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3 3 I ( ) ( ) Rayleigh-Jeans Einstein Planck Debye α
4 Zeeman α Bohr Bohr de Broglie de Broglie
5 I
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7 X 19 ( ) ( ) MRI ( ) (Planck 1 ) (1900) E = nhν (1.1) 1 Max Karl Ernst Ludwig Planck ( ) 1918
8 8 1 (E ν n 0 h ) h ( [ ] [ ] 2 [ ] 1 = [ ] [ ] = [ ] [ ] ) h = J s (1.2) 1. Stefan-Boltzmann J. Stefan (1879) L. Boltzmann 2 (1884) σ Stefan-Boltzmann 2. Wien (1893) j = σt 4 (1.3) W m 2 K 4 W. Wien 3 λ max λ max = b T (1.4) 2 Ludwig Eduard Boltzmann ( ) 3 Wilhelm Wien ( ) 1911
9 Wien (1896) Wien ( ) U(ν)dν = 8πk Bβ c 3 exp( βν/t )ν 3 dν (1.5) k B Boltzmann k B = J/K β 4. Rayleigh-Jeans (1900,1905) Rayleigh 4 U(ν)dν = 8πk BT c 3 ν 2 dν (1.6) 5. Planck Planck Wien Rayleigh-Jeans U(ν)dν = 8πh c 3 1 exp(hν/k B T ) 1 ν3 dν (1.7) (1900 ) 4 John William Strutt Rayleigh ( ) 1904
10 Planck Planck ( ) U(ν)dν = 8π ( ν ) c F ν 3 dν (1.8) 3 T F Wien Planck F (x) = k B β exp( βx) F (x) = h = k B β k B β exp(βx) 1 7. Planck Planck ( ) 8. Rayleigh-Jeans Wien Planck 2 (Einstein 5, 1905) E = 1 2 mv2 = hν W (1.9) : 5 Albert Einstein ( ) 1921
11 Hertz 6 (1887) 2. Hallwacks (1888) Hallwacks 7 3. J.J.Thomson 8 (1899) (e/m) 4. Lenard (1902) Lenard 9 (a) 1 (b) (c) 1 () ( ) 5. Einstein (1905) 6 Heinrich Hertz ( ) ( ) 7 Wilhelm Ludwig Franz Hallwachs ( ) 8 Joseph John Thomson ( ) Philipp Eduard Anton von Lénárd ( ) 1905
12 Millikan (1916) Millikan 10 Einstein Einstein (1.9) h Planck h Einstein p = h/λ (1.10) (p λ ) Compton 11 X X (Compton) (1923) CCD Wilson, Bothe Dulong 12 -Petit 13 (1819) C V = 3R (1.11) 10 Robert Andrews Millikan ( ) Arthur Holly Compton( ) Pierre Louis Dulong ( ) ( ) 13 Alexis Thérèse Petit ( )
13 Einstein (1907) Dulong-Petit 3. Debye 14 (1912) Einstein Balmer 15 (1885) h, h, h, h, h = m λ = n2 h n = 3, 4, 5, 6, (1.12) n Peter Joseph William Debye ( ) Johann Jakob Balmer ( ) ( )
14 14 1 Rydberg 16 1 λ = 1 R n: (1.13) λ (n + b) 2 (1888) λ Rydberg R (Rydberg ) R = m 1 Balmer R = 4/h b Rydberg 1 λ = R ( 1 (m + a) 2 1 (n + b) 2 ) n, m: (1.14) Rydberg (1890) Rydberg a = b = 0 ( 1 1 λ = R m 1 ) 2 n 2 (1.15) Balmer m = 2 m = 1 Lyman 17 (1906) m = 3 Paschen (1908), C. H. F. Paschen) m = 4 Blackett (1922) ( ) (de Broglie) (1923) (Davisson, Germer) (Ni 16 Johannes Robert Rydberg ( ) 17 Theodore Lyman ( )
15 )(1927) (1928) p k(= 2π λ ) = p ħ (1.16) k ( λ) ( ) (Bonse, 1974) (Zeeman) (1896)( ) (Stern, Gerlach) (1922) : ( ) NMR,MRI:
16
17 q 1, q 2,, q f p 1, p 2,, p f (f ) {q} {p} {q} {p} E(q 1, q 2,, q f ; p 1, p 2,, p f ) q 1 q 1 + dq 1 2 q 2 q 2 + dq 2, f q f q f + dq f p 1 p 1 + dp 1, f p f p f + dp f
18 18 2 P (q 1, q 2,, q f ; p 1, p 2,, p f )dq 1 dq 2 dq f dp 1 dp 2 dp f ( = A exp 1 ) k B T E(q 1, q 2,, q f ; p 1, p 2,, p f ) dq 1 dq 2 dq f dp 1 dp 2 dp f (2.1) ( Liouville ) A ( A 1 = exp 1 ) k B T E(q 1,, q f ; p 1,, p f ) dq 1 dq f dp 1 dp f (2.2) T k B k B = (13) JK ( ) T k B T/2 E kin = α 1 p α s p 2 s + + α f p 2 f (2.3) α s p 2 s s α s ( ) s
19 α s p 2 s α s p 2 s = A α s p 2 s exp ( = A α s p 2 s exp ( 1 ) k B T E(q 1,, q f ; p 1,, p f ) ( f )) 1 k B T s=1 α s p 2 s + V dq 1 dq f dp 1 dp f dq 1 dq f dp 1 dp f (2.4) V {q} ( I = α s p 2 s exp 1 ) k B T α sp 2 s dp s (2.5) I = k BT p s exp 2 p s = k BT exp 2 ( 1 ( 1 k B T α sp 2 s ) dp s k B T α sp 2 s ) dp s (2.6) ( ) α s p 2 s = A k ( ( f )) BT exp 1 α s p 2 s + V dq 1 dq f dp 1 dp f 2 k B T s=1 A (2.2) (2.7) α s p 2 s = 1 2 k BT (2.8) (2.8) 1 f s T
20 20 2 ( E = E kin = N A i=1 1 2m (p2 ix + p 2 iy + p 2 iz) (2.9) m p ix, p iy, p iz i x, y z N A 1 ( ) N A = (27) mol 1 (2.10) (2.9) (2.3) 1 2m p2 ix = 1 2m p2 iy = 1 2m p2 iz = 1 2 k BT (2.11) E kin = 1 2m ( p2 ix + p 2 iy + p 2 iz ) = 3 2 k BT (2.12) 1 U = 3 2 N Ak B T (2.13) R = N A k B R U = 3 RT (2.14) 2 C V = 3 2 R (2.15)
21 ( ) 2 2 x, y, z θ ϕ 5 p x, p y, p z, p θ, p ϕ I H = 1 2m (p2 x + p 2 y + p 2 z) + 1 ( ) p 2 θ + p2 ϕ 2I sin 2 (2.16) θ (2.3) 1 E = 1 2m ( p2 x + p 2 y + p 2 z ) + 1 ( p 2 ) p 2 ϕ 2I θ + sin 2 θ = 5 2 k BT (2.17) U = 5 RT (2.18) 2 C V = 5 2 R (2.19) s q s, p s E s = a s q 2 s + b s p 2 s (2.20)
22 : 1 2 b s p 2 s = 1 2 k BT (2.21) E s = a s q 2 s + b s p 2 s = k B T (2.22) 1 1 f E = fk B T (2.23) 1 N A f = 3N A 6 (2.24)
23 N A N A 6 3, 3 N 1 f = 3N A (2.25) 1 U = 3N A k B T = 3RT (2.26) C = 3R (2.27) Rayleigh-Jeans Maxwell f = 3N A ν ν + dν Z(ν)dν L L L 3 1 L ν ν + dν L 2L, 2L/2, 2L/3,, 2L/s, c ν s = s c, s = 1, 2, (2.28) 2L
24 24 2 = c/2l ν ν + dν Z(ν)dν = dν = 2L c dν (2.29) 3 3 s x, s y, s z (2.28) c ν sx,s y,s z = s 2 x + s 2 y + s 2 z (2.30) 2L x, y, z x = c 2L s x, y = c 2L s y, z = c 2L s z (2.31) s x, s y, s z c 3 /(2L) 3 r = c x 2 + y 2 + z 2 = s 2 x + s 2 y + s 2 z (2.32) 2L r ν sx,s y,s z ν ν + dν xyz ν ν + dν 2 s x, s y, s z 1/8) 2 Z(ν)dν 3 = c 3 /(2L) 3 4πν 2 dν/8 Z(ν)dν = πν2 dν 2 3 = 4πL3 c 3 ν 2 dν (2.33) c 2 Z(ν) (2.33) 2 Z(ν)dν = 8πL3 c 3 ν 2 dν (2.34)
25 k B T ν ν + dν E(ν)dν = Z(ν)k B T dν = 8πL3 c 3 k B T ν 2 dν (2.35) Rayleigh-Jeans U(ν) = 8πk BT c 3 ν 2 dν (2.36) 2.2 A exp( E n /k B T ) (2.37) A 1 = n exp( E n /k B T ) (2.38) Einstein N 3 3 3N ν (2.20) ν s = a s b s /π ϵ n = nhν (n = 0, 1, 2 ) (2.39)
26 : Einstein ϵ = A nhν exp( nhν/k B T ) n=0 = A x (exp( hνx)) n n=0 = A 1 x 1 exp( hνx) hν exp( hνx) = A (1 exp( hνx)) 2 (1/k B T = x ) = A hν exp( hν/k BT ) (1 exp( hν/k B T )) 2 (2.40) n k=0 ar k = a(1 rn+1 ) 1 r (r 1) 1/A = exp( nhν/k B T ) = n=0 1 1 exp( hν/k B T ) (2.41)
27 ϵ = hν exp( hν/k BT ) 1 exp( hν/k B T ) = hν exp(hν/k B T ) 1 (2.42) 1 E = 3N A ϵ = 3N A hν exp(hν/k B T ) 1 (2.43) 1. T exp(hν/k B T ) 1 + hν/k B T E 3N A hν (1 + hν/k B T ) 1 = 3N Ak B T (2.44) Dulong-Petit 2. T 0 exp(hν/k B T ) 1 E 3N A hν exp(hν/k B T ) = 3N Ahν exp( hν/k B T ) 0 (2.45) (Debye ) Einstein Planck ν ν + dν (2.34) Z(ν)dν = 8πL3 c 3 ν 2 dν (2.46)
28 28 2, (2.42) ν ν + dν E(ν)dν = hν Z(ν)dν (2.47) exp(hν/k B T ) 1 U(ν)dν = 8πh c 3 1 exp(hν/k B T ) 1 ν3 dν (2.48) Planck Debye
29 Loschmidt 1 (1865) (Avogadro constant) A. Einstein (1905) J. Perrin 2 (1908) J. J. Thomson e/m (1897) Millikan Fletcher 3 e ( ) Goldstein 4 (1886) J. J. Thomson Goldstein 1 Johann Josef Loschmidt ( ) 2 Jean Baptiste Perrin ( ) (1926) 3 Harvey Fletcher ( ) 4 Eugen Goldstein ( )
30 30 3 ( ) W. Wien e/m (1898) E. Rutherford α α α Rutherford 1898 α β 1899 α β 1902,1903 Rutherford α Chadwick Zeeman Zeeman 6 Zeeman D ( ) H. A. Lorentz Larmor 7 e/m e/m 5 Ernest Rutherford ( ) Pieter Zeeman ( ) Joseph Larmor ( )
31 α Geiger 8 Marsden 9 Rutherford 1909 α (ZnS) α α α Rutherford J = pdq (3.1) ( ) Delaunay 10 1, m F = kq (3.2) H = p2 2m + k 2 q2 (3.3) dq dt = H p = p m, (3.4) dp dt = H = kq q (3.5) 8 Johannes (Hans) Wilhelm Geiger ( ) 9 Ernest Marsden ( ) 10 Charles-Eugène Delaunay ( )
32 32 3 q = A sin(ωt + α), (3.6) p = mωa cos(ωt + α) (3.7) A α ω( 2πν) ω = k m (3.8) E = p2 2m + k 2 q2 (mωa cos(ωt + α))2 = 2m + mω2 (A sin(ωt + α))2 2 = 1 2 mω2 A 2 (3.9) ( ) pdq = mωa cos(ωt + α)d(a sin(ωt + α)) 2π/ω = mω 2 A 2 cos(ωt + α) 2 dt = mω 2 A = 2πE ω 0 = E ν 2π ω (3.10) 3.3 Bohr Bohr Bohr 11 (1913) 11 Niels Henrik David Bohr ( ) 1922
33 3.3. Bohr E 1, E 2, E 3, ν hν = E m E n (3.11) (quantum condition) pdq = nh (n = 0, 1, 2, 3, ) (3.12) q p Bohr 1913 de Broglie ( (3.29) ) Bohr (3.12) (3.10) E ν = nh (3.13)
34 34 3 E = nhν (3.14) ħ h 2π (3.15) E = nħω (3.16) ω r p q pdq = p 2πr = nh (3.17) e e e 2 4πϵ 0 r 2 (3.18) m v2 r = p = mv 2πpr = nh n = 1 e2 4πϵ 0 r 2 (3.19) r = 4πϵ 0ħ 2 me 2 n2 (3.20) a = 4πϵ 0ħ 2 me 2 (3.21)
35 3.4. de Broglie 35 E p 2 /2m e 2 /4πϵ 0 r E = p2 2m e2 4πϵ 0 r (3.22) r = n 2 a e2 E = (3.23) 8πϵ 0 an 2 m n ν ( hν = e2 1 8πϵ 0 a n 1 ) 2 m 2 (3.24) Rydberg R = e 2 8πhϵ 0 ac = me4 8h 3 ϵ 2 0c (3.25) 3.4 de Broglie ν λ) E p E = hν, p = h λ (3.26) ( h = [J s] 0 E = c p (1) ν = E h, λ = h p (3.27) de Broglie 12 (1924) 12 Louis de Broglie ( ) 1929
36 de Broglie Bohr de Broglie r de Broglie λ λ = h/p 2πr = nλ (n = 0, 1, 2, 3, ) (3.28) pr = nħ (3.29) pr ħ Bohr Bohr Heisenberg 13 (1925) de Broglie Schrödinger 14 (1926) (?) Lanczos 15 ( ) Schrödinger,Lanczos,Pauli 16 Schrödinger Heisenberg 13 Werner Heisenberg ( ) Erwin Rudolf Josef Alexander Schrödinger ( ) Cornelius Lanczos ( ) 16 Wolfgang Ernst Pauli ( ) 1945
37 Heisenberg Feynman Richard Phillips Feynman ( ) 1965
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More information) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
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72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
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