II

II 28 5 31

3 I 5 1 7 1.1.......................... 7 1.1.1 ( )................ 7 1.1.2........................ 12 1.1.3................... 13 1.1.4 ( )................. 14 1.1.5................... 15 1.2.......................... 15 2 17 2.1......................... 17 2.1.1................. 18 2.1.2................. 19 2.1.3 2.................. 20 2.1.4.......................... 21 2.1.5 Rayleigh-Jeans.......... 23 2.2......................... 25 2.2.1 Einstein............... 25 2.2.2 Planck.............. 27 2.2.3 Debye.................... 28 3 29 3.1.......................... 29 3.1.1..................... 29 3.1.2.......................... 29 3.1.3.......................... 29 3.1.4 α......................... 30

4 3.1.5......................... 30 3.1.6 Zeeman..................... 30 3.1.7 α.................... 31 3.2.................... 31 3.3 Bohr.......................... 32 3.3.1 Bohr...................... 32 3.3.2...................... 33 3.3.3........................... 33 3.4 de Broglie......................... 35 3.4.1 de Broglie.............. 36 3.4.2................... 36 3.5......................... 36

I

7 1 19 X 19 ( ) ( ) MRI 1.1 1.1.1 ( ) (Planck 1 ) (1900) E = nhν (1.1) 1 Max Karl Ernst Ludwig Planck (1858-1947) 1918

8 1 (E ν n 0 h ) h ( [ ] [ ] 2 [ ] 1 = [ ] [ ] = [ ] [ ] ) h = 6.626068 10 34 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) 5.670400 10 8 W m 2 K 4 W. Wien 3 λ max λ max = b T (1.4) 2 Ludwig Eduard Boltzmann (1844-1906) 3 Wilhelm Wien (1864-1928) 1911

1.1. 9 3. Wien (1896) Wien ( ) U(ν)dν = 8πk Bβ c 3 exp( βν/t )ν 3 dν (1.5) k B Boltzmann k B = 1.3806505 10 23 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 (1842-1919) 1904

10 1 6. 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 (1879-1955) 1921

1.1. 11 1. 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 (1857-1694) ( ) 7 Wilhelm Ludwig Franz Hallwachs (1859-1922) 8 Joseph John Thomson (1856-1940) 1906 9 Philipp Eduard Anton von Lénárd (1862-1947) 1905

12 1 6. 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 1.1.2 1. Dulong 12 -Petit 13 (1819) C V = 3R (1.11) 10 Robert Andrews Millikan (1868-1953) 1923 11 Arthur Holly Compton(1892-1962) 1927 12 Pierre Louis Dulong (1785-1338) ( ) 13 Alexis Thérèse Petit (1791-1820)

1.1. 13 0 2. Einstein (1907) Dulong-Petit 3. Debye 14 (1912) Einstein 1.1.3 1 Balmer 15 (1885) 4 9 5 h, 16 12 h, 25 21 h, 36 32 h, h = 3.6456 10 7 m λ = n2 h n = 3, 4, 5, 6, (1.12) n 2 4 14 Peter Joseph William Debye (1884-1966) 1936 15 Johann Jakob Balmer (1825-1898) ( )

14 1 Rydberg 16 1 λ = 1 R n: (1.13) λ (n + b) 2 (1888) λ Rydberg R (Rydberg ) R = 1.09737 10 7 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) 1.1.4 ( ) (de Broglie) (1923) (Davisson, Germer) (Ni 16 Johannes Robert Rydberg (1854-1919) 17 Theodore Lyman (1874-1954)

1.2. 15 )(1927) (1928) p k(= 2π λ ) = p ħ (1.16) k ( λ) ( ) (Bonse, 1974) 1.1.5 (Zeeman) (1896)( ) (Stern, Gerlach) (1922) 1.2 1. 2. : ( ) 3. 4. NMR,MRI:

17 2 2.1 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 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 = 1.3806488(13) 10 23 JK 1 2.1.1 ( ) T k B T/2 E kin = α 1 p 2 1 + + α s p 2 s + + α f p 2 f (2.3) α s p 2 s s α s ( ) s

2.1. 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 2.1.2 1 1 T

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 = 6.02214129(27) 10 23 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) 2.1.3 2 2

2.1. 21 3 ( ) 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) 2.1.4 0 s q s, p s E s = a s q 2 s + b s p 2 s (2.20)

22 2 2.1: 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)

2.1. 23 3N 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) 2.1.5 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 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 3 3 3 3 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)

2.2. 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) 2.2.1 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 2 2.2: 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)

2.2. 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 2.2.2 Planck ν ν + dν (2.34) Z(ν)dν = 8πL3 c 3 ν 2 dν (2.46)

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 2.2.3 Debye

29 3 3.1 3.1.1 Loschmidt 1 (1865) (Avogadro constant) A. Einstein (1905) J. Perrin 2 (1908) 3.1.2 J. J. Thomson e/m (1897) Millikan Fletcher 3 e (1909-1913) 3.1.3 Goldstein 4 (1886) J. J. Thomson Goldstein 1 Johann Josef Loschmidt (1821-1895) 2 Jean Baptiste Perrin (1870-1942) (1926) 3 Harvey Fletcher (1884-1981) 4 Eugen Goldstein (1850-1930)

30 3 ( ) W. Wien e/m (1898) E. Rutherford 5 1918 α 3.1.4 α α 2 2 1 Rutherford 1898 α β 1899 α β 1902,1903 Rutherford α 3.1.5 Chadwick 1932 3.1.6 Zeeman Zeeman 6 Zeeman D (1896-1897) H. A. Lorentz Larmor 7 e/m e/m 5 Ernest Rutherford (1871-1937) 1908 6 Pieter Zeeman (1865-1943) 1902 7 Joseph Larmor (1857-1942)

3.2. 31 3.1.7 α Geiger 8 Marsden 9 Rutherford 1909 α (ZnS) α α α Rutherford 1911 3.2 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 (1882-1945) 9 Ernest Marsden (1889-1970) 10 Charles-Eugène Delaunay (1816-1872)

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 2 1 2 = 2πE ω 0 = E ν 2π ω (3.10) 3.3 Bohr 3.3.1 Bohr Bohr 11 (1913) 11 Niels Henrik David Bohr (1885-1962) 1922

3.3. Bohr 33 1. E 1, E 2, E 3,... 2. ν hν = E m E n (3.11) 3. 3.3.2 (quantum condition) pdq = nh (n = 0, 1, 2, 3, ) (3.12) q p Bohr 1913 de Broglie ( (3.29) ). 3.3.3 1 1 Bohr (3.12) (3.10) E ν = nh (3.13)

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)

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 = 6.626 10 34 [J s] 0 E = c p (1) ν = E h, λ = h p (3.27) de Broglie 12 (1924) 12 Louis de Broglie (1892-1987) 1929

36 3 3.4.1 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 3.4.2 3.5 Bohr Heisenberg 13 (1925) de Broglie Schrödinger 14 (1926) (?) Lanczos 15 ( ) Schrödinger,Lanczos,Pauli 16 Schrödinger Heisenberg 13 Werner Heisenberg (1901-9976) 1932 14 Erwin Rudolf Josef Alexander Schrödinger (1887-1961) 1933 15 Cornelius Lanczos (1893-1974) 16 Wolfgang Ernst Pauli (1900-1958) 1945

3.5. 37 Heisenberg Feynman 17 17 Richard Phillips Feynman (1918-1988) 1965