LNSC2010_HK.dvi
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1 Jeans CMB nebula) (Frederick William Herschel, ) (galaxy) (Edwin Powell Hubble, ) 1924 [28] 1.1 (H. Leavitt) 1916
![2: [12] 1: SDSS [SDSS homepage] 1910 (V.M.](/docs-images/103/160913505/images/2-0.jpg "Slipher) v (M.")
![Humanson 24 (1929 [12] 2): v = H 0 d.](/docs-images/103/160913505/images/2-2.jpg "(1) H 0 2Mpc 100 3: 3 (66% H 0 =74.2 ± 3.")
2 2: [12] 1: SDSS [SDSS homepage] 1910 (V.M. Slipher) v (M. Humanson 24 (1929 [12] 2): v = H 0 d. (1) H 0 2Mpc 100 3: 3 (66% H 0 =74.2 ± 3.6km/s/Mpc (2) [23] 1929 H 0 1/H 0 2
![2 (cosmic distance ladder) ( 1) 1,000km/s 15Mpc 4: CfA [Huchra homepage] 100Mpc H 0 H 0 10% H 0 =](/docs-images/103/160913505/images/3-1.jpg "100hkm/s/Mpc = 70h 70 km/s/mpc (3) h h 70 v d = v 100km/s h 1 Mpc = v 70km/s h 1 70 Mpc (4) h 70 (2) v")
3 Cepheid Tully-Fisher SNIa 0 100pc 100pc 10kpc 100pc 50kpc 10kpc 25Mpc 10Mpc 200Mpc 60Mpc 4000Mpc 1: (Baade) [2] δ-cephei ( I) RR Lyrae ( II) 1.1 Hubble v = H 0 d a (cosmic distance ladder) ( 1) 1,000km/s 15Mpc 4: CfA [Huchra homepage] 100Mpc H 0 H 0 10% H 0 = 100hkm/s/Mpc = 70h 70 km/s/mpc (3) h h 70 v d = v 100km/s h 1 Mpc = v 70km/s h 1 70 Mpc (4) h 70 (2) v z = λ λ = Δλ (5) λ λ λ λ z >0 c v z = v/c 3
4 CfA CfA % 18,000 SSRS -1998? 13% 5,400 LCRS % 26,000 2dFGRS % 220,000 6dFGRS ,000 SDSSI % 657,000 SDSSII % 790,000 2: 6: 2dFRS( SDSS( ) [2dF- GRS homepage] 5: 7: 2dF [2dFGRS homepage] CfA John Huchra) (Marc Davis) (Margaret Geller) 4 2 SDSS(Sloan Digital Sky Survey) 80 CfA z<0.05 v <15, 000km/s) 5 5 CfA z < 0.25 d < 1, 000h 1 70 Mpc) 1.3 Hubble 1.3 CfA 2dFGRS 22h 1 70 Mpc [11] 40Mpc) 4 10h 1 Mpc 4
![8: [Chadra homepage] 100h 1 Mpc CfA (Great Wall) [21] SSRS Southern Wall) [7]](/docs-images/103/160913505/images/5-0.jpg "1 7 200Mpc 10 15 M 4 50 50 (M31) 30 8 0.")
5 8: [Chadra homepage] 100h 1 Mpc CfA (Great Wall) [21] SSRS Southern Wall) [7] Mpc M (M31) Mpc 1,500km/s R M v 2 GM R ( )( ) R M v M 3Mpc 1, 500km/s N M M [14] R <3Mpc - M/L B 6.43(M/L B ) 2% M X 13% M 15% 85% (F. Zwicky) 70 (1933) [26, 30] 1.4 5
6 9: Hickson [Gemini Obs. homepage] 9 10 (E (S (Irr v r M v 2 r = f GM r 2 GM = f 1 rv 2. (6) f v M r r M 1/r 1/2 11: [27] 8kpc 65kpc SDSS 11 [27] 15kpc M 1/4 21cm [5] 12 [26, 30] 10: 1 [16, 17, 18, 4, 25, 24] a 0 a 0 6
7 t 0 t 0 O P t t 13: 12: [25] 1.4 V j 0 =(1.7 ± 0.6) 10 8 hl Mpc 3 (7) M/L M/L 10h(M/L) M = g) 2 100h 1 Mpc 100h 1 Mpc ( 1.1 ) 2.1 O O 13 t t = T (t)(t (t) Σ(t) t t 7
8 k ds 2 = dr2 1 kr 2 + r2 (dθ 2 +sin 2 θdφ 2 ) (8) [31] 4πr 2 r k =0 r (k >0) (k <0) ±(X 0 ) 2 +(X 1 ) 2 +(X 2 ) 2 +(X 3 ) 2 = ±R 2 (9) k R k = ±1/R 2 (r, θ, φ) (X 1,X 2,X 3 )=r(cos φ sin θ, sin φ sin θ, cos θ) (10) ds 2 = ±(dx 0 ) 2 +(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2 (11) (8) r x, k =0 r = Rf(χ/R); f(x) = sin x, k =1/R 2 > 0 sinh x, k = 1/R 2 < 0 (12) χ ds 2 = dσk 2 dχ2 + R 2 f(χ/r) 2 (dθ 2 +sin 2 θdφ 2 ) (13) Σ(t) t t = t 0 K a(t) t a(t) 2 dσk 2 Σ(t) t ds 2 = c 2 dt 2 + a(t) 2 dσ 2 K (14) (c dt 2 g tt = c 2 g ti =0(x i ) g ti - (Robertson-Walker metric) a(t) (comoving coordinates) t = t 0 a(t 0 )= (9) (R, 0, 0, 0) χ D(χ) V (χ) S(χ) 2.2 (χ, θ, φ) O:χ =0 P t d(t) =a(t)χ (15) O P v(t) v(t) = d(t) = ȧ(t)χ = H(t)d(t) (16) H(t) := ȧ(t) a(t) (17) 8
9 t t = t 0 v = H 0 d, H 0 = ȧ(t 0) a(t 0 ) (18) H 0 P O O (θ, φ) ds 2 =0 cdt = ±adχ. (19) t t 0 χ = t0 t cdt a(t ) (20) dt dt 0 dt a(t) = dt 0 a(t 0 ) (21) T λct λ λ λ = a(t 0) a(t) λ (22) a(t) <a(t 0 ) 5 z z = a(t 0) a(t) 1 (23) z t a(t 0 )=1 (20) z cdz χ = (24) H 0 H H(t) z z 1 z cz = H 0 χ (25) d = χ (z 1) χ = c(t 0 t) χ z H z H 2.3 (Einstein) G μν R μν 1 2 Rg μν = 8πG c 4 T μν (26) 9
10 g μν (14) { (ȧ ) } 2 G tt =3 + K a a 2, (27a) [ (ȧ ) ] 2 G i j = 2ä a + + K a a 2 δj i (27b) ρ(t), P (t) T 00 = ρ(t), T i j = P (t)δ i j (28) T μν ρ P (ȧ ) 2 = 8πG a 3c 2 ρ c2 K a 2, (29a) ä a = 4πG c 2 (ρ +3P ). (29b) ν T ν μ = 0 (30) μ =0 ρ = 3(ρ + P )ȧ a (31) V (t) E(t) S(t) Q(t) T Ṡ Q = Ė + P V = 0 (32) T Ṡ = Q (Bianchi) (29a) (29b) (29a) (31) (29b) 2.2 R M t dr/dt R(t) 2.4 (29a) (31) a(t) ρ(t) a(t) ρ(t) a(t) a(t 0 )=1 ρ(t) ρ 0 = ρ(t 0 ) ρ 0 H 0 Ω M = 8πG 3c 2 H0 2 ρ 0 (33) Ω M Ω K = c2 K H 2 0 (34) 10
11 Ω M +Ω K = 1 (35) K >0 Ω M > 1 (Friedmann) (P =0) (31) ρ = ρ 0 /a 3 1 H 2 0 ( ) da 2 = Ω M dt a +Ω K (36) K =0 a(t) =(t/t 0 ) 2/3, t 0 = 2 3H 0 (37) (Einstein-de Sitter) K 0 K <0 Ω K =1 Ω M > 0 a = H 0 t = Ω M sinh 2 θ 1 Ω M 2, Ω M (sinh θ θ) (38) 2(1 Ω M ) 3/2 t 0 a =1 θ 1 Ω M 1+ 1 ΩM H 0 t 0 = ln. 1 Ω M (1 Ω M ) 1/2 ΩM (39) 2 Ω M =0, Ω K =1 Minkowski Minkowski Milne 2 a = Ω M θ Ω M 1 sin2 2, Ω M H 0 t = (θ sin θ), (40) 2(Ω M 1) 3/2 { } 1 Ω M tan 1 (Ω M 1) 1/2 H 0 t 0 = Ω M 1 (Ω M 1) 1/2 1. (41) 14 t =0 a =0 t = t 0 l H (t) 2 1 K 0 t m = πω M (1 Ω M ) 3/2 1 H 0 (42) t =2t m (P = ρ/3) P = ρ/3 ρ = ρ 0 a 4 (43) 11
12 ds(k>0) ds(k=0) a inflation ds(k<0) AdS F(K<0) F(K>0) F(K=0) 14: a 2 ȧ 2 = 8πGρ 0 3c 2 (44) ( ) t 1/2 a = ; t 0 = t 0 (de Sitter) 3c 4 2πGρ 0 (45) P = ρ ρ ρ = P = c4 8πG Λ (46) G μν +Λg μν = 8πG c 4 T μν (47) T μν Λ ȧ 2 = c2 Λ 3 a2 c 2 K (48) H = c Λ /3 (49) Λ > 0 a 0 e H t ; K =0, a = 3K/Λcosh(H t) ;K>0, 3 K /Λsinh(H t) ;K<0 (50) t 14 ds t K =0 t K >0 K =0 K <0 (48) K< 0 a = 3 K / Λ sin(h t) (51) c/h 0 Hubble c/h 0 aχ L d F F = L/(4πd 2 ) 12
13 d 2 L = L (52) 4πF (13) χ dt de de = ω dn dn ω ω 0 = ω/(1+z) z de 0 = de/(1+z) dt 0 =(1+z)dt (12) f(x) 4πR 2 f(χ/r) 2 L F = 4πR 2 f(χ/r) 2 (1 + z) 2 d L =(1+z)Rf(χ/R) (53) χ z (24) H a z d L z H 2 H 2 0 = Ω M a 3 +Ω K a 2 +Ω Λ =Ω M (1+z) 3 +Ω K (1+z) 2 +Ω Λ (54) Ω Λ Ω M +Ω K +Ω Λ = z Einstein-De Sitter : (Ω M, Ω K, Ω Λ )=(1, 0, 0), De Sitter (Ω M, Ω K, Ω Λ )=(0, 0, 1), Milne (Ω M, Ω K, Ω Λ )=(0, 1, 0). 15: HST Ia 2.5 Einstein-de Sitter 2 z = z L F L Ia 15 Hubble Ia [22] d L 5log 10 (d L /10pc) 13
14 Ω Λ No Big Bang BAO SNLS 1st Year Closed Flat Open Ω M Accelerating Decelerating 16: Einstein-de Sitter (Ω M =1, Ω K = Ω Λ =0) Milne (Ω M =Ω Λ =0, Ω K =1) Ω M =0.29, Ω K =0, Ω Λ =0.71 (55) (29b) 2.6 ) 16 99% Ω Λ > 0 [8] 2.6 (d 2 a/dt 2 ) (Ω M, Ω K, Ω Λ )=(0.26, 0, 0.74) Z X =0.74 Y =0.25 Z =0.013 [10, 1] I I I II I 1/10 1/1000 [29, 5] III Y p Y p = (56) [20] 10 (BBN) George Gamov) WMAP CMB 14
15 h 2 Ω IR/opt/UV (60) CMB 1/50 X h 2 Ω X (61) 17: CMB T =2.728K Planck [13] h 2 Ω b = (57) (Cosmic Microwave Background) 2.7K G. Gamov 1965 A.A. Penzias R.W. Wilson J.C. Mather COBE FIRAS T CMB =2.728 ± 0.004K (58) Planck 17 [9] 2006 CMB h 2 Ω CMB = (T/2.73K) 4 (59) CMB ( 1.87mm 160GHz) 1 M/L j 0 M/L (M/L) clusters = (295 ± 53)h(M/L) (62) Ω =1 M/L (M/L) cr = (1025 ± 140)h Ω M =0.24 ± 0.14 (63) [6] M/L [3] Ω M = (64) 10% CMBIa 15
![2.6.2 CMB CMB 18: WMAP 5 h 2 Ω CDM =0.1131 ± 0.0034 (65) [13] 2.5 Ia P < ρ/3 Ω DE =0.726 ± 0.015 (66) 18 2.7 Planck (G =1,c =1, =1) (t pl =(G /c 5 ) 1/2 =5.](/docs-images/103/160913505/images/16-0.jpg "4 10 44 s, 1yr=3 10 7 s) s CMB = 4 ( ) kb T 3 ( ) 3 TCMB = 150 cm 3 45 c 2.73K (67) ( n b =2.46 10 7 Ωb h 2 ) cm 3 (68) 0.")
16 2.6.2 CMB CMB 18: WMAP 5 h 2 Ω CDM = ± (65) [13] 2.5 Ia P < ρ/3 Ω DE =0.726 ± (66) Planck (G =1,c =1, =1) (t pl =(G /c 5 ) 1/2 = s, 1yr= s) s CMB = 4 ( ) kb T 3 ( ) 3 TCMB = 150 cm 3 45 c 2.73K (67) ( n b = Ωb h 2 ) cm 3 (68) CMB s CMB /n b (69) (s/n)star 30 (70) CMB 1/10 5 CMB a 3 s CMB const T 1/a (71) 1 ω ω ω 1/a H 16
17 10 20 GeV GeV GeV GeV 10 8 GeV 100TeV 100GeV 100MeV 100keV 100eV 0.1eV 10-4 ev T rh? reheating T PQ Peccei-Quinn symetry breaking T WS Electroweak phase transition T QH Quark-Hadron transition T Neutrino decoupling T BBN BBN T eq Trec Tdec H Rec T 0 =2.734 K [s] x10 4 3x10 10 [yr] 19: e + + e 2γ u +ū, 100GeV 3 ( 3 6 (u,d,c,s,t,b) Λ 100MeV s/n b (69) 1MeV 0.5MeV 10 (BBN) BBN 7000K T rec 3800K T dec 3000K CMB 19 [29] 2.8 CMB (59) z 2.9 t l H 1/H Jeans L 17
18 μ dv = P μ φ (72) dt GμM/L 2 Gμ 2 L P/L c 2 sμ/l Gμ 2 L c2 s μ (73) L c s L J = (Gμ) 1/2 (74) Jeans L<L J L J L >L J μ μ L J Jeans δμ Gδμμ/L 2, c 2 sδμ/l δμ Jeans L J L J L J Jeans L <L J 3.1 (r) (b) P b /P r Ω CMB = ,Ω b =0.046, k B T CMB = ev, m p = 940MeV/c ρ P c s = c( P/ ρ) 1/ : CMB Jeans t rec c s c/ 3 Jeans χ J := a 1 L J c s ah 2ct 3a (75) χ J Hubble c/(ah) c 2 P s = ρ = 1 3 4ρ r (76) 4ρ r + ρ b Jeans T rec Jeans P g = n b k B T P g /P r n b /s 10 9 Jeans 4/
19 k/a d 2 X da 2 + k2 c 2 ( ) r H 1/2 a 4 H 2 X 0; X = a 3 ρ r Δ r 1+w r (77) Δ r δρ r /ρ r WKB Δ r = A ( ) 1/2 ( ) 1+wr kc r a 2 sin c r ρ r 0 a 2 H da (78) 21: SDSS Jeans χ Jm 1/(aH) dec χ Jm Jeans χ Jm χ J c/(ah) λ/a CMB 20 (Baryon Acaustic Oscillation) BAO SDSS 21[8]) 2 ξ(s) s s 100h 1 Mpc BAO t = t dec Δ r 2 A 2 ( ) (k) k ( a) 2 sin2 γ (79) (ah) dec γ = 1 0 ( ) 2 1/2 dx 0.5 (80) (1 + x)(4 + 3x) Δ r (t dec ) 2 k n = π (2n 1) (2n 1)π (81) (ah) dec 2γ (ah) dec CMB t = t dec t = t dec r plc (t dec ) χ plc (t dec ) χ plc = t0 t dt ( a =3t 0 1 a 1/2) (82) χ plc (t dec ) 3t 0 = l H (t 0 ), r plc (t dec )=3t 0 a(t dec ) r plc (t dec ) l H (t dec ) (83a) (83b) r plc (t dec ) l H (t dec ) t 0a dec = z 1/2 t dec 33. (84) dec (ah) dec /(ah) 0 c (ah) dec c 33H 0 100h 1 Mpc (85) 19
20 22: WMAP CMB (79) Δ r 2 k (ah/c) dec Δ r 2 Fourier ξ(s) s c/(ah) dec 1 [15] 21 BAO 3.3 CMB CMB Δ r =4δT/T CMB CMB t dec 2.723K 10 5 G.F.Smoot COBE DMR 2006 Boomerang MAXIMA-1 WMAP 22 WMAP 23 [19] ξ(s) 23: WMAP CMB Yl m (s) 2 l LFRW CMB t = t dec 2πr plc (t dec ) t = t dec 2l H (t dec ) 2πr plc (t dec ) (86) 1Doppler peak l l = k 1 r plc (t dec )= = 200γ π 200 k 1 2r plc (t dec ) (ah) dec l H (t dec ) 23 Ω K Ω K < 0.1 (87) [13]2 SNIa BAO 20
21 Ω Λ Cluster fgas SNIa CMB Ω m 24: WMAP, BAO X 24)(57), (65), (66) CMB 4.1 c Planck, Newton G Planck :L pl (Planck t pl (Planck :E pl ) CMB Planck t = t pl s Planck c, G, k pl Planck L pl cm k pl 1/L 2 pl H 2 = 8πG 3c 2 ρ c2 K a 2 (88) Ω K ( ) Ω K ρk (89) ρ m Planck Ω K < 0.1 Planck t = t pl ) Planck > L pl. (90) 1/a CMB l H (t dec ) 33 ((84) Planck t = t pl L pl
22 Planck Planck L pl Planck L L pl L pl ɛ L δφ N =(L/L pl ) 3 δφ ɛ N(L/L pl ) 1 = ɛ(l/l pl ) 1/2 (91) l H (t dec ) l H (t pl ) a(t pl) a(t dec ) t dec t pl T dec T pl ( tdec t eq ) 1/3 ( teq t pl ) 1/2 t = t dec δφ ɛ < 10 5 ɛ < (92) (91) L 2 δρ ρ L1/2 (93) CMB Harrison-Zeldovich t = t s t = t e H ȧ a = H a = a se H(t ts) (94) 25: Hubble c/h t = t e Hubble 25 t s <t<t e Hubble Hubble Λ H (Λ/3) 1/2 de Sitter 22
23 ( 26) V (φ) 1 φ P = 1 2 φ 2 V (φ), ρ = 1 2 φ 2 + V (φ) (95) φ 0 P ρ V (φ) (96) φ φ H Hubble d a dt (1/H) = d(ah) =ä>0 (97) dt d K/a 2 dt H 2 = 2K ä ȧ 2 < 0 (98) ä>0 4.3 H t = t e LFRW LFRW T r GeV H 0.165g(T r /E pl ) 4 E pl /L 3 pl (g = 100) 26: 4.4 L t = t e Hubble c/h 4.5 c/h Δt HΔt 4.6 HΔt Hubble Hubble 23
24 Harrison-Zeldvich CMB B- E- CMB B- B δg L h = κ 1 δg h =1/L Hubble [1] Asplund, M., Grevesse, N., Sauval, A., Allende Prieto, C. and Kiselman, D.: Astron. Astrophys. 417, (2004). [2] Baade, W.: Astrophys. J. 100, 137 (1944). [3] Bahcall, N. and Fan, X.: Astrophys. J. 504, 1 (1998). [4] Bekenstein, J. and Milgrom, M.: Astrophys. J. 286, 7 14 (1984). [5] Binney, J. and Merrifield, M.: Galactic Astronomy, Princeton Univ. Press (1998). [6] Carlberg, R., et al.: Astrophys. J. 462, 32 (1996). [7] Da Costa, L. N., et al.: Astrophys. J. 424, L1 (1994). [8] (D.J. Eisenstein et al), S. C.: Astrophys. J. 633, (2005). 24
25 [9] Fixsen, D., et al.: Astrophys. J. 473, 576 (1996). [10] Grevesse, N. and Sauval, A.: Adv. Space Res. 30, 3 11 (2002). [11] Hoyle, F. and Vogeley, M.: Astrophys. J. 607, (2004). [12] Hubble, E.: Proc.Nat.Acad.Sc.(USA)15, (1929). [28] I, (2007). [29], (1991). [30], (1992). [31], (1997). [13] Komatsu, E., et al.: Ap. J. Supple. 180, (2009). [14] Lokas, E. and Mamon, G.: Mon. Not. R. Astr. Soc. 343, (2003). [15] Matsubara, T.: Astrophys. J. 615, (2004). [16] Milgrom, M.: Astrophys. J. 270, 365 (1983). [17] Milgrom, M.: Astrophys. J. 270, 371 (1983). [18] Milgrom, M.: Astrophys. J. 270, 384 (1983). [19] Nolta, M., et al.: Ap. J. Suppl. 180, (2009). [20] Porter, R., Ferland, G., MacAdam, K. and Storey, P.: arxiv: (2008). [21] Ramella, M., Geller, M. and Huchra, J.: Astrophys. J. 384, (1992). [22] Riess, A., et al.: Astrophys. J. 659, (2007). [23] Riess, A., et al.: Astrophys. J. 699, (2009). [24] Sanders, R.: arxiv: (2008). [25] Sanders, R. and McGaugh, S.: Ann. Rev. Astron. Astrophys. 40, 263 (2002). [26] Trimble, V.: Ann. Rev. Astron. Astrophys. 25, (1987). [27] Xue, X.-X., et al.: Astrophys. J. 684, (2008). 25
C10-075 26 2 12 1 1 4 1.1............................. 4 1.2............................ 5 1.3................................... 5 2 6 2.1............................ 6 2.2......................... 6
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
![6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2](/thumbs/92/109118076.jpg "6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2")
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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20 11 1 KEK 2 (cosmological perturbation theory) CMB R. Durrer, The theory of CMB Anisotropies, astro-ph/0109522; A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
( ) Note Ω m = 1 Ω m : ( ) r-process α 1: 2 32T h(t 1/2 = y) 2 38U(t 1/2 = y) 2 35U(t 1/2 = 7.038
( ) Note 4 19 11 22 6 6.1 1 Ω m = 1 Ω m.3 6.1.1 : ( ) r-process α 1: 2 32T h(t 1/2 = 1.45 1 1 y) 2 38U(t 1/2 = 4.468 1 9 y) 2 35U(t 1/2 = 7.38 1 8 y) 2 44Pu(t 1/2 = 8.26 1 7 y) β / (J.A.Johnson and M.Bolte:
Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2
12 Big Bang 12.1 Big Bang Big Bang 12.1 1-5 1 32 K 1 19 GeV 1-4 time after the Big Bang [ s ] 1-3 1-2 1-1 1 1 1 1 2 inflationary epoch gravity strong electromagnetic weak 1 27 K 1 14 GeV 1 15 K 1 2 GeV
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2 = 8πG a 3c 2 ρ Kc2 a 2 + Λc2 3 (3), ä a = 4πG Λc2 (ρ
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................
A 99% MS-Free Presentation
A 99% MS-Free Presentation 2 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem
2 1 ds 2 = a 2 (η) ( dη 2 + γ ij dx i dx j ) (1.2) ( dt ) conformal time η η = a(t) a(t) (scale factor) t =const (3) R ijkl = K a 2 (t) (γ ikγ jl γ il
1 1.1 Robertson-Walker [ ds 2 = dt 2 + a 2 dr 2 ] (t) 1 Kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) sin 2 χ = dt 2 + a 2 (t) dχ 2 + χ 2 (dθ 2 + sin 2 θdφ 2 ) = dt 2 + sinh 2 χ a 2 (t) (1 + K 4 r2 ) [d r 2 + r (dθ
Lagrange.dvi
Lagrange 1 2006.2.10 1 Introduction CfA survey 2dF SDSS high-z Euler Lagrange Lagrange CDM Lagrange Lagrange 2 Newton Friedmann Newton Newton Newton [1,2,3,4,5] ρ + r ρu = 0, 1 t r u +u r u = 1 t r ρ rp
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N
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The Structure and Evolution of Stars I. Basic Equations. M r r =4πr2 ρ () P r = GM rρ. r 2 (2) r: M r : P and ρ: G: M r Lagrange r = M r 4πr 2 rho ( ) P = GM r M r 4πr. 4 (2 ) s(ρ, P ) s(ρ, P ) r L r T
( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e
![( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e](/thumbs/98/136160482.jpg "( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e")
( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
all.dvi
I 1 Density Matrix 1.1 ( (Observable) Ô :ensemble ensemble average) Ô en =Tr ˆρ en Ô ˆρ en Tr  n, n =, 1,, Tr  = n n  n Tr  I w j j ( j =, 1,, ) ˆρ en j w j j ˆρ en = j w j j j Ô en = j w j j Ô j emsemble
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
main.dvi
SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
2 g g = GM R 2 = 980 cm s ;1 M m potential energy E r E = ; GMm r (1.4) potential = E m = ;GM r (1.5) r F E F = ; de dr (1.6) g g = ; d dr (1.7) g g g
1 1 (gravitation) 1.1 m F a ma = F (1.1) F a m F 1.1 m F a (1.1) m a F m F a m a F F a m 0 0 1.2 (universal gravitation) (potential) M m gravitational force F r F = ; GMm r 2 (1.2) G = 6:67 10 ;8 dyn cm
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
cm λ λ = h/p p ( ) λ = cm E pc [ev] 2.2 quark lepton u d c s t b e 1 3e electric charge e color charge red blue green qq
![cm λ λ = h/p p ( ) λ = cm E pc [ev] 2.2 quark lepton u d c s t b e 1 3e electric charge e color charge red blue green qq](/thumbs/89/98710373.jpg "cm λ λ = h/p p ( ) λ = cm E pc [ev] 2.2 quark lepton u d c s t b e 1 3e electric charge e color charge red blue green qq")
2007 2007 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 2007 2 4 5 6 6 2 2.1 1: KEK Web page 1 1 1 10 16 cm λ λ = h/p p ( ) λ = 10 16 cm E pc [ev] 2.2 quark lepton 2 2.2.1 u d c s t b + 2 3 e 1 3e electric charge
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
keisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
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--- = ---- 16 Z 8 0 8 8 0 Big Bang 8 8 s-process 50 r-process 8 50 N r-process s-process Hydrogen 71% Helium 8% Others 1.9% Heay 4-4% lements(>ni p-process (γ process? r-process s-process Big Bang H,He
B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................
gr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
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20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
all.dvi
72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)
1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )
: 8.2: A group (i.e. a very small cluster) of galaxies superimposed on a x-ray image from the ROSAT satellite
1 8 8.1 8.1.1 8.1: ( Ω = ρ/ρ c ) (Fukugita, M. et al., APJ 503 (1998) 518) ( 15%) (z 0 ) 1.................. 0.0026 h 1 0.0043 h 1 0.0014 h 1 A 2..................... 0.00086 h 1 0.00129 h 1 0.00051 h
O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
Formation process of regular satellites on the circumplanetary disk Hidetaka Okada Department of Earth Sciences, Undergraduate school of Scie
Formation process of regular satellites on the circumplanetary disk Hidetaka Okada 22060172 Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group
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3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0
![5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0](/thumbs/99/141438380.jpg "5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0")
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â = Tr Âe βĥ Tr e βĥ = dγ e βh (p,q) A(p, q) dγ e βh (p,q) (5.2) e βĥ A(p, q) p q Â(t) = Tr Â(t)e βĥ Tr e βĥ = dγ() e βĥ(p(),q())
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,