1 (Contents) (4) Why Has the Superstring Theory Collapsed? Noboru NAKANISHI 2 2. A Periodic Potential Problem
|
|
- あつとし ひろなが
- 5 years ago
- Views:
Transcription
1
2 1 (Contents) (4) Why Has the Superstring Theory Collapsed? Noboru NAKANISHI 2 2. A Periodic Potential Problem in Quantum Mechanics (4) Kenji SETO Anti-commutativity among Linearly Independent Imaginary Units Katsusada MORITA The Parity Operator Minoru YONEZAWA The Parity Operator A Comment Noboru NAKANISHI Editorial Comments Shozo NIIZEKI, Tadashi YANO 29
3 Why Has the Superstring Theory Collapsed? * 1 Noboru NAKANISHI * *3 30 *4 72, 345 (1986), ( , 48, 44 (1993). superalgebra graded algebra indecomposable extension SUSY 1 x µ ( ) 1 ( ) *5 SUSY *1 *2 nbr-nak@trio.plala.or.jp *3 *4 3 woit/wordpress P. Woit Some Early Criticism of String Theory, October 30, 2006 *5 3 2
4 SUSY 2 SUSY 2 2 SU(3) SU(2) U(1) *6 S SUSY SUSY SUSY S 2 SUSY SUSY SUSY *7 SUSY SUSY SUSY 1/2 SUSY *6 *7 SUSY SUSY SUSY 3
5 S (NG) SUSY 1/2 NG SUSY NG NG SUSY SUSY (SUGRA) NG SUSY S SUSY SUSY SUSY 1 SUSY 2 1 SUSY CERN LHC SUSY is dead SUSY SUSY 4
6 1/2 *8 BRS *9 x µ x µ NG 2 SU(2) U(1) SU(2) U(1) U(1) U(1) 20 SUSY SUSY SUSY ( *8 *9 FP BRS 5
7 x µ SUSY x µ SUSY SUSY SUSY 3 S 1 2 S S s t 2 2 s s S * 10 l * 11 l s l = α(s) α(s) * 12 α(s) = α 0 + α s *10 *11 *12 6
8 α s 1 t t t α(s) (s 0) log t 1 α 0 1 α 0 = 1 1 s < 0 s * 13 N 2 2 N 4 N = 4 B( α(s), α(t)) = Γ( α(s))γ( α(t)) Γ( α(s) α(t)) = 1 0 dx x α(s) 1 (1 x) α(t) 1 * 14 N N 3 * 15 3 SU(3) *13 *14 N = *15 N 3 7
9 2 2 D 2 * 16 D 2 * 17 2 * 18 D = D * 19 4 SUSY 2 α 0 = 2 2 c h 1/2π ħ 1 20 *16 *17 S *18 *19 D = 26 8
10 α 2 s [l = ] 1 α * /2 * n + 2 (n = 0, 1, 2, ) n = SO(32) E 8 E 8 SU(3) SU(2) U(1) SU(5) SO(10) * 22 * *21 2 *22 9
11 p + 1 Dp 11 M * 23 4 * (2010) 10
12 * * 25 * 26 6 a s (a = (1/x) α > 1) s = p µ p µ = p 2 0 p 2 p µ p 0 p 2 0 *24 D 3 * (2012) *26 L. Smolin The Troubles with Physics (2006) p.279-p.282 S. Mandelstam Mandelstam 1 11
13 2 T T T T T* T T* T T* * T T* 2 * 28 5 The only game in town * 29 The only Game in Town K. Vonnegut * 30 A guy with the gambling sickness loses his shirt every night in a poker game. Somebody tells him that the game is crooked, rigged to send him to the poorhouse. And he says, haggardly, I know, I know. But it s the only game in town. *27 Φ = 0 Φ 1 T Φ T* T* *28 M. Abe and N. Nakanishi, Prog. Theor. Phys. 115, 1151 (2006) 113, 76 (2006) *29 Not Even Wrong (2006) P. Woit *30 B. Schroer, String theory and the crisis in particle physics, special volume of I. J. M. P. D (2006) 12
14 (4) A Periodic Potential Problem in Quantum Mechanics (4) Kenji Seto 1 Kronig- Penney Schrödinger [ ħ2 d 2 ] 2m dx 2 + V (x) Ψ = EΨ (1.1) V (x) 2l V (x + 2l) = V (x) l x < l V (x) = V 0 ( x l ) 2 (1.2) V 0 x, V 0, E x l x, 2ml 2 ħ 2 V 0 V 0, 2ml 2 ħ 2 E E (1.3) [ d 2 ] dx 2 + E V (x) Ψ = 0 (1.4) V (x) 2 V (x + 2) = V (x) 1 x < 1 V (x) = V 0 x 2 (1.5) 2 µ µ = (4V 0 ) 1/4 (2.1) seto@pony.ocn.ne.jp 13
15 x z E κ z = µx, κ = E µ (2.2) 1 x < 1 [ d 2 dz 2 + κ z2 4 ] Ψ = 0 (2.3) Weber (Weber ) D κ (z) 1 F 1 D κ (z) = 2 κ/2 [ π e z2 /4 1 ( Γ ((1 κ)/2) 1 F 1 κ 2, 1 2 ; z 2 ) 2 2 z ( 1 κ Γ ( κ/2) 1 F 1, ; z 2 )] 2 (2.4) *1 Weber Hermite H n (z) κ n D n (z) = e z2 /4 H n (z) Hermite Weber 1 D κ 1 (iz) = 2 (κ+1)/2 [ π e z2 /4 1 ( κ + 1 Γ ((κ + 2)/2) 1 F 1, 2 1 ) 2 ; z2 2 2 iz ( κ )] Γ ((κ + 1)/2) 1 F 1, 2 2 ; z2 2 (2.5) Floquet (2.3) ( S 1 (z) =e z2 /4 κ F 1, 2 ( S 2 (z) =ze z2 /4 κ F 1, 2 1 ) 2 ; z ; z2 2 (2.3) 2 *2 S 1 (z) S 2 (z) S 1, S 2 Wronskian ) (2.6) W (z) = S 1 (z)s 2(z) S 1(z)S 2 (z) (2.7) z S 1, S 2 (2.3) Wronskian z = 0 W (z) 1 (2.8) (1.4) 1 x < 1 A, B Ψ(x) = AS 1 (µx) + BS 2 (µx) (2.9) *1 3 ( ) p λ κ *2 Kummer ( p.67) 1 F 1 (α, γ; z) = e z 1F 1 (γ α, γ; z) S 1, S 2 (2.4)
16 Floquet 1 K (0 K π) e ik e ik 1 x < 3 Ψ(x) = e ik[ ( ) ( )] AS 1 µ(x 2) + BS2 µ(x 2), or Ψ(x) = e ik [ ( ) ( )] AS 1 µ(x 2) + BS2 µ(x 2) (2.10) K 2 1 K = 0, π 2 K 1 K 0 < K < π (2.9) (2.10) x = 1 AS 1 (µ) + BS 2 (µ) = e ik[ AS 1 (µ) BS 2 (µ) ] AS 1(µ) + BS 2(µ) = e ik[ AS 1(µ) + BS 2(µ) ] (2.11) S 1, S 2 S 1, S 2 ( ) ( ) ( ) (1 e ik )S 1 (µ) (1 + e ik )S 2 (µ) A 0 (1 + e ik )S 1(µ) (1 e ik )S 2(µ) = B 0 (2.12) A, B cos(k) = S 1 (µ)s 2(µ) + S 1(µ)S 2 (µ) (2.13) (2.8) Wronskian 1 K 1 S 1, S 2 κ, E A, B (2.11) 1 A = (1 + e ik )S 2 (µ), B = (1 e ik )S 1 (µ) (2.14) x n 2n 1 x < 2n + 1 Ψ(x) = e ikn[ (1 + e ik ( ) )S 2 (µ)s 1 µ(x 2n) (1 e ik ( )] )S 1 (µ)s 2 µ(x 2n) (2.15) 3 2 E, E κ, K κ, κ, K, K S 1, S 2 Ψ(x, E), S i (z, κ), i = 1, 2 15
17 (2.15) Ψ(x, E)Ψ(x, E )dx = n= 2n+1 2n 1 ( Ψ(x, E)Ψ(x, E )dx = n= lim M M n= M ) 1 e i(k K )n Ψ(x, E)Ψ(x, E )dx 1 (3.1) e i(k K )n = 2πδ(K K ), 0 < K, K < π (3.2) E, E (1.4) [ d 2 ] dx 2 + E V (x) Ψ(x, E) = 0, [ d 2 ] dx 2 + E V (x) Ψ(x, E ) = 0 (3.3) E 1 Ψ(x, E ) 2 Ψ(x, E) d [ Ψ(x, E) dψ(x, E ) dx dx Ψ(x, E)Ψ(x, E )dx = dψ(x, E) ] Ψ(x, E ) (E E )Ψ(x, E)Ψ(x, E ) = 0 (3.4) dx 1 [ E E Ψ(x, E) dψ(x, E ) dx dψ(x, E) ] 1 Ψ(x, E ) dx 1 n = 0 (2.15) 16 S 1, S 2 Wronskian (2.8) (2.13) 1 1 Ψ(x, E)Ψ(x, E )dx = 2µ E E [ S 1 (µ, κ)s 2 (µ, κ) [ (1 + e i(k K ) ) cos(k ) (e ik + e ik ) ] (3.5) S 1 (µ, κ )S 2 (µ, κ ) [ (1 + e i(k K ) ) cos(k) (e ik + e ik ) ]] (3.6) (3.2) (3.6) (3.1) E, E (2.2) κ, κ Ψ(x, E)Ψ(x, E )dx = 4π [S µ(κ κ 1 (µ, κ)s 2 (µ, κ) [ (1 + e i(k K ) ) cos(k ) (e ik + e ik ) ] ) S 1 (µ, κ )S 2 (µ, κ ) [ (1 + e i(k K ) ) cos(k) (e ik + e ik ) ]] δ(k K ) (3.7) 1 κ (2.13) K K κ κ κ κ K = K κ κ κ κ 0/0 l Hôpital κ κ κ Ψ(x, E)Ψ(x, E )dx = 8π µ S 1(µ, κ)s 2 (µ, κ) sin(k) dk dκ δ(k K ) (3.8) 16
18 0 < K < π sin(k) S 1 (µ, κ)s 2 (µ, κ) dk/dκ E Ψ(x, E)Ψ(x, E )dx = N 2 (E)δ(E E ), N 2 (E) = 8πµ S 1 (µ, κ)s 2 (µ, κ) sin(k) (3.9) N(E) Ψ(x, E)/N(E) K (2.10) 2 sin(k), dk/dκ 4 (2.6) S 1 (z, κ), S 2 (z, κ) κ S 1 (z, 0) = e z2 /4, S 2 (z, 1) = ze z2 /4 (4.1) κ 1-1, 1-2 z κ S 1 ( 1-1) S 2 ( 1-2) 0 z 5, 0 κ 10 S 1 κ 1.6 z S 2 0 κ < 1 z κ 1 2 z S 1, S 2 κ 4 z S S 1 (z, κ) 1-2 S 2 (z, κ) 17
19 (2.13) V 0 E K V 0 = 50 2 K (3.8) S 1 (µ, κ)s 2 (µ, κ) S 1 (µ, κ)s 2 (µ, κ) dk/de = dk/µ 2 dκ 2 K E (V 0 = 50) 3 V 0 V 0 - E (2.13) V 0 - E 5 4 (2.6) S 1 (z), S 2 (z) 50 18
20 100 z 5.5 z z z z 5.5 V 0 (2.1) µ (2.2) z, κ V 0 < 0 [ ] 19
21 1 1) Anti-commutativity among Linearly Independent Imaginary Units Katsusada Morita 2) n a 1 a n a 2 n a 2 = a 2 z z 2 = z 2 1 x, y x, y xy = x y 2 z = x + iy z 2 = x 2 + y 2 3) z 1 z 2 = z 1 z 2 3 [1] a = a 0 + ia 1 + ja 2, b = b 0 + ib 1 + jb 2 {1, i, j} 1 4) ( ) ab : ab = (ab) 0 + i(ab) 1 + j(ab) 2 + ija 1 b 2 + jia 2 b 1 (ab) 0 = a 0 b 0 a 1 b 1 a 2 b 2 (ab) 1 = a 0 b 1 + a 1 b 0 (ab) 2 = a 0 b 2 + a 2 b 0 (1.1) i 2 = j 2 = 1 ij + ji = 0 (1.2) 5) (1.1) ij(a 1 b 2 a 2 b 1 ) ij(ab) 3 ab 2 = (ab) (ab) (ab) (ab) 2 3 = a 2 b 2 a 2 z 2 = x 2 + y 2 a 2 = a a a 2 2 ab 2 4 ( {1, i, j, ij} ) ij {1, i, j} ij = α + βi + γj, 1) 2) kmorita@cello.ocn.ne.jp 3) z = 0 z 2 = 0 z = 0 x = y = 0. {1, i} a = a 0 + ia 1 + ja 2 = 0 a 2 = a a2 1 + a2 2 = 0 a = 0 a 0 = a 1 = a 2 = 0 {1, i, j} 1 4) 1 Hamilton Dickson [2] 1 {i, j, k} Frobenius 5) ij = ji 0 = i 2 j 2 = (i j)(i+j) j = ±i a a = a 0 +i(a 1 ±a 2 ) ij ji ij = αji, α( +1) R (ij) 2 = (ji) 2 = α α 2 = 1 α 1 α = 1 (1.2) Hamilton [3] 20
22 α, β, γ (ij) 2 = 1 6) i(ij) = (ij)i, j(ij) = (ij)j ij {i, j} {i, j, ij} 1 7) ab 2 ab {1, i, j, ij} 3 a 2 b 2 ab 2 = a 2 b 2 Hurwitz a = b (1.2) (a 2 ) 3 = 0 a 2 2 = a 4 {i, j} (1.2) (1.2) {1, i, j} 1 a a 2 2 = a 4 {1, i, j} 1 2 (1.2) {1, i, j} 1 3 n (n 1) 1 n a a 2 = a 2 4 Hamilton [1] 2 1 Hamilton a [1] a = a 0 + ia 1 + ja 2, i 2 = j 2 = 1, a 0, a 1, a 2 R (2.1) b = b 0 + ib 1 + jb 2, b 0, b 1, b 2 R (1.1) ij ji 2 ( ) z 1 z 2 = z 1 z 2 a = b a µ = b µ (µ = 0, 1, 2) a 2 a 2 = a 2 0 a 2 1 a ia 0 a 1 + 2ja 0 a 2 + (ij + ji)a 1 a 2 (2.2) (1.2) a 2 a 2 2 = a 4 : a 2 2 = (a 2 0 a 2 1 a 2 2) 2 + (2a 0 a 1 ) 2 + (2a 0 a 2 ) 2 = (a a a 2 2) 2 = a 4 (2.3) (2.3) {i, j} (1.2) Hamilton [1] Hamilton ) Hamilton [1] [3] a = a 0 + ia 1 + ja 2 a = (a 0, a 1, a 2 ) 1 = (1, 0, 0), i = (0, 1, 0), j = (0, 0, 1) {1, i, j} α 0 + iα 1 + jα 2 = 0, α 0, α 1, α 2 R (2.4) 6) Hamilton k = ij, k 2 = 1 7)
23 α 0 = α 1 = α 2 = 0 {i, j} 1 iα 1 + jα 2 = 0, α 1, α 2 R (2.5) α 1 = α 2 = 0 (2.5) 2 (α α 2 2) + (ij + ji)α 1 α 2 = 0, ij + ji R (2.6) {i, j} (1.2) α 1 = α 2 = 0 {i, j} 1 {1, i, j} 1 (i ± j) 2 < 0 (i ± j) w = 1 w = i ± j {1, i, j} 1 (i ± j) 2 < 0 ij + ji = 2α R, α < 1 8) α 0 [2] {i, j} α < 1 a, b I = i, J = ai + bj, a = ±α/ 1 α 2, b = ±1/ 1 α 2 (2.7) I 2 = J 2 = 1, IJ + JI = 0 (2.8) I = i, J = j ( (2.1) a 1,2 ) (1.2) 1 {i, j} 3 (n 1) 1 [4] a, b {1, e i } i=1,2,3 l a = a 0 e 0 + a i e i + a 4 l a 0, a i, a 4 R b = b 0 e 0 + b i e i + b 4 l b 0, b i, b 4 R e 2 0 = e 0, l 2 = 1, le i = e i l (i = 1, 2, 3) ab = (ab) 0 e 0 + (ab) i e i + (ab) (4i) e i l + (ab) 4 l (i, j, k = 1, 2, 3 l [4] {e i, le i, l} {e A } A=1,2,3,,7 ): (ab) 0 = a 0 b 0 a i b i a 4 b 4 (ab) i = (a 0 b i + a i b 0 ) + ϵ jki a j b k (ab) 4i = a i b 4 a 4 b i (ab) 4 = a 0 b 4 + a 4 b 0 (2.9) (2.10) ab 2 = (ab) (ab) 2 i + (ab)2 4 + (ab) 2 4i ab 2 = a 2 b 2 a = b (a 2 ) 4i = 0 a 2 a 2 2 = a 4 8) ij + ji = 2β R, β < 1 (2.6) α 1,2 α 1 = α 2 = 0 22
24 3 n a = a 0 + a 1 e 1 + a 2 e a n 1 e n 1, a µ (µ = 0, 1,, n 1) R (3.1) 1 {e i } i=1,2,,n 1 e i e j + e j e i = 2δ ij (3.2) e i e j = δ ij + c ijk e k, c jik = c ijk {e i } {E k } e i e j = δ ij + c ijk E k, c jik = c ijk, E k E k = δ kk a 9) 1 1 e 1 = (0, 1, 0, 0,, 0, 0) e 2 = (0, 0, 1, 0,, 0, 0). e n 1 = (0, 0, 0, 0,, 0, 1) (3.3) α 1 e 1 + α 2 e α n 1 e n 1 = (0, α 1, α 2,, α n 1 ) = (0, 0, 0,, 0) α 1 = α 2 = = α n 1 = 0 e i e 2 1 = e 2 2 = = e 2 n 1 = ( 1, 0,, 0) e 0 ( ) (α 1 e 1 + α 2 e α n 1 e n 1 ) 2 = i i j αi α i α j (e i e j + e j e i ) = 0 2 i j e i e j + e j e i = 0 (i j ) (3.4) i α i = 0 {e i } i=1,2,,n ) 1 (3.4) 11) i, j(i j) {1, e i, e j } 1 (e i ± e j ) 2 < 0 e i e j + e j e i = 2α ij, α ij < 1 {e i, e j } {I i, I j } {I i, I j } (2.7), (2.8) {I i, I j } = {e i, e j } (3.4) {e i } i=1,2,,n 1 1 (3.4) (3.4) ( Einstein ) a 2 2 = (a 0 + a 1 e 1 + a 2 e a n 1 e n 1 ) 2 2 = (a 0 + a i e i ) 2 2 = a a 0 a i e i + a i a j e i e j 2 = a a 0 a i e i + a i a j (e i e j + e j e i )/2 2 = (a 2 0 a 2 i ) 2 + (2a 0 a i ) 2 = (a a 2 i ) 2 = a 4 (3.5) 9) e 1 = i, e 2 = j E 3 = e 1 e 2, E3 2 = 1 10) j 1 e 1 e j = e j e 1 n a = a 0 + a i e i a = a 0 ± i(a a n), i e 1 11) [4] A 23
25 a 2 = a 2 (3.6) (2.3) n 1 a = (a 0, a 1,, a n 1 ) 2 a 2 ab = a b a = b (3.6) (3.6) (3.6) n a 2 a 2 n (3.6) (3.4) a {1, e 1,, e n 1 } 1 ( ) 4 Hamilton [1] ( {i, j}(i 2 = j 2 = 1) ) {1, i, j} 3 {1, i, j} 1 Hamilton a a 2 = a 2 {i, j} a a aa Hamilton Dickson [2] a 1 a 2 = a 2 n a 2 a 2 n n a {e 0, e 1,, e n 1 } 1 Cayley-Dickson a 2 = a 2 ab = a b R, C, H, O Hurwitz, ( ) [1] W. R. Hamilton, Quaternions, Proc. Roy. Irish Acad. vol. L(1945), [2] L. E. Dickson, Linear Algebras, Cambridge: at the University Press, [3] ( 2014). [4] K. Morita, Quasi-Associativity and Cayley-Dickson Algebras, PTEP, 2014, 013A03 (19 pages). 24
26 1 The Parity Operator Minoru Yonezawa 2 1 (Parity operator) P 3 2 [ ] [ 1 n P = ( 2x)n n! x n n=0 m=0 ] [ 1 m ( 2y)m m! y m l=0 ] 1 l ( 2z)l l! z l (2.1) 3 1 (Taylor expansion) f(x + a) = n=0 ( f( x) = exp a d ) f(x) dx = a= 2x ( 1 dn an f(x) = exp a d ) f(x) (2.2) n! dxn dx n=0 1 n! dn ( 2x)n f(x) (2.3) dxn (2.3) (2.3) a f(x) x a = 2x 4 P f(x) = f( x), 1 dn P = ( 2x)n n! dx n (2.4) n=0 1 2 m-yonezawa@mtc.biglobe.ne.jp
27 3 p x = iħ d dx x x + a (unitary operator) U = e i a ħ p x f(x + a) = Uf(x) (2.5) 3 x x pp (translation operator) 2 1 (2.2) a = 2x n 2 f( x) = n=0 ( 1 dn ( 2x)n f(x) = exp 2x d ) f(x) (3.1) n! dxn dx ( exp 2x d ) = dx (3.1) n=0 ( 1 2x d ) n (3.2) n! dx ( ( 2x) n dn dx n 2x d ) n (3.3) dx (2.2) (2.2) a = 2x f(x + a) = f( x) a = a = 2x 26
28 (2.3) 1 (2.3) 2 Taylor Taylor f(x+h) x+h x f(x) f (n) (x), (n = 1, 2, 3, ) f(x + h) h f(x + h) = a 0 + a 1 h + a 2 h 2 + a 3 h 3 + a 4 h a n h n + (3.4) h a 0, a 1, a 2, a 3, a 4, (3.4) h = 0 a 0 = f(x) (3.4) h f (x + h) = a 1 + 2a 2 h + 3a 3 h 2 + 4a 4 h na n h n 1 + (3.5) (3.5) h = 0 a 1 = f (x) (3.5) h f (x + h) = 2a a 3 h + 3 4a 4 h (n 1)na n h n 2 + (3.6) (3.6) h = 0 a 2 = 1 2! f (x) (3.6) h f (x + h) = 1 2 3a a 4 h + + (n 2)(n 1)na n h n 3 + (3.7) (3.7) h = 0 a 3 = 1 3! f (x) a n = 1 n! f (n) (x) (3.4) f(x + h) = f(x) + hf (x) + h2 2! f (x) + h3 3! f (x) + + hn n! f n (x) + (3.8) ( ) 27
29 The Parity Operator A Comment 1 Noboru NAKANISHI 2 P P f(x) = d dx P a x dyf(y) (1) dyk(x, y)f(y) (2) K(x, y) = δ(x + y) (3) ( ) 1 2 nbr-nak@trio.plala.or.jp 28
30
1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More information2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
More information[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)
. D............................................... : E = κ ............................................ 3.................................................
More information1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin
More information30 3..........................................................................................3.................................... 4.4..................................... 6 A Q, P s- 7 B α- 9 Q P ()
More informationII 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
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
More informationChebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ
More informationphs.dvi
483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....
More informationQMI13a.dvi
I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More information) ] [ 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
More informationJuly 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
More informationQCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
More information4 14 4 14 4 1 1 4 1.1................................................ 4 1............................................. 4 1.3................................................ 5 1.4 1............................................
More information6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civit
6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civita ɛ 123 =1 0 r p = 2 2 = (6.4) Planck h L p = h ( h
More informationI ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
More informationDVIOUT-fujin
2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2 1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2..........................
More informationNo δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
More informationHilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2
hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................
More information4. 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
More information‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í
Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I
More information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,
More informationLINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
More informationkawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2
Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................
More informationBlack-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W
003 7 14 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson e-mail: takatoshi-tasaki@nifty.com kabutaro@mocha.freemail.ne.jp
More informationSO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
More information( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
More informationSO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α
SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij
More informationall.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
More informationsimx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =
II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [
More informationE 1/2 3/ () +3/2 +3/ () +1/2 +1/ / E [1] B (3.2) F E 4.1 y x E = (E x,, ) j y 4.1 E int = (, E y, ) j y = (Hall ef
4 213 5 8 4.1.1 () f A exp( E/k B ) f E = A [ k B exp E ] = f k B k B = f (2 E /3n). 1 k B /2 σ = e 2 τ(e)d(e) 2E 3nf 3m 2 E de = ne2 τ E m (4.1) E E τ E = τe E = / τ(e)e 3/2 f de E 3/2 f de (4.2) f (3.2)
More informationp = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
More informationn 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.........................
More informationmeiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
More information³ÎΨÏÀ
2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p
More information<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
電気電子数学入門 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. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9 ii 1... 1 1.1 1 1.
More informationPart () () 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
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
More information2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m
2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2
More information006 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......................
More informationnewmain.dvi
数論 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. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
More information(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
More information( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1
( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S
More informationHanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
More information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)
More informationV(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
More informationDecember 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
More informationEinstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More information30 I .............................................2........................................3................................................4.......................................... 2.5..........................................
More information( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
More information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
More information1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1
1 1.1......... 1............. 1.3... 1.4......... 1.5.............. 1.6................ Bownian Motion.1.......... Einstein.............. 3.3 Einstein........ 3.4..... 3.5 Langevin Eq.... 3.6................
More information1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
More information.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
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
More information9 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)
More informationii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
More information振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
More information25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
More informationʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)
(2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)
More information211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
More information(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w
S = 4π dτ dσ gg ij i X µ j X ν η µν η µν g ij g ij = g ij = ( 0 0 ) τ, σ (+, +) τ τ = iτ ds ds = dτ + dσ ds = dτ + dσ δ ij ( ) a =, a = τ b = σ g ij δ ab g g ( +, +,... ) S = 4π S = 4π ( i) = i 4π dτ dσ
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More informationv v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
More information第86回日本感染症学会総会学術集会後抄録(I)
κ κ κ κ κ κ μ μ β β β γ α α β β γ α β α α α γ α β β γ μ β β μ μ α ββ β β β β β β β β β β β β β β β β β β γ β μ μ μ μμ μ μ μ μ β β μ μ μ μ μ μ μ μ μ μ μ μ μ μ β
More information量子力学3-2013
( 3 ) 5 8 5 03 Email: hatsugai.yasuhiro.ge@u.tsukuba.ac.jp 3 5.............................. 5........................ 5........................ 6.............................. 8.......................
More informationMacdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,
More information医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 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. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More information( ) ) AGD 2) 7) 1
( 9 5 6 ) ) AGD ) 7) S. ψ (r, t) ψ(r, t) (r, t) Ĥ ψ(r, t) = e iĥt/ħ ψ(r, )e iĥt/ħ ˆn(r, t) = ψ (r, t)ψ(r, t) () : ψ(r, t)ψ (r, t) ψ (r, t)ψ(r, t) = δ(r r ) () ψ(r, t)ψ(r, t) ψ(r, t)ψ(r, t) = (3) ψ (r,
More informationeto-vol1.dvi
( 1) 1 ( [1] ) [] ( ) (AC) [3] [4, 5, 6] 3 (i) AC (ii) (iii) 3 AC [3, 7] [4, 5, 6] 1.1 ( e; e>0) Ze r v [ 1(a)] v [ 1(a )] B = μ 0 4π Zer v r 3 = μ 0 4π 1 Ze l m r 3, μ 0 l = mr v ( l s ) s μ s = μ B s
More information() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
More informationD-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane
D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]
More information,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
More information2 4 202 9 202 9 6 3................................................... 3.2................................................ 4.3......................................... 6.4.......................................
More informationx (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
More informationx A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
More informationZ: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
More informationNote.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..................................
More information( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
More informationMathematical Logic I 12 Contents I Zorn
Mathematical Logic I 12 Contents I 2 1 3 1.1............................. 3 1.2.......................... 5 1.3 Zorn.................. 5 2 6 2.1.............................. 6 2.2..............................
More informationI A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
More informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
More informationN cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
More information1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) (
August 26, 2005 1 1 1.1...................................... 1 1.2......................... 4 1.3....................... 5 1.4.............. 7 1.5.................... 8 1.6 GIM..........................
More information50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
More information数学の基礎訓練I
I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
More informationφ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
More informationI. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
More informationI
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
More informationA
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
More information, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. main.tex 2011/08/13( )
81 4 2 4.1, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. 82 4.2. ζ t + V (ζ + βy) = 0 (4.2.1), V = 0 (4.2.2). (4.2.1), (3.3.66) R 1 Φ / Z, Γ., F 1 ( 3.2 ). 7,., ( )., (4.2.1) 500 hpa., 500 hpa (4.2.1) 1949,.,
More information. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More informationi
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
More informationω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
More information