II

II 28 5 31

3 I 7 1 9 1.1.......................... 9 1.1.1 ( )................ 9 1.1.2........................ 14 1.1.3................... 15 1.1.4 ( )................. 16 1.1.5................... 17 1.2.......................... 17 2 19 2.1......................... 19 2.1.1................. 20 2.1.2................. 21 2.1.3 2.................. 22 2.1.4.......................... 23 2.1.5 Rayleigh-Jeans.......... 25 2.2......................... 27 2.2.1 Einstein............... 27 2.2.2 Planck.............. 29 2.2.3 Debye.................... 30 3 31 3.1.......................... 31 3.1.1..................... 31 3.1.2.......................... 31 3.1.3.......................... 31 3.1.4 α......................... 32

4 3.1.5......................... 32 3.1.6 Zeeman..................... 32 3.1.7 α.................... 33 3.2.................... 33 3.3 Bohr.......................... 34 3.3.1 Bohr...................... 34 3.3.2...................... 35 3.3.3........................... 35 3.4 de Broglie......................... 37 3.4.1 de Broglie.............. 38 3.4.2................... 38 3.5......................... 38 II 41 4 43 4.1.......................... 43 4.1.1 Newton Galilei........ 43 4.1.2.......................... 45 4.1.3..................... 46 4.1.4........................ 47 4.1.5..................... 48 4.2 Michelson-Morley................... 49 4.3 Lorentz....................... 51 4.4 Lorentz................ 55 4.4.1.................... 55 4.4.2 Lorentz..................... 57 4.4.3................ 58 4.4.4................ 58 4.4.5..................... 60 4.5.......................... 61 5 63 5.1 Lorentz..................... 63 5.2............... 66

5.2.1........................ 67 5.2.2..................... 67 5.2.3..................... 67 5.2.4........................ 68 5.3........................ 69 5.3.1........................... 69 5.3.2.................... 70 5.3.3...................... 70 5.3.4.................... 70 5.3.5.......................... 70 5.3.6.......................... 71 5.3.7............... 71 5.3.8.................... 72 5.3.9......................... 72 5.3.10................. 73 6 75 6.1........................ 75 6.2.............. 77 6.3................... 78 6.3.1.................... 79 6.3.2 4................... 80 6.3.3........................ 81 7 83 7.1 Maxwell................. 83 7.1.1.............. 84 7.1.2 Lorentz..................... 84 7.1.3...................... 85 7.1.4 Lorentz.............. 85 7.2 Maxwell........... 86 7.2.1...................... 86 7.2.2..................... 86 7.2.3 B, E..................... 87 7.2.4......................... 88 7.3 Maxwell (II)......... 88 5

6 7.3.1......... 89 7.3.2......... 90

I

9 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

10 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. 11 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

12 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. 13 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

14 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. 15 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) ( )

16 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. 17 )(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:

19 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

20 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. 21 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

22 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. 23 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)

24 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. 25 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

26 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. 27 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)

28 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. 29 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)

30 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

31 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)

32 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. 33 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)

34 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 35 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)

36 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 37 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

38 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. 39 Heisenberg Feynman 17 17 Richard Phillips Feynman (1918-1988) 1965

II

43 4 19 Maxwell 1 Hertz (aether) Michelson 2 -Morley 3 Lorentz 4 20 Einstein GPS 4.1 4.1.1 Newton Galilei Newton 5 3 1 James Clerk Maxwell (1831-1879) ( ) 2 Albert Abraham Michelson (1852-1931) 1907 3 Edward Morley (1838-1923) 4 Hendrik Antoon Lorentz (1853-1928) 1902 5 Isaac Newton (1643-1727) ( )

44 4 1. 2. f = m a (4.1) 3. Newton (inertial system) Galilei 6 Newton 2 S S S S x x. t = 0 t = 0 t = 0 2 S S x v S t () x, y, z S x, y, z ( 4.1) x = x vt, y = y, z = z, t = t (4.2) S S dx dt d 2 x dt 2 = dx dt v, dy dt = dy dt, dz dt = dz dt, (4.3) = d2 x dt, d 2 y 2 dt 2 = d2 y dt, d 2 z 2 dt 2 = d2 z dt 2 (4.4) 6 Galileo Galilei (1564-1642) ( )

4.1. 45 4.1: S Newton m d2 x dt 2 = f (4.5) f S. S f f = f (4.4) (4.5) m d2 x dt 2 = f (4.6) S (4.5) Galilei () 4.1.2 1. R. Hooke 7 (1660 ) aether 2. A. Newton 7 Robert Hooke (1635-1703)

46 4 1670-1672 Opticks (1704) (corpuscular theory of light) 3. C. Huygens 8 (1678) 4. T. Young 9 Young (1805 ) 5. A.J.Fresnel 10 ( ). ( ) 4.1.3 1. Rømer 11 (1676). Bradley 2. Bradley 12 (1728) 8 Christiaan Huygens(1629-1695) 9 Thomas Young(1773-1629) 10 Augustin-Jean Fresnel (1788-1827) () 11 Ole Christensen Rømer (1644-1710) 12 James Bradley (1693-1762)

4.1. 47 3. Fizeau 13 (1849) 4. Foucault 14 (1850) 1862 ( 0.6 % ) 4.1.4 1. Cavendish 15 Cavendish 1773 2 1873 Maxwell 2. Coulomb 16 (1780 ) Coulomb Cavendish Cavendish Maxwell 3. Ampère 17 (1820) 2 4. Ampère (1826) 13 Armand Hippolyte Louis Fizeau (1819-1896) 14 Jean Bernard Léon Foucault (1819-1868) 15 Henry Cavendish (1731-1870) ( ) 16 Charles-Augustin de Coulomb (1736-1806) 17 André-Marie Ampère (1775-1836)

48 4 5. Faraday 18 6. Maxwell (1864), Faraday 1 7. Hertz (1886-1887) ( ) 4.1.5 (aether) Fresnel Maxwell Hertz (Doppler 19 effect). Maxwell ( 1c ) 2 2 t + 2 2 x + 2 2 y + 2 E = 0 (4.7) 2 z 2 Galilei. 18 Michael Faraday (1791-1867) ) 19 Christian Andreas Doppler (1803-1853)

4.2. Michelson-Morley 49 30 km/s 30km/s. 4.2 Michelson-Morley (A. A. Michelson).. 1. Michelson (1881) ( ) Maxwell () (v/c) 2 2. Michelson-Morley (1887) (). 1 1.5m 30cm... 4.2. L M M 2 M 2 M M L M M 1 M 1 M M L 2, L 1 L 1, L 2. L 1, L 2.

50 4 4.2: : M. MM 1 v. L 1 MM 1 T 1 = l 1 c v + l 1 c + v = 2l 1/c 1 β 2, β = v c (4.8) L 2 MM 2 T 2 = 2 (l 2 ) 2 + (vt 2 /2) 2 c l 2 /c T 2 = 2 1 β 2 (4.9) (4.10) L 1 L 2 ( ) l 1 = c(t 1 T 2 ) = 2 1 β l 2 2 1 β 2 (4.11)

4.3. Lorentz 51 M 90 L 1 L 2 ( ) = c(t 1 T 2) l 1 = 2 l 2 (4.12) 1 β 2 1 β 2 δ = (l 1 + l 2 )β 2 (4.13) λ ϕ = 2π δ λ 2π (l 1 + l 2 ) β 2 λ. 1/40 Michleson-Morley 2 (l 1 = l 2 ).... 4.3 Lorentz Michelson-Morley. Galilei. Galilei. 1. 2. S S S S 4.1.

52 4 1. S 1 S x, y, z, t x, y, z, t 1 2. xy x y xz x z z = 0 z = 0 y = 0 y = 0 x, t y = κ(v)y, z = κ(v)z (4.14) κ x, y, z, t v y, z. v v κ. κ(v) v S S S x v y = κ( v)y, z = κ( v)z (4.15). (4.14), (4.15) (κ( v )) 2 = 1 κ( v ) = ±1 (4.16) v 0 y y, z z κ( v ) = 1 y = y, z = z (4.17) 3.. t = t = 0 S O t O c S P ( (x, y, z)) t s 2 x 2 + y 2 + z 2 (ct) 2 = 0 (4.18) S P S (x, y, z ) P S t s 2 x 2 + y 2 + z 2 (ct ) 2 = 0 (4.19)

4.3. Lorentz 53 x, y, z, t x, y, z, t 1 ( ) (4.18) (4.19) ( ) s 2 0 s 2 = α(v)s 2 (4.20) α(v) x, y, z, t v s 2 x 2 + y 2 + z 2 (ct) 2 = x 2 + y 2 + z 2 (ct ) 2 = s 2 (4.21) 4. (x, t) (x, t ) x 2 (ct) 2 = x 2 (ct ) 2 (4.22) x = ax + bt, t = fx + gt (4.23) a, b, f, g v (4.23) (4.22) x, t a 2 c 2 f 2 = 1 ab c 2 fg = 0 (4.24) g 2 b 2 /c 2 = 1 a = ± cosh θ, b = c sinh θ = ca tanh θ, () (4.25) f = ± 1 c sinh θ, g = cosh θ = cf tanh θ θ tanh θ = b ca () (4.26)

54 4 θ S O S v x O x = y = z = 0 ax + bt = 0, y = 0, z = 0 (4.27) x = b a t (4.28) b a = v (4.29) tanh θ = v c β = v c (4.30) cosh θ = (1 tanh 2 θ) 1/2 = 1 1 β 2 sinh θ = β 1 β 2 (4.31) x = ± x vt 1 β 2, (4.32) t = ± t (v/c2 )x 1 β 2 (4.33) v 0 x x, t t x = x vt, 1 β 2 t = t (v/c2 )x 1 β 2 (4.34)

4.4. Lorentz 55 Lorentz Poincaré 20 Einstein Einstein 2 2 Lorentz (4.34) x = x + vt 1 β 2, t = t + (v/c 2 )x 1 β 2 (4.35). (4.34) v v, (t, x, y, z) (t, x, y, z ). 4.4 Lorentz S S (4.34) Lorentz Lorentz.. Lorentz... (event). 1 (world point). (world line). 4.4.1 (4.34). 2.. x y, z. x ct ( 4.3). S x t = 0 (4.34) ct = βx 20 Jules Henri Poincaré (1854-1912)

56 4. ct x = 0 x = βct. x tan θ = β. P,Q. S. S P,Q (t, x = a) (t, x = b) S (t P, x = a ) (t Q, x = b ). (4.34) P,Q 4.3: : S P,Q S. ct P = ct βa 1 β 2, ct Q = ct βb 1 β 2. (4.36) t Q t P = (a b)β c 1 β 2 (4.37) a > b t Q > t P. S Q P.

4.4. Lorentz 57 4.4.2 Lorentz S x. 4.4 A,B. S A,B x x = a, x = b. S l 0 = a b. S x v. S A,B x x A (t) = a, x B (t) = b. S l l = a b. 4.4: :. (4.34) 2 A,B a = a vt, 1 β 2 b = b vt 1 β 2 (4.38) l 0 = a b = a b 1 β 2, (4.39) l = a b = l 0 1 β 2 (4.40) Lorentz.

58 4 4.4.3 W 1, W 2 S,S O,O. W 1 t W 2 x x = vt. S W 2 (x = 0, t ). t t. (4.34) 2 t = t (v/c2 )vt 1 β 2 = t 1 β 2 < t (4.41).. W 2 x = f(t) x. t t t + t W 2 v(t) = df/dt t = t W 1 0 T W 2 T = T 0 dt = T. f(t) 0 1 1 ( ) 2 df (4.42) c 2 dt T T 1 1 c 2 ( df dt ) 2 dt (4.43)... 4.4.4 A ν λ ( ) 1 A sin 2π n x νt + α λ (4.44)

4.4. Lorentz 59 n α S,S 1 λ n x νt = 1 λ n x ν t (4.45) ν, λ, n S k = 1 λ n, k = 1 λ n (4.46) k x νt = k x ν t (4.47) (4.35) x, t ν = ν vk x, 1 β 2 k x = k x (v/c 2 )ν, k y = k y, k z = k z (4.48) 1 β 2 S xy xy x θ S x θ n = (cos θ, sin θ, 0) n = (cos θ, sin θ, 0). νλ = c. ν = ν 1 β cos θ 1 β 2, ν cos θ = ν cos θ β 1 β 2, ν sin θ = ν sin θ (4.49) tan θ = sin θ 1 β 2 cos θ β (4.50) tan θ 2 = tan θ 1 + β 2 1 β (4.51) (4.49) v

60 4 v (4.49) S x. (4.49) 2 cos θ = β (4.49) 1 ν = ν 1 β 2 (4.52). (4.50). 4.4.5 Galilei... S 1 x = x (t ), y = y (t ), z = z (t ) (4.53). S u x = dx (t ), u dt y = dy (t ), u dt z = dz (t ) (4.54) dt u = (u x) 2 + (u y) 2 + (u z) 2 (4.55). S u x = dx(t), u y = dy(t) dt dt u =.. (4.35), u z = dz(t) dt (4.56) (u x ) 2 + (u y ) 2 + (u z ) 2 (4.57) dx = dx + vdt 1 β 2, dy = dy, dz = dz, dt = dt + (v/c 2 )dx 1 β 2 (4.58)

4.5. 61. 1,2,3 4 u x = u x + v 1 + (u xv/c 2 ), u y = u y 1 β 2 1 + (u xv/c 2 ), u z = u z 1 β 2 1 + (u xv/c 2 ). u (4.59) u 2 = u 2 x + u 2 y + u 2 z = (u x + v) 2 + (u 2 y + u 2 z)(1 v 2 /c 2 ) (1 + (u xv/c 2 )) 2 = c2 (1 + (u xv/c 2 )) 2 c 2 (1 u 2 /c 2 )(1 v 2 /c 2 ) (1 + (u xv/c 2 )) 2 (4.60) 1 u2 c 2 = (1 u 2 /c 2 )(1 v 2 /c 2 ) (1 + (u xv/c 2 )) 2 (4.61) u < c, v < c u u < c.. u, v c u = c. 4.5 Kennedy-Thorndyke Michelson-Morley Fitzgerald-Lorentz. 2 Lorentz. R.J. Kennedy and E.M. Thorndike, Experimental Establishment of the Relativity of Time Phys. Rev. 42 400-418 (1932) 2 %. 2. H.E. Ives and G.R. Stilwell, An Experimental Study of the Rate of a Moving Atomic Clock J. Opt. Soc. Am. 28 215-226 (1938)

62 4 Michelson-Morley 400 1 (Brillet- Hall experiment ) 2 2 100 1.

63 5 5.1 Lorentz ( ) P S (t, x, y, z) x 0 = ct, x 1 = x, x 2 = y, x 3 = z 4 x µ (µ = 0, 1, 2, 3) (x 0, x 1, x 2, x 3 ) P 4 P S (x 0, x 1, x 2, x 3 ) S S x µ x µ 1 x µ = 3 a µ νx ν + b µ (µ = 0, 1, 2, 3) (5.1) ν=0 a µ ν x µ 16 b µ S S 2 0 3 (Einstein ) (5.1) x µ = a µ νx ν + b µ S S b µ = 0 x µ = a µ νx ν (5.2) (5.1) (x 0 ) 2 + 3 (x k ) 2 = (x 0 ) 2 + k=1 3 (x k ) 2 (5.3) k=1

64 5 0 (µ ν) η µν = η νµ = 1 (µ = ν = 0) 1 (µ = ν = 1, 2, 3) (5.4) (5.3) η µν x µ x ν = η µν x µ x ν (5.5) (5.2) (5.2) x η µν = η ρσ a ρ µa σ ν (5.6) (5.2) x µ x µ a µ ν (5.6) Lorentz. (5.2) (5.1) (5.6) Lorentz (5.6) µ, ν 10. a µ ν 16 (5.6) 10 16-10=6. 3 3. Lorentz a µ ν { 1 (µ = ν) δ µ ν = (5.7) 0 (µ ν) x µ = x µ Lorentz Lorentz x ν = b ν µx µ (5.8) b ν µ Lorentz a µ ν.

5.1. Lorentz 65 a µ ν 4 4 A. (A) µν a µ ν (5.9). Y (Y ) µν η µν (5.10). A A T (A T ) µν (A) νµ = a ν µ (5.11) (5.6) (Y ) µν = (A T Y A) µν (5.12) det(y ) = det(a T Y A) = det(a T ) det(y ) det(a) (5.13) det(a T ) = det(a), det(y ) = 1 (5.14) det(a) = ±1 (5.15) A. η µν 0 (µ ν) η µν 1 (µ = ν = 0) 1 (µ = ν = 1, 2, 3) (5.16) Y Y 1 (Y 1 ) µν η µν (5.17) (5.12) 1 = Y 1 A T Y A (5.18)

66 5. 2 δ µ ν = η µλ a ρ λη ρσ a σ ν (5.19) δ µ ν = b µ σa σ ν (5.20) b µ σ = η µλ a ρ λη ρσ (5.21) η µλ a ρ λη ρσ a µ σ (5.22). (5.6). (5.6) µ = ν = 0 3 1 = (a 0 0) 2 + (a k 0) 2 (5.23) a 0 0 = ± 1 + k=1 3 (a k 0) 2 (5.24) a 0 0 1 a 0 0 1 Lorentz det(a) = ±1 a 0 0 1 a 0 0 1 4 det(a) = 1 a 0 0 1 Lorentz (proper Lorentz transformation). Lorentz Lorentz. k=1 5.2 S S P 4 x µ, x µ x µ = a µ νx ν + b µ (5.25) Lorentz η µν = η ρσ a ρ µa σ ν (5.26)

5.2. 67 5.2.1 S S C, C C = C (5.27) C (scalar) S S Θ(x) Θ(x 0, x 1, x 2, x 3 ) Θ(x ) Θ(x 0, x 1, x 2, x 3 ) Θ(x) = Θ (x ) (5.28) Θ(x) (x x S,S ) 5.2.2 4 S S A µ, A µ A µ = x µ x ν Aν = a µ νa ν (5.29) (contravariant vector) 2 P,Q x µ = x µ Q xµ P A µ x S S A µ (x), A µ (x ) A µ (x) A µ (x ) = x µ x ν Aν (x) (5.30) 5.2.3 4 S S B µ, B µ B µ = xν x µ B ν (5.31)

68 5 (covariant vector) Θ(x) Θ µ (x) Θ(x) x µ (5.32) 5.2.4 2 A µ (x), B µ (x) C µν (x ) C µν (x) C µν (x) A µ (x)b ν (x) (5.33) C µν (x ) A µ (x )B ν (x ) = x µ x ρ x ν x σ Aρ (x)b σ (x) = x µ x ρ x ν x σ Cρσ (x) (5.34) C µν (x ) = x µ x ρ x ν x σ Cρσ (x) (5.35) C 2 (contravariant tensor field of the 2nd rank) T (x) S S T αβ γ δϵ ζ (x), T λµ ν ρσ τ(x ) (5.36) T λµ ν ρσ τ(x ) = x λ x µ x α x x ν x δ x ϵ xζ β x γ x ρ x σ x T αβ γ τ δϵ ζ (x), (5.37) T (x) (tensor field) (λµ ν) r (ρσ τ) s

5.3. 69 T r s (mixed tensor) r (contravariant tensor of rank r) s (covariant tensor of rank s) 2 F µν F µν = F νµ (5.38) F. F µν = F νµ (5.39) F. 1.. Kronecker δ µ ν η µν 2 δ µ ν = x µ x α x β x ν δα β (5.40) η µν = x µ x α x ν x β ηαβ (5.41) 5.3 5.3.1 2 A µν λ, B µν λ C µν λ = A µν λ + B µν λ (5.42) C µν λ A µν λ, B µν λ

70 5 5.3.2 S A µν λ SA µν λ (5.43) 5.3.3 A µν λ B αβ C µν λαβ Aµν λ B αβ (5.44) 5 C A, B 5.3.4 A µν λ (x) x ρ (5.45) 4 ρ / x ρ A µν λ (x) 5.3.5 A αβ γ δϵ ζ 1 0,1,2,3 2 (contraction) B β γ δ ζ = Aαβ γ δα ζ (5.46)

5.3. 71 5.3.6 2 A µν λ, B αβ C µν λαβ A µν λb αβ (5.47) C µ λα C µβ λαβ = A µβ λb αβ (5.48) D ν λα = A βν λb αβ, E ν λβ = A αν λb αβ 3 A, B A B C = A µ B µ (5.49) C A, B 5.3.7 η µν A µ η µν B ν η µν A µ B ν η µν η µν B ν = η µν η νλ A λ = δ µ λa λ = A µ A µ B ν A µ A µ η µν A ν (5.50) T µν = η µα η νβ T αβ (5.51)

72 5 5.3.8 A µ (A) 2 A µ A µ = η µν A µ A ν = (A 0 ) 2 + 3 (A k ) 2 (5.52) A A µ (A) 2 > 0 A µ (space-like vector) (A) 2 < 0 A µ (time-like vector) (A) 2 = 0 (null vector) (A) 2 Lorentz A 3 k=1 5.3.9 5.1 P O x µ. x µ P 3 (x 0 ) 2 + (x k ) 2 = 0 (5.53) k=1. (space-like).. Q O y µ Q. (time-like).. O O. R O z µ R. (light-cone).

5.3. 73 5.3.10 5.1: η µν x µ x = 1 2 ν c 2 t + 2 2 x + 2 2 y + 2 (5.54) 2 z 2 ( (D Alembertian) c

75 6 6.1 Newton Galilei Lorentz Newton Lorentz Newton x k (k = 1, 2, 3) t Lorentz λ x µ (µ = 0, 1, 2, 3) λ 4 x µ x µ + x µ ( x µ ) 4 ( 2 ) s 2 = η µν x µ x ν = c 2 ( t) 2 + ( x) 2 ( λ) 2 3 v = x/ t c x µ s 2 s 2 = c 2 ( τ) 2 Lorentz τ ( ) τ c 2 ( τ) 2 η µν x µ x ν (6.1) τ λ 4 x µ (τ) (6.1) τ = t 1 ( ) 2 v (6.2) c

76 6 v = d x/dt 3 τ (proper-time) S x µ (τ) τ x k (τ) = x k (τ + τ) (k = 1, 2, 3) x 0 (τ + τ) x 0 (τ) = x 0 = c t c 2 ( τ) 2 = η µν x µ x ν = c 2 ( t ) 2. τ = t t τ τ Lorentz x µ (τ) u µ dxµ dτ (6.3) x µ 4 (four-velocity) 4 Lorentz S S S x µ (τ) = (x 0, x = 0) S x µ (τ) Lorentz x µ = a µ νx ν, x ν = b ν µx µ = b ν 0x 0 (6.4) 2 τ u ν = b ν 0c dt dτ = cbν 0 (6.5) 4 u µ 3 v (6.2) u k = v k 1 β 2, u0 = c 1 β 2. (6.6) k = 1, 2, 3, β = v /c η µν u µ u ν = c 2 (6.7). (6.1) ( τ) 2.

6.2. 77 4 a µ duµ dτ = d2 x µ (6.8) dτ 2 (6.7) a µ u µ η µν u µ a ν = 0 (6.9) 6.2 Lorentz S S c Galilei c Newton S S Newton m d2 x k dt 2 = F k, dx k dt = 0 (k = 1, 2, 3) (6.10) m F k S k m d2 x 0 dt 2 = F 0 (6.11) d 2 (ct )/dt 2 = 0 F 0 = 0 m d2 x µ dt 2 = F µ, (µ = 0, 1, 2, 3) (6.12) t Lorentz S t τ m d2 x µ dτ 2 = F µ, (6.13)

78 6 Lorentz F µ. Newton (4.5), Lorentz. S S Lorentz (6.13) S m d2 x µ dτ 2 = F µ (6.14) F µ S F i S S x ν = b ν µx µ F µ = b µ νf ν = 3 b µ k F k k=1 (6.15) F µ 4 4 (four-force) u µ F µ = 0 (6.16) (6.14) u µ mu µ a µ = u µ F µ (6.9) 0 (6.16) 6.3 (6.14 ) d dτ pµ = F µ (6.17) p µ 4 (four-momentum) p µ mu µ = m dxµ dτ (6.18)

6.3. 79 p µ (F µ = 0) d dτ pµ = 0 (6.19) p µ 6.3.1 p 0 p = (p 1, p 2, p 3 ) (6.2) d p dt = F 1 ( ) 2 v c K F 1 ( ) 2 v (6.20) c Newton (6.17) d p dt = K (6.21). (6.17). (6.16) (6.6) ( v ) 2 cf 0 1 = ( v K) (6.22) c. (6.17) ( ) 2 d v dt (cp0 ) = cf 0 1 = ( K c v) (6.23) cp 0 = +

80 6. Einstein 0. 4 p µ. Einstein. m v p = 1 ( v/c) = 2 cp 0 mc = 2 = W = (6.24) 1 ( v/c) 2 v = 0 p = 0 W mc 2 m (rest energy) 6.3.2 4 u µ u µ = c 2 (6.25) p µ p µ = (mc) 2 (6.26) W p W = c ( p) 2 + (mc) 2 (6.27) v c W = mc 2 + 1 2m ( p)2 +. 1 2 Newton

6.3. 81 6.3.3 Cockroft-Walton (1932) Cockroft 1 Walton 2 2 α ( ) 7 3Li + 1 1 H 4 2 He + 4 2 He α 8.6Mev M = 7.014368 + 1.007277 2 4.001506 = 0.018633( ) Mc 2 = 0.018633 931.48 = 17.3(MeV) 2 8.6 = 17.2(MeV) Einstein 9.1093826 10 31 kg 511 kev ( positron ( Dirac 3 (1928) Anderson 4 (1932)) 2 γ 1 John Douglas Cockcroft (1897-1967) 1951 2 Ernest Walton (1903-1995) 1951 3 Paul Adrien Maurice Dirac (1902-1984) 1933 4 Carl David Anderson (1905-1991) 1936

83 7 Maxwell 7.1 Maxwell Maxwell ρ j Maxwell B = 0 (7.1) B t + E = 0 (7.2) D = ρ (7.3) H D t = j (7.4) D, E. B, H D = ϵ 0 E, B = µ0 H (7.5) ϵ 0, µ 0 c ϵ 0 µ 0 = 1 c 2

84 7 7.1.1 (7.1),(7.2) B, E A, ϕ B = A, E A = ϕ (7.6) t ( V ) = 0 ( V 3 ) ( f) = 0 (f ) (7.6) Maxwell (7.3),(7.4) V = ( V ) 2 V ϕ = ρ ϵ 0 (7.7) A = µ 0 j (7.8) ( (5.54) ) A + 1 c 2 ϕ t = 0 (7.9) 7.1.2 Lorentz (7.7),(7.8) (7.9) (7.7),(7.8) (7.9) (7.7),(7.8) ( A + 1 ) ( ϕ = µ c 2 0 j + ρ ) (7.10) t t Maxwell j + ρ t = 0 (7.11)

7.1. Maxwell 85 ( A + 1 ) ϕ = 0 (7.12) c 2 t (7.7),(7.8) (7.9) (7.11) (7.9) Lorentz Lorentz ( ) 7.1.3 B, E (7.6) A, ϕ. A, ϕ A = A + λ, ϕ = ϕ λ t (7.13) A, ϕ λ( x, t) ( x, t) A, ϕ (7.6) A = A = B, A t ϕ = A t ϕ = E (7.14) B, E A, ϕ λ A, ϕ B, E A, ϕ A, ϕ (gauge transformation) B, E 7.1.4 Lorentz A, ϕ Lorentz λ A, ϕ Lorentz 0 = A + 1 ϕ c 2 t = λ + A + 1 ϕ (7.15) c 2 t λ Lorentz.

86 7 7.2 Maxwell 7.2.1 (7.11) j 1 = j x, j 2 = j y, j 3 = j z, j 0 = cρ (7.16) (7.11) S µ j µ (x) = 0 (7.17) ( µ / x µ ) j µ ( ) S j µ (x ) µj µ (x ) = 0 (7.18) µ j µ j µ 4 (four-current) 7.2.2 A 0 = 1 c ϕ, A1 = A x, A 2 = A y, A 3 = A z, (7.19) (7.7),(7.8) A λ = µ 0 j λ (7.20) (5.54) Lorentz A λ

7.2. Maxwell 87 Lorentz (7.9) λ A λ = 0 (7.21) A µ = η µν A ν A µ Āµ + µ λ (7.22) λ λ = 0 Āµ A µ A µ, A µ 4 (four-potential) 7.2.3 B, E B, E f µν f µν µ A ν ν A µ = f νµ (7.23) f µν B, E f 00 f 01 f 02 f 03 0 1 f 10 f 11 f 12 f E c x 1E c y 1E c z 1 13 f 20 f 21 f 22 f 23 = E c x 0 B z B y 1 E c y B z 0 B x 1 f 30 f 31 f 32 f 33 E c z B y B x 0 (7.24) f µν. f µν f µν = µ Ā ν ν Ā µ = f µν (7.25) f µν B, E f 00 f 01 f 02 f 03 1 0 f 10 f 11 f 12 f 13 E 1 c x E 1 c y E c z f 20 f 21 f 22 f 23 = 1 E c x 0 B z B y 1E c y B z 0 B x f 30 f 31 f 32 f 33 1E c z B y B x 0 (7.26)

88 7 7.2.4 Maxwell (7.20) Lorentz (7.21) 7.3 Maxwell (II) 4 Maxwell f µν = f νµ (7.24) B, E 4 (7.16) Maxwell (7.3), (7.4) (7.1) (7.2) ν f λν = µ 0 j λ (7.27) λ f µν + µ f νλ + ν f λµ = 0 (7.28) (7.28) 64 = 4 4 4 4 (7.28) F λµν λ f µν + µ f νλ + ν f λµ (7.29) f µν F λµν λ, µ, ν 2 F = 0 λ, µ, ν F λµν F 123, F 012, F 023, F 031 4 F 123 = 0 0 = F 123 = 1 f 23 + 2 f 31 + 3 f 12 = B (7.30) Maxwell (7.1)

7.3. Maxwell (II) 89 7.3.1 (7.27) f λρ (LHS) LHS = f λρ ν f λν = ν (f λρ f λν ) f λν ν f λρ (7.31) 2 f λν = f νλ f λν ν f λρ = 1 2 f λν ( ν f λρ λ f νρ ) = 1 2 f λν ( ν f λρ + λ f ρν + ρ f νλ ) + 1 2 f λν ρ f νλ (7.32) (7.28) f λν ν f λρ = 1 4 ρ(f λν f λν ) (7.33) LHS = ν ( f λν f λρ 1 4 δν ρf αβ f αβ ) (7.34) ν ( f λν f λρ 1 4 δν ρf αβ f αβ ) = µ 0 j λ f λρ (7.35) T ν ρ 1 µ 0 ( f λν f λρ 1 4 δν ρf αβ f αβ ) (7.36) ν T ν ρ = f ρλ j λ (7.37) T νµ = η ρµ T ν ρ = 1 µ 0 ( η λσ f λν f σµ 1 4 ηνµ f αβ f αβ ) = T µν (7.38) T νµ B, E T 00 T 01 T 02 T 03 w cg x cg y cg z T 10 T 11 T 12 T 13 T 20 T 21 T 22 T 23 = 1 S c x M xx M xy M xz 1S c y M yx M yy M yz T 30 T 31 T 32 T 33 1Sz M c zx M zy M zz (7.39)

90 7 w = 1 2 ( D E + B H) = g = 1 c 2 ( E H) = S = ( E H) = (7.40) S Poynting vector M Maxwell M ik = ϵ 0 E i E k + µ 0 H i H k 1 2 δ ik( D E + B H) (7.41) T µν (energy momentum tensor) 7.3.2 (7.37) (7.39) w t = S + ( E j) (7.42) g k 3 t = l M lk (ρe + j B) k (k = 1, 2, 3) (7.43) l=1 t 3 V Gauss wd 3 x = S d σ + ( E t j)d 3 x (7.44) V F V 3 g k d 3 x = M lk dσ l (ρe t + j B) k d 3 x (k = 1, 2, 3) V l=1 F V (7.45) V F F dσ n d σ ndσ (7.44) 1 F 2 V. V. (7.45) 2

7.3. Maxwell (II) 91 V 1 F V V. V. ρ = 0, j = 0 0 V P 0 P k T 00 d 3 x = T 0k d 3 x = c wd 3 x (7.46) g k d 3 x (k = 1, 2, 3) (7.47) P µ 4 (ρ 0, j 0) P µ ( )