2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) (1 8 ) erg, (2.6) /k B 1 11 / K
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1 2 2.1? [ ] L 1 ε(p) = 1 ( p 2 2m x + p 2 y + pz) 2 = h2 ( k 2 2m x + ky 2 + kz) 2 n x, n y, n z (2.1) (2.2) p = hk = h 2π L (n x, n y, n z ) (2.3) n k p 1 i (ε i ε i+1 )1 1 g = 2S /2 g = 2 ( p F k F ) (= p 2 F/2m) v F = p F /m 1 ε 1 N(ε) N(ε) = 2 4π 3 p3 F V 1 (2π h) 3 = V 3π 2 ( ) pf 3 V = h 3π 2 k3 F. (2.4) 1 (spinless fermion) 2 13
2 2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) (1 8 ) erg, (2.6) /k B 1 11 / K ε < g/cm 3 n = cm 3 (2.7) 14
3 k F = ( 3π 2 n ) 1/3 = ( ) 1/3 = ( ) 1/3 1 7 = cm 1 (2.8) p F = hk F = = g cm/sec (2.9) v F = p F m e = = cm/sec (2.1) 1km/sec = h2 2m k2 F = ( ) ( ) 2 = = erg (2.11) 3.1eV(= / ) 3 T F = k B = [ ] E = 2V (2π h) 3 = V 5π 2 ( pf h pf : E = N p2 2m = N 2m = K (2.12) pf p 2 V 2m 4πp2 dp = 2π 2 h 3 p 4 dp = 1 V p 5 F m 1π 2 h 3 m ) 3 p 2 F 2m = 3 5 N. (2.13) pf pf p 2 p 2 dp p 2 dp = N p 5 F/5 2m p 3 F/3 = N 2m 3 5 p2 F. (2.14) E = 3 ( 5 N h2 3π 2 N ) 2/3 (2.15) 2m V : P = E V = [ 3 V 5 N h2 2m ( 3π 2 N ) 2/3 ] V = 2 h 2 5 2m ( ) 3π 2 2/3 ( ) N 5/3 V = 2 E 3 V = 2 N 5 V = 2 5 n. (2.16) 15
4 P = 2 E 3 V (2.17) 3 κ 1 κ P = V V = V V (3π2 ) 2/3 5 (2.18) h 2 ( ) N 5/3 (2.19) m V = (3π2 ) 2/3 h 2 ( ) N 5/3 3 m V (2.2) = 2 h 2 ( 3π 2 n ) 2/3 2 n = 3 2m 3 n (2.21) ( ) 1 κ = = erg/cm 3. (2.22) C 11 C κ = C C 12 3 [ ] = 1 3 ( ) = dyn/cm 2 (2.23) (white dwarf) M R U = 3GM 2 5R GM 2 R (2.24) 3 4 (1968)
5 (a) (b) 2.2: (a) B (b) : ( science.php?gallery VignVCMId=ba8d9f48ddfe411VgnVCM1ee2a8cRCRD&no=1) ( K 2K = i r i ṗ i = i r i i U (2.25) ( -1) 2K = U (2.26) m N K = E = 3 ( 5 N h2 3π 2 N ) 2/3 2m V (2.27) h2 M 5/3 mm 5/3 N R 2 (2.28) (2.24) (2.28) M 1/3 R h 2 Gmm 5/3 N ( ) ( ) 5/3 1 2 g 1/3 cm (2.29) 17
6 ρ M/R 3 M 2 2 (2.28) (2.24) ( ) E(R) = h2 M 5/3 mm 5/3 N R GM 2 2 R (2.3) 2 R 1 R (2.29) (2.28) ( ) E = 3 ( M 4 Ncp F M V ) 1/3 M 4/3 R (2.31) p + e n + ν e (2.28) m m N m n (2.29) M 1/3 R 1 17 g 1/3 cm (2.32) (2.29) (2.31) g 1 4 km 1km 5 (Chandrasekhar) 18
7 f Μ 2.3: k B T/µ = 1/1, k B T/µ = 1/1 k B T/µ = Ε [ ] f(ε) N = i f(ε i ), E = i ε i f(ε i ) (2.33) 1 i dε 1 S V i (2π h) S = S 3 d 3 p (2.34) g = 2S + 1 ε = p2 2m, i dε dp = p m = = = 2mε m = 2ε m (2.35) gv 4πp 2 dp (2π h) 3 gv 2π 2 h 3 2mε dp dε dε gv m3/2 dε 2 1/2 3 ε1/2 π2 h dεd(ε) (2.36) 19
8 D f Μ 2.4: 3 D(ε) D(ε)f(ε) k B T/µ = 1/1 µ D(ε) = 1 (density of states) 1 N = E = gv m3/2 2 1/2 π 2 h 3 ε1/2 (2.37) D(ε)f(ε)dε, (2.38) εd(ε)f(ε)dε (2.39) 1 1 ε(p) D( ) = 2V 4πp2 dp (2π h) 3 = V dε p=pf π 2 h 3 mp F = 3 V p 3 F 2m 2 3π 2 h 3 = 3N (2.4) 2 N(ε) ε 2/3 1 A n A A n n A = e βen A n n e βen 1 i N i {N i } n {N i } p 2 F Ε N = i N i, E n = i N i ε i 2
9 f i N i f i = f(ε i ) = N i A 1 a a i a 1 i A = i a i N i = i a i N i = i a i f i A L 1 i p = hk = 2π h L n = 2π h L (n x, n y, n z ) (n x, n y, n z ) n 1 n p n 2π h/l i (??) S S = S [ ] d 3 n S S = S ( ) L 3 2π h ε 1 1 N(ε) = ε D(ε )dε dn(ε) dε d 3 p (2.37) dn(ε) N = f(ε)dε dε = N(ε)f(ε) + N(ε) ( ddε ) f(ε) dε ( ddε ) f(ε) = N(ε) = D(ε) (2.41) dε (2.42) N() = N(ε)f(ε) (as ε ) f (ε) O(k B T ) (k B T/ ) 2 N = N(µ) + π2 6 D (µ)(k B T ) 2 + (2.43) = N(µ ) + D(µ )(µ µ ) + π2 6 D (µ )(k B T ) 2 + (2.44) 21
10 f Ε Μ 2.5: f (ε) k B T/µ = 1/1, k B T/µ = 1/1 k B T/µ = 1 N(µ ) = N µ µ = π2 6 µ = π2 6 (2.36) D (µ ) D(µ ) (k BT ) 2 + (2.45) D ( ) D( ) (k BT ) 2 + (2.46) D (ε) D(ε) = d ln D dε = 1 2ε (2.47) µ = π2 (k B T ) ε F ( ) 2 ( ) 4 = 1 π2 kb T kb T + O (2.48) 12 2 ( ) 22
11 Sommerfeld ( F (ε) df(ε) ) dε = F (µ) + 2 C 2n (k B T ) 2n d2n F (ε) dε n=1 dε 2n (2.49) ε=µ f(ε) = 1/[exp{(ε µ)/k B T } + 1], F (ε) C 2n 1 x 2n 1 dx (2.5) Γ(2n) e x + 1 = ( n) ζ(2n) (2.51) Γ(m) x m 1 e x dx, (2.52) n! = Γ(n + 1) Riemann ζ(m) r=1 (2.48) 1 r m (2.53) F (ε)f (ε)dε = F (µ) + π2 6 (k BT ) 2 F (µ) + 7π4 36 (k BT ) 4 F (µ) + (2.54) F (ε) = ϕ(ε) ϕ(ε)f(ε)dε = µ ϕ(ε)dε + π2 6 (k BT ) 2 ϕ (µ) + 7π4 36 (k BT ) 4 ϕ (µ) + (2.55) [ ] (2.54) ϕ(ε) = εd(ε) µ = µ µ E = = µ + µ εd(ε)f(ε)dε εd(ε)dε + π2 6 (k BT ) 2 (D(µ) + µd (µ)) + = E + µ D(µ ) µ + π2 6 (k BT ) 2 (D(µ ) + µ D (µ )) + = E + π2 6 (k BT ) 2 D(µ ) +. (2.56) 23
12 (2.39) E = E + π2 6 ( ) 3N 2 (k B T ) 2 + = E + π2 4 N (k BT ) 2 + ε F = 3 ( ) 2 ( ) 4 5 N 1 + 5π2 kb T kb T + O. (2.57) 12 C = de dt = π2 2 Nk k B T B (2.58) k B T / k B T / k B T k B T (2.47) G = Nµ = N 1 π2 12 ( ) 2 ( ) 4 kb T kb T + O. (2.59) P V = (2/3)E G = E + P V T S = 5 3 E T S (2.6) : S = (5/3)E G T = π2 2 Nk k B T B + (2.61) (2.57) [ ] F = E T S = 3 5 N 1 5π2 12 ( ) 2 ( ) 4 kb T kb T + O (2.62) 2 24
13 2.6: T = D(ε)f(ε) ( ) ( ) (Lande g ) (Bohr ) ( )=1/2 2 µ B H = µ B H εp+ = p2 2m µ BH (2.63) εp = p2 2m + µ BH (2.64) M + M H = ε = M = M + M = µ B (N + N ) D(ε + µ B H) D(ε µ B H) = µ B f(ε)dε µ B f(ε)dε 2 2 D(ε) = µ B (f(ε µ B H) f(ε + µ B H)) dε 2 ( ε ) f(ε) µ 2 BH = µ 2 BH ( D(ε) dε D(µ) + π2 6 D (µ )(k B T ) 2 + ). (2.65) D(ε) 25
14 χ = M H = µ 2 BD( ) (2.66) H Pauli (paramagnetism) ( ) M = µ 2 BH D(µ ) + D (µ )(µ µ ) + π2 6 D (µ )(k B T ) 2 + [ ( = µ 2 BH D(µ ) + D (µ ) π2 D ) ] (µ ) 6 D(µ ) (k BT ) 2 + π2 6 D (µ )(k B T ) 2 + [ ( = µ 2 BH D(µ ) + π2 D (µ ) (D (µ )) 2 ) ] (k B T ) D(µ ) = µ 2 BH D(µ ) + D(µ ) π2 d 2 ln D(ε) 6 dε 2 (k B T ) 2 + ε=µ ( ) 2 ( ) 4 = µ 2 BHD(µ ) 1 π2 kb T kb T + O. (2.67) 12 D(ε) ε 1/2 χ = χ 1 π2 12 ( ) 2 ( ) 4 kb T kb T + O. (2.68) 26
[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3
![[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3](/thumbs/103/161261933.jpg "[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3")
4.2 4.2.1 [ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 S i S j (4.39) i, j z 5 2 z = 4 z = 6 3 z = 6 z = 8 zn/2 1 2 N i z nearest neighbors of i j=1
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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
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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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
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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
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![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())
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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
(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou
(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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