 4σ + i p0 + p ) x i p0 x Ep 0 )t ) + i x vg t ) p a p } = A { i exp p0 x Ep 0 )t )} { exp x v gt) } π a { dp exp a p i x v ) gt }, a = A { i exp p0 x Ep 0 )t )} { exp x v gt) } a, 3) a v g de dp p 0) = p 0 m, a 1 σ + it m, 33) ψ = A { a exp x v gt) } σ a, 34) v g v g Ep 0 )/p 0 x = v g t 34) x = v g t x = σ a ψ p = p 0 p = σ x p = σ a = σ t ) m, 35)

15 4. 1 Schrodinger V x) E = p + V x), 36) m 5) 36) i ψt, x) = Ĥψt, x), t Ĥ = + V x), 37) m x V x) Schrodinger ψt, x) ψ dx ψt, x) = 1, 38) ρ = ψ x Born t x ρt, x) = ψt, x) ψt, x) Schrodinger ρ ρ t ψ = ψ t + ψ t ψ = i ψ ψ m x ψ ) x ψ = { i x m ψ ψ x ψ x ψ )}, 39) ) j i ψ ψ m x ψ ) x ψ, 40) 39) ρ t + j = 0, 41) x ψ ψ x 15

16 4.3 ψt, x) Born t xt) = dx ψ t, x)ˆxψt, x), ˆx = x, 4) ψt, x) pt) = dx ψ t, x)ˆpψt, x), ˆp = i x. 43) Fourier pt) = 1 π = = dx dp dp dp ψ t, p )p ψt, p)e i p p )x dp ψ t, p )p ψt, p)δp p ) dp ψ t, p)p ψt, p), 44) x p ˆx ψt, p) = i p ψt, p), ˆp ψt, p) = p ψt, p), 45) [ˆx, ˆp] = i, 46) x p x) x x ) = x x, p) p p ) = p p, 47) ẑ = tˆx x ) + iˆp p ) z z = t x) t + p) 0 x p, 48) 16

17 Fourier ψt, x) Fourier ψt, x) = A dp ψt, p)e i px, 49) A p Dirac δx x ) = 49) ψt, p) = 1 πa dp π e i px x ), 50) dx ψt, x)e i px, 51) A dx ψ t, x)ψt, x) = dp ψ t, p) ψt, p), 5) A = 1 π Gauss Gauss P x) = 1 πσ e x x 0 ) σ 53) Gauss x x x = x x 0 ) = x x 0 ) x x 0 ) = dx P x) = 1, 54) dx xp x) = x 0, 55) dx x x 0 ) P x) = 1 πσ dx x e x σ = σ, 56) x x 0 x x x 0 ) = σ 17

18 dp e ap = π a.. 3. t = 0 dp p e ap = 1 π a a. ψx) = Ae x 4σ, A x = x 4. Fourier ψp) = 1 π p = p dx ψx)e i px, { i ) } { i ) } ψt, x) = A exp px Ep)t + B exp px Ep)t, ) 7. ψt, x) = 1 π dp ψp) { i ) } exp px Ep)t, ) 8. Schrodinger d dt x = 1 m p 9. Schrodinger d dv dt p = dx Ehrenfest ) 18

19 5 5.1 E ψt, x) Ĥψ = Eψ ψt, x) = e i Et ϕx), 57) Schrodinger 37) ϕx) d ϕ dx = m E V )ϕ, 58) E 40) x jx) = i ϕ dϕ m dx dϕ ) dx ϕ, 59) V = V 0 ) E V 0 > 0 58) ϕx) = Ae i px + A e i px, p = me V 0 ), 60) A A 59) jx) = p m A p m A, p = me V 0 ), 61) A x p/m A x p/m V 0 E E V 0 < 0 ϕx) = Be ρx + B e ρx, ρ = mv0 E), 6) B B 59) jx) = iρ m B B B B ) mv0 E), ρ =, 63) x ± ϕ 0 B = B = 0 19

20 5. [ V 0 < E < 0 ] 6 E 6: Schrodinger 58) Ae ρx + A e ρx, x 0, ϕx) = Be ipx + B e ipx, 0 x a, 64) Ce ρx + C e ρx, a x, A, A, B, B, C, C ρ p me me + V0 ) ρ =, p =, 65) Born ψ = ϕ ) x x = ± A = C = 0 x = 0 x = a A = B + B, 66) Be ipa + B e ipa = C e ρa, 67) x = 0 x = a = ρa = ipb B ), 68) ipbe ipa B e ipa ) = ρc e ρa, 69) 0

21 66) 68) A B A = 1 1 i ρ ), p B A = i ρ ), 70) p 67) 69) A C A ρ = cos pa + ρ p sin pa ρ p sin pa = C A e ρa, 71) cos pa = C A ρ p ρ p ρ p e ρa, 7) ±1 cos pa =, 73) sin pa mv 0 p p E : 73) mv 0a = 8π) [0 < E ] 6 E x = x = Schrodinger 58) Ae ikx + A e ikx, x 0, ϕx) = Be ipx + B e ipx, 0 x a, 74) Ce ikx + C e ikx, a x, 1

22 A, A, B, B, C, C k p k = me, p = me + V0 ), 75) x = x = C = 0 x = 0, a A + A = B + B, 76) Be ipa + B e ipa = Ce ika, 77) x = 0, a 76) 78) ka A ) = pb B ), 78) pbe ipa B e ipa ) = kce ika, 79) B = A B = A 77) 1 + k p 1 k p ) + A ) + A 1 k ), p 1 + k ), 80) p C = A cos pa + i k )e p sin pa ika + A cos pa i k ) p sin pa e ika, 81) B, B, C 79) A i sin pa + k ) p cos pa + A i sin pa k ) p cos pa = k p A cos pa + i k p sin pa ) + k p A cos pa i k p sin pa ), 8) A A = 81) C A = ) i 1 k p sin pa k p cos pa i ), 83) 1 + k p sin pa k p e ika k p cos pa i 1 + k p ) sin pa, 84)

23 ) 1 k p sin pa j r j i = j t j i = 4k p 4k p cos pa k p ) sin pa, 85) 4k p cos pa k p ) sin pa, 86) x = pa = nπ n = 1,, ) p λ p = π/λ a = nλ/ 5.3 [0 < E < V 0 ] 8 0 < E < V 0 x = 8: Schrodinger 58) Ae ikx + A e ikx, x 0, ϕx) = Be ρx + B e ρx, 0 x a, Ce ikx + C e ikx, a x, 87) A, A, B, B, C, C ρ k ρ = mv0 E), k = me, 88) 3

24 x = 40) E m A A ), x 0, V jx) = i 0 E) m B B B B ), 0 x a, E m C C ), a x, x 0 x j i = E/m A x j r = E/m A a x x j t = E/m C x = C = 0 0 x a x = 0 x = a 89) A + A = B + B, 90) Be ρa + B e ρa = Ce ika, 91) x = 0 x = a 90) 9) ika A ) = ρb B ), 9) ρbe ρa B e ρa ) = ikce ika, 93) B = A B = A 91) 1 + i k ρ 1 i k ρ ) + A ) + A 1 i k ), ρ 1 + i k ), 94) ρ C = A cosh ρa + i k )e ρ sinh ρa ika + A cosh ρa i k ) ρ sinh ρa e ika, 95) B, B, C 93) A sinh ρa + i k ) ρ cosh ρa + A sinh ρa i k ) ρ cosh ρa = i k ρ A cosh ρa + i k ρ sinh ρa ) + i k ρ A cosh ρa i k ρ sinh ρa ), 96) A A = k ρ + 1 ) sinh ρa k 1 ) sinh ρa + i k 97) ρ ρ cosh ρa, 4

25 95) C A = j r j i = i k ρ e ika k 1 ) sinh ρa + i k 98) ρ ρ cosh ρa, k + 1 ) ρ sinh ρa j t j i = k ρ 1 ) sinh ρa + 4k ρ cosh ρa, 99) 4k ρ k ρ 1 ) sinh ρa + 4k ρ cosh ρa, 100) 0 α α ) 8 V 0 < E 5. V 0 V a V x) V x) = σ 0 δx na), 101) n= 9 ) 58) V x) = V x a) ϕx) = ϕx a) ϕ ϕx) = e iθ ϕx a), 10) θ Bloch 7 Bloch a x a Ae ipx + A e ipx, a x 0, ϕx) = e iθ Ae ipx a) + A e ipx a)), 0 x a, 103) 7 ϕx) ϕx a) 1 ϕx) ϕx a) ϕ Bloch 10) 5

26 9: 1 x = 0 A + A = Ae iθ e ipa + A e iθ e ipa, 104) x = 0 ϵ x ϵ ϵ 0 dϕ dx 0 +) dϕ dx 0 ) = mσ 0 ϕ0), 105) 103) Ae iθ e ipa A e iθ e ipa A + A = i mσ 0 p A + A ), 106) 104) 106) A A A = 1 + eiθ e ipa 1 e iθ e ipa A = 1 eiθ e ipa mσ i 0 p 1 e iθ e ipa + i mσ A, 107) 0 p e iθ + 1 = e iθ cos pa + mσ 0 p sin pa ), cos θ = cos pa + mσ 0a sin pa pa, 108) 1 cos θ 1 pa 1 cos pa + mσ 0a sin pa 1, 109) pa 10 pa E ) 6

27 : pa mσ 0a = Ĥϕ 1 = Eϕ 1 Ĥϕ = Eϕ ϕ 1 = cϕ c ). V x) = V x) 3. 11a) 4. 11b) V x) = σ 0 δx) x = c) 0 < E < V V 0 < E 8. a) b) c) 11: 7

28 6 6.1 V x) = 1 mω x, 110) m ω Shrodinger 37) 58) d ϕ me dx + m ω x ) ϕ = 0, 111) 111) 111) y = mω ) 1 x, 11) d ϕ dy + E ω y) ϕ = 0, 113) y ± 0 y ± ϕy) e 1 y ϕy) ϕy) = Hy)e 1 y, 114) 113) Hy) d H dy dh E ) y dy + ω 1 H = 0, 115) 0 Hy) e 1 y Hy) Hermite Hermite ϕx) = 1 n πn! E n = ω n + 1 ), 116) mω ) 1 4 H n y)e 1 y, y = mω ) 1 x, 117) 8

29 : n = 0, 1,, 3 d H n dy Hermite y dh n dy + nh n = 0, 118) Hermite n = 0, 1,, H 0, H 1, H, H n y) = i=0 c iy i i + )i + 1)c i+ = n i)c i n c i c n 0 dn H n y) = 1) n e y dy n e y ), 119) H 0 = 1, H 1 = y, H = 4y, 10) H n dyh m y)h n y)e y = δ mn n πn!, 11) 9

30 6. V x) = 1 mω x E n = ωn + 1 ) Shrodinger ϕx) = C n H n y)e 1 y C n y = mω/) 1/ x Hermite H n n dyh my)h n y)e y = δ mn n πn! 1. H n y) Hermite d H n dy ) y dh n dy + nh n = 0. H n y) = a=0 c ay a c a+ = a n) a+)a+1) c a n = 0, 1,, c a Hy) e y 3. C n 4. Hermite H 0 y) E 0 x) = ˆx ˆx 5. H 1 y) x) 6. H y) x) 7. H 3 y) x) 8. H n y) = 1) n e y d n dy n e y ) ) yh n+1 = n + 1)H n + H n+ 9. H n y) Hermite 10. m < n dyym H n y)e y = dyyn H n y)e y = πn! 1. Hermite 30

31 x m V x) mẍ = dv dx, 1) 1) Lagrangian Lx, ẋ) Lx, ẋ) = 1 mẋ V x), 13) action S[x] Lagrangian S[x] = t t 1 dt Lx, ẋ), 14) action S[x] x 1) x xt) xt)+δxt) δxt 1 ) = δxt ) = 0 13 ) t L L ) t { L 0 = δs[x] = dt δx + t 1 x ẋ δẋ = dt t 1 x d L )} δx, 15) dt ẋ δx L x d L ) = 0, 16) dt ẋ Euler-Lagrange Lagrangian 13) 1) 13: 31

32 x, ẋ) Euler-Lagrange x, p) Hamiltonian x p Hamiltonian Hx, p) Hamiltonian δh = δpẋ + pδẋ L x p L ẋ, 17) Hx, p) pẋ Lx, ẋ), 18) δx L ẋ L δẋ = δpẋ δx = δpẋ ṗδx, 19) x H x p 16) 17) ẋ = H p, ṗ = H x, 130) x p Ox, p) Ȯ = O x ẋ + O p ṗ = O H x p H O x p {O, H} P.B., 131) Poisson 7. Schrodinger Heisenberg Schrodinger ψt, x) Ô Schrodinger 37) ψt, x) = e i Ĥt ψ0, x), 13) Ô O = dx ψt, x) Ôψt, x) = ÔH dx ψ0, x) e i Ĥt Ôe i Ĥt ψ0, x), 133) Ô H e i Ĥt Ôe i Ĥt, 134) d = i [ÔH, Ĥ], 135) dtôh Heisenberg ψ H x) ψ0, x), 136) 3

33 ÔH O = dx ψ H x) Ô H t)ψ H x), 137) Heisenberg 131) 135) Heisenberg {O 1, O } P.B. i [Ô1, Ô], 138) {x, x} P.B. = 0 i [ˆx, ˆx] = 0, {x, p} P.B. = 1 i [ˆx, ˆp] = 1, 139) {p, p} P.B. = 0 i [ˆp, ˆp] = 0, ˆx ˆp Heisenberg Born ˆx ˆp Heisenberg {x n, p } P.B. [ˆx n, ˆp ]. Heisenberg 3. [ˆx, ˆp] = i1 ˆx ˆp 33

34 Schrodinger 8 Dirac x ˆx x = x x, 140) x x x x = δx x ), 141) ψ ψ = dx x ψt, x), 14) ψ = x x ψ = dx ψ t, x) x, 143) dx x x ψt, x ) = ψt, x), 144) x ψ x = dx ψ t, x ) x x = ψ t, x), 145) 8 34

35 dx x x ψ = dx x ψt, x) = ψ, 146) dx x x = 1 147) ˆp p ˆp p = p p, p p = δp p ), ψ = dp p ψt, p), ψ = ψt, x) = 49) dp x p p ψ = dp p p = 1, dp ψ t, p) p, 148) dp x p ψt, p), 149) x p = 1 π e i px, 150) ϕ ψ ϕ ψ = = dx dx ϕ t, x ) x x ψt, x) dx ϕ t, x)ψt, x) = ψ ϕ, 151) ϕ λ 1 ψ 1 + λ ψ ) = λ 1 ϕ ψ 1 + λ ϕ ψ, λ 1 ϕ 1 + λ ϕ ) ψ = λ 1 ϕ 1 ψ + λ ϕ ψ, 15) ψ ψ = ψ = 0 dx ψ t, x)ψt, x) 0, 153) 35

36 8. Ôˆx, ˆp) ˆx ˆp Ôˆx, ˆp) ψ = = dx x Ox, i x )ψt, x) 154) dp p Oi p, p) ψt, p), 155) ψ Ôˆx, ˆp) = = dx O x, i x )ψ t, x) x 156) dp O i p, p) ψ t, p) p, 157) Ô Ô Schrodinger i ψ = Ĥ ψ, t Ĥ = ˆp + V ˆx), 158) m x ψ Ô O ψ = ψ Ô ψ = dx ψ t, x)ox, i x )ψt, x), 159) O ψ = dx O x, i x )ψ t, x)ψt, x) = ψ Ô ψ, 160) Ô Ô Ô = Ô ˆx ˆp Ĥ ˆp = i x ψ p ψ = = dx i x ψ t, x))ψt, x) dx ψ t, x) i x ψt, x)) = ψ p ψ, 161) ˆp Ô1 Ô Ô1Ô) = Ô Ô 1, 16) 36

37 ψ Ô1Ô) = = = ψ Ô dx O 1 O ψ) x dx O 1O ψ) x )Ô 1, 163) Ô1 Ô [Ô1, Ô] = 0, 164) m Ô1 n Ô 1 m j = m m j, j = 1,, n 165) Ô 1 Ô m j ) = Ô Ô 1 m j = m Ô m j ), 166) Ô m j Ô1 m Ô m j m j Ô m j = n m i C i j, 167) i=1 C C i j n n P 1 CP = D D P Ô m j P j ) i = m k P k jp 1 CP ) j i = d i m j P j i), 168) j j,k j Ô Ô1 Ô Ô1 Ô 8.3 Schrodinger Heisenberg Schrodinger 158) ψt) = e i Ĥt ψ0), 169) Ô O = ψt) Ô ψt) = ψ0) e i Ĥt Ôe i Ĥt ψ0), 170) 37

38 Ô H t) = e i Ĥt Ôe i Ĥt, ψ H = ψ0), 171) O = H ψ ÔHt) ψ H, 17) Schrodinger Heisenberg Heisenberg d = i [ÔH, Ĥ], 173) dtôh d dt O = {O, H} P.B. O H x p H O x p, 174) 8.4 ˆx ˆp mω 1 â = ˆx + i mω ˆp, â = mω 1 ˆx i ˆp, 175) mω Ĥ = ω â â + 1 ), 176) â â ˆN = â â ˆN n [â, â ] = 1, 177) [ ˆN, â] = â, [ ˆN, â ] = â, 178) ˆN n = n n, 179) 38

39 n Ĥ n = ω n + 1 ) n, 180) ω n + 1 ) 178) ˆNâ n = n 1)â n, ˆNâ n = n + 1)â n, 181) â ˆN 1 â ˆN 1 â â â ω â 0 179) 0 â 0 = 0, n = 1 n! â ) n 0, n = 0, 1,,, 18) 0 0 = 1 ϕ n x) = x n â = 1 y + d ), â = 1 y d ) = 1 e 1 d y dy dy dy e 1 y, 183) y = mω x x â 0 = 0 ϕ 0 x) 1 y + d ) ϕ 0 x) = 0, 184) dy ϕ 0 x) = C 0 e 1 y, C 0 = mω π ) 1 4, 185) C 0 dxϕ 0 x) = 1 ϕ n x) = 1 n! x â ) n 0 n ϕ n x) ϕ n x) = 1 1 e 1 d y n! dy e 1 y) n ϕ0 x) = 1 n n! C 0 1) n e 1 y dn dy n e y, 186) 117)

40 3. n â = 1 y d dy ) â = 1 y + d dy ) y = mω x x â 0 = 0 ϕ 0 x) = x = â 0 1 ϕ 1 x) = x 1 5. n = 1 n! â ) n 0 n ϕ n x) = x n â = 1 y d dy ) = 1 e 1 y d dy e 1 y ) 40

41 A SI c = m s 1 h = ) J s e = ) C c = ) MeV fm m e = ) MeV/c = ) kg m p = ) MeV/c = ) 10 7 kg a 0 = 4πϵ 0 /m e e ) = ) m k = ) 10 3 J K 1 = ) 10 5 ev K 1 µ B = e/m e ) = ) MeV/T B Bloch 1 V x) = V x a) ϕx) 1 ϕx) = me V x)) ϕx) dx = m E V x a)) 1 d = ϕx a), 187) ϕx a) dx d 0 = d { ϕx a) dϕx) dx dx a) } ϕx)dϕx, 188) dx C = ϕx a) dϕx) dx dϕx + a) = ϕx) dx ϕx)dϕx a) dx ϕx + a) dϕx) dx, 189) 1 dϕx + a) + ϕx a)) = 1 dϕx) ϕx + a) + ϕx a) dx ϕx) dx, 190) 41

42 ϕx + a) + ϕx a) = Dϕx) ϕx + a) λ ϕx) = λ + ϕx) λ ϕx a) ), λ ± D ± D 4, φx) = λ + φx a), φx) ϕx + a) λ ϕx), 191) D φx) φx) = λ n +φx na) λ + = 1 λ + = e iθ Bloch φx) = e iθ φx a), 19) C 3 ) ds = dr + r dθ + r sin θdϕ, 193) g ij g ij = 0 r 0, g ij = r sin r, 194) 1 θ 0 0 r sin θ i, j = r, θ, ϕ g ij g ij g g = r sin θ = 1 g i gg ij j ), 195) = 1 { 1 r r r sin θ r ) + θ sin θ θ ) + ϕ sin θ = r + r r + 1 r θ + cos θ r sin θ θ + )} sin θ ϕ 1 r sin θ ϕ. 196) 4

43 D Hermite Hermite d H n dy y dh n dy + nh n = 0, 197) H n y) = a=0 c ay a 0 = aa 1)c a y a ac a y a + n c a y a = a= a=1 { } a + 1)a + )ca+ + n a)c a y a, 198) a=0 a=0 n a) c a+ = a + 1)a + ) c a, 199) n 0, 1,, a c a+ a c a H n e y n = 0, 1,, Hermite Hermite Hermite dn H n y) = 1) n e y dy n e y ), 00) yh n+1 = n + 1)H n + H n+, dh n dy = yh n H n+1, 01) dye y y m H n y) = 0, m < n), dye y y n H n y) = πn!, 0) dyh m y)h n y)e y = δ mn n πn!, 03) 43

44 E 3 3 Ĥ = 3 i=1 ˆp i m + mω ) ˆx i, 04) Ψx i ) 3 i=1 m i + mω ) x i Ψ = EΨ, 05) Ψx i ) = ϕ 1 x 1 )ϕ x )ϕ 3 x 3 ) m d dx i + mω ) x i ϕ i x i ) = E i ϕ i x i ), i = 1,, 3, 06) E = E 1 + E + E 3 ϕ i x i ) 1 ϕ i x i ) = B n ini! H n i Ax i )e A x i mω, A ) 1, B mω π ) 1 4, 07) E n = ω n + 3 ), n = n 1 + n + n 3, n i = 0, 1,, 3,, 08) n 1, n, n 3 ) E 0 = 3 ω Ψ 0,0,0) 1 E 1 = 5 ω Ψ 1,0,0), Ψ 0,1,0), Ψ 0,0,1) 3 E n n 3 n+c = n + 1)n + ), 09) 44

Planck Bohr

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