I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1
(d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c) (d) (e) Dirac bra-ket (f) 4. (a) (b) 5. (a) Schrödinger (b) Heisenberg Heisenberg 6. Schrödinger (a) 2
(b) (c) Bloch (d) 7. : 3
[ ] 1. L. I. Schiff; Quantum Mechanics McGraw-hill Kogakusha, okyo (1955) :, ( ) J. J. Sakurai; Modern Quantum Mechanics (Ed. S. F. uan) Benjamin, Menlo Park (1985) :, A. Messiah; Quantum Mechanics (2 vol.) Dover, New York (1981) :, P.A.M. Dirac : he Principles of Quantum Mechanics (4th ed.) Oxford-Misuzu, (1958) :, 2. 1969 R. P. Feynman, R. B. Leighton, M. L. Sands : he Feynman lectures on Physics III Quantum Mechanics (Addison-Wesley, 1965) 4
2010 3. I, Schiff J. J. Sakurai 4.. 5
1 1.1 L U u= L 3 u (1) u( ) 0 ρ(ν, )dν (2) ν ν +Δν ρ(ν, )Δν (3) 1.2 Rayleigh - Jeans 1.2.1 Φ(r) e ik r (4) 2πν = c k ( ) (5) L k = ( ) 2π (n x,n y,n z ) (6) 2L n x,n y,n z =1, 2, 3, 4,... (7) 1.2.2 ρ(ν, )Δν = g(ν) <ɛ>δν (8) g(ν) ν ν +Δν <ɛ> g(ν) (5) (6) 6
( c ) 2 ν 2 = (n 2 2L x + n 2 y + n 2 z) νx 2 + νy 2 + νz 2 (9) Δn i = 1 Δν =(c/2l) (i = x, y, z) (10) Δν = 1 Δn i =(2L/c) (11) g(ν)δν = (ν ) (ν ) = ν (1/8) 2 (2/c) 3 = (4πν 2 ) (Δν) (1/8) 2 (2/c) 3 = 8πν2 Δν. (12) c3 (11) L =1 1/8 ν 2 < ɛ> ɛ e ɛ/kb ɛe ɛ/k B dɛ <ɛ>= e ɛ/k B dɛ = k B (13) Rayleigh-Jeans ρ(ν, )=g(ν) <ɛ>= 8πν2 c 3 k B. (14) 1.2.3 Rayleigh - Jeans 1.3 Wien Wien displacement law ( ) 1.3.1 P = 1 3 u = 1 U 3 (15) 7
) < xx > = E x D x 1 2 E D + H xb x 1 2 H B (16) = 1 { 1 3 2 E D + 1 } 2 H B (17) = 1 3 u (18) xx Maxwell xx x x Stephan - Boltzmann law u = (const.) 4 a 4 (19) ) ΔS = ΔU + P Δ (20) 2 S 2 S = 1 du d, (21) = 4 du 3 d 4u 3 2. (22) du d =4u. (23) L ν 1 L = 1/3. (24) ( ) ν = 1 S 3 ν. (25) ( ) U S ( ) U S U = u= a 4 (26) ( ) = a 4 +4a 3, (27) S = P = 1 3 u = 1 3 a 4. (28) ( ) = 1 S 3. (29) 8
1.3.2 +Δ ν ν + δν +Δ +Δ ν ν +Δν δν δν +Δ(δν) ρ(ν, )δν = P Δδν+( +Δ )ρ(ν +Δν, +Δ )(δν +Δ(δν)). (30) (25) (29) Δ = Δ 3, (31) Δν = ν Δ 3, (32) Δ(δν) = δν Δ 3, (33) ρ(ν +Δν, +Δ ) = ρ(ν, )+ ρ ρ Δν + Δ (34) ν = ρ(ν, ) ν ρ Δ 3 ν ρ Δ 3. (35) ρδν= ρ 3 δνδ +( +Δ ) {ρ ( ν 3 ν ρ 3 ν + ρ 3 ρ(ν, )= (37) m=0 n=0 ρ ν + 3 ) } ( ρ Δ δν 1 Δ ). 3 (36) = ρ, (37) a mn ν m n (38) m + n 3 = 0 (39) ( ν ) m ρ = ν 3 a m (40) m=0 ν 3 φ( ν ). (41) ρ(ν, )=ν 3 φ( ν ). (ρ(ν, ) ) (42) 9
1.3.3 Displacement law Stephan - Boltzmann law (42) ρ(ν, ) ν max ν max (42) displacement law 1.3.4 Wien ρ(ν, )=( ν N(ν)) (ν F (ν)) ρ(ν, )=F(ν)N(ν). (43) E E = f(ν) (44) N(ν) = (const.) e f(ν)/kb (45) ρ = F (ν)e f(ν)/kb = ν 3 φ( ν ). (46) F (ν) = (const.) ν 3, (47) f(ν) = λν. (48) ρ(ν, ) = (const.) ν 3 e λν/kb. (49) ρ(ν, )= 8πν3 c 3 k Bβe βν/. (β = λ/k B ) (50) Wien 1.3.5 Wien Rqyleigh-Jeans Maxwell 10
1.4 Planck S (Rayleigh-Jeans) U = k B ( ) (51) ( ) S U ( 2 ) S U 2 = k B U = 1, (52) = k B U 2 (53) (Wien) ( ) S U ( 2 ) S U 2 = k B hν ln U hν = 1, (54) = k B 1 hν U (55) Planck (53) (55) ( 2 ) S k B U 2 = U(U + hν), (56) ( ) S = k B U + hν ln = 1 U hν U, (57) U Planck 1.5 Wien (57) [ S = k B ln U + hν + U ] U + hν ln. (58) hν hν U Wien (U hν) ( U S k B 1 ln U ). (59) hν hν E ν E(ν, )=8π ν2 U (60) c3 S(ν, ) = 8π ν2 c 3 k U B hν = k B E hν [ 1 ln ( 1 ln U ) (61) hν ( c 3 )] E 8πhν 3. (62) 11
, 0 ( 0 ) E S S 0 = k B hν ln ( ) = k B n ln 0 0 ( ) n = k B ln. (63) 0 n E hν (64) W W 0 = ( 0 S = k B ln W (65) ( ) W S S 0 = k B ln. (66) W 0 ) n. (67) Wien Rayleig-Jeans 1.5.1 (21) (22) ( ) S ( ) S ) ΔS = 1 ΔU + P Δ (68) = 1 ( ) U = 1 ( ) U {( ) S {( ) S } } = = 1 du, )U = u( ) (69) d = u + P = 4u 3. (70) = 4 3 du, (21) (71) d du d 4u. (22) (72) 3 2 4u 3 2 = 1 du 3 d. (73) du d =4u. (74) 12
2 δ 2.1 Dirac δ Dirac δ (1) δ(x) = 0 for x 0, (75) (2) δ(x)dx =1. (76) h(x) x =0 (3) h(x)δ(x)dx = f(0). (77) Dirac δ f(x) x 0 sin gx δ(x) = lim g πx. (78) f(x) f(x) sin gx gx sin gx πx. (79) gx πx = g π (80) g f(0).x 0, f(x) x = π/g g x =0 (75) (76) A e iz I(z) dz (81) z 0 sin x x dx = π 2 (82) (88) B I(α) = 0 e αx sin x dx (83) x 13
di dα = = 0 ( e αx sin x α x 0 = 1 2i 0 1 = α 2 +1 sin xe αx dx ) dx [e (i α)x dx e (i+α)x dx] (84) I(α) = tan 1 α + C. (C : ) (85) (83) I( ) =0 0= π + C. (86) 2 I(α) = tan 1 α + π 2 (87) α =0 (82) sin gx dx = π (88) x f(x)dx = 1 (89) g (76) (78) Dirac δ 2.2 δ ϕ k (x) Ne ikx (90) g ϕ k (x)ϕ k(x)dx = N 2 lim e i(k k )x dx g g = N 2 lim g N =1/(2π) 1/2 2 sin g(k k )x k k (78) = 2πN 2 δ(k k ). (91) ϕ(x) = 1 (2π) 1/2 eikx. (92) 14
3 Dirac bra ket 3.1 x > x >: x ψ(x) < x ψ > ψ > x > ψ(x) (93) ψ x (x) x > ψ x (x) =<x x > (94) ψ x (x) 2 x > x x x ψ x (x) =0(x = x ) ψ x (x) δ(x x ). ψ x (x) =<x x >= δ(x x ). (ket x >) (95) dx x ><x = 1 (96) ψ > = 1 ψ > = = dx x ><x ψ > dxψ(x) x >. (97) ψ x ψ(x) {φ n (x)} F (x) F (x) = c n φ n (x). (98) n=1 c n = = dxφ (x)f (x) dx < φ n x><x F > = <φ n F>. (99) 15
3.2 bra-ket Schrödinger A A ψ > A ψ >= a ψ >. (100) < x <x A ψ > = <x A dx x >< x ψ > ) (96) (100) = = dx <x A x >< x ψ> dx <x A x >ψ(x ). (101) <x A ψ >= a<x ψ >= aψ(x) (102) (101) (102) A = p x = i x ( x ) dx <x A x >ψ(x )=aψ(x). (103) <x p x x >= i x δ(x x ) (104) dx <x p x x >ψ(x )= i ψ(x) =( k)ψ(x) x ) ψ(x) eikx (105) A = H ( ) <x H x >= ( 2 2 ) 2m x 2 + (x) δ(x x ) (106) < x H x > x xx 16