nosenote3.dvi
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- せせら さだい
- 5 years ago
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1
2 i
3 Verlet
4 1 1 MD t N r 1 (t), r 2 (t),, r N (t) ṙ 1 (t), ṙ 2 (t),, ṙ N (t) MD a 1, a 2, a 3 r i (i =1,,n)
5 1 2 T =0K r i + m 1 a 1 + m 2 a 2 + m 3 a 3 (m 1,m 2,m 3 =0, ±1, ±2,, ± ), (i =1,,n) (1.1) 1.1 a 1 =(a 1x,a 1y,a 1z ), a 2, a 3 r i =(x i,y i,z i )(i =1,,n) a 1, a 2, a 3 m 1,m 2,m 3 (3,N) R(i, j) i =1 X i =2 Y i =3 Z j N N = n m 1 m 2 m 3 n m 1,m 2,m 3 parameter N 256 MKS 1A=10 10 m LJ A σ 1.2 MD
6 1 3 (1/2, 1/2, 1/2) (1/4, 1/4, 1/4) (3/4, 3/4, 3/4) (1/4, 1/4, 1/4) (1/4, 3/4, 3/4) (3/4, 1/4, 3/4) (3/4, 3/4, 1/4) MD MD MD ξ 1 ξ x i = x i + ξδ (1.2) x i x i δ 10 % 0 ξ 1 ξ =2ξ 1 MD
7 1 4 32bit 2 32 IR(IY)=513*IY R(IY)=DFLOAT(IY)/ D0 IR(IY) IR(IY) R(IY) IY IY IY IY=IR(IY) R(IY) IY=IR(IY) X=R(IY) X 1 x 1 MD 4σ σ
8 5 2 g(r) r r r + dr ρ4πr 2 g(r)dr 4πr 2 dr ρ g(r) =1 g(r)?? 0 δ r =0 r g(r) =0 n(r) = r 0 ρ 4πr 2 g(r)dr (2.1) r g(r) X n(r) g(r) g(r) r r + dr ρ4πr 2 g(r)dr g(r) R r c dr NG(0:1000) 0 r ij NN=INT(r ij /dr) (NN-1)*dr NN*dr NN
9 2 6 NG(NN)=NG(NN)+1 NG NN 1 NN=INT(r ij /dr +0.5) g(r) NN*dr NG(NN) N/2*NT ANG(NN)=4πr 2 g(r)dr N NT NT NT=1 n(r) ANG(NN) 0 g(r) ANG(NN)/4πr 2 drρ g(r) minimum image convention MD r h r h g(r) g(r) MD i j MD j j MD j i j i j g(r) R MD j j i R minimum image convention x MD L x L y L z x L x CLX CLXH=CLX*0.5d0
10 2 7 DX=R(1,I)-R(1,J) IF(DX.GT.CLXH) DX=DX-CLX IF(DX.LT.-CLXH) DX=DX+CLX MD DX L x /2 L X /2 x MD IF FORTRAN INT(X) MOD(X,1) DX=DX-CLX*INT(DX/CLXH) IF IF IF 1/4 g(r) 40 g(r) 0.1 CHARACTER*1 A(40)/40* / CHARACTER*1 ASTER/ * /,ONE/ 1 /,ZERO/ 0 /,BLANK/ / g(r) GR DO 200 J=1,NR GR=G(J) IK=INT( GR*10.0D D ) IF(IK.GT.40) IK=40 A(1)=ZERO A(11)=ONE A(IK)=ASTER WRITE(*,110) X(J),G(J),N(J),A
11 FORMAT(1H,3F6.2,3X,40A1) A(IK)=BLANK 200 CONTINUE WRITE X(I) FORMAT g(r) 10.0D0 g(r) d0 g(r) g(r) =1 11 1
12 9 3 Verlet MD MD q p dq dt = p m, dp dt = mω2 q, m d2 q dt = 2 mω2 q (3.1) E = p2 2m mω2 q 2 dq dt = P, dp dt = Q, d 2 Q = Q (3.2) dt2 E = 1 2 P Q2 P P t =0 P Q P 0 Q 0 P = P 0 cos t Q 0 sin t, Q= P 0 sin t + Q 0 cos t (3.3) t t n = n t Q P Q n P n Q n P n Q n+1 P n+1
13 3 Verlet 10 t t = n t t (3.3) MD Verlet leap frog t n = n t Q n t n 1/2 P n 1/2 (1) n Q n F n F n = Q n (2) dp/dt = F P n+1/2 = P n 1/2 + F n t (3.4) t n+1/2 P n+1/2 P n+1/2 = P n 1/2 Q n t (3.5) leap frog (3) n +1 Q n+1 dq dt = P Q n+1 = Q n + P n+1/2 t (3.6)
14 3 Verlet 11 (4) Q n P n P n 1/2 P n+1/2 P n P n = 1 2 (P n+1/2 + P n 1/2 ) (3.7) Q 0 P 1/2 (3.4) (3.7) P n 1/2 P n P n = P n 1/2 + F n t 2 (3.8) n =0 Q 0 P 0 P 1/2 = P 0 F 0 t 2 (3.9) 3.1 Verlet t Q 0 P 0 Q n P n E n = 1 2 (P n 2 + Q 2 n) Verlet S n = 1 ( [P 2n )] 4 ( t)2 Q 2 n (3.10) Q 0 =1.0, P 0 =1.0 t = Q n,p n,e n,s n ˆP n, ˆQn, Ê n E n S n?? E n S n
15 3 Verlet 12 S n 10 E n t 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, t =0.2 Q, P, E t t/ t δx x ˆx δx = x ˆx x Q P E t t δx = A( t) k ln δx =lna + k ln t k MD t t t
16 3 Verlet 13 t Verlet t t t =0.001, , , , , , , , , Q, P, E t t/ t 1 Verlet 3 t t t
17 3 Verlet 14 t t t = 1 E i+1 E i (3.11) N i Verlet 0 t t 20
18 3 Verlet Q 0 =1.0, P 0 = Verlet Q n P n Verlet P n+1/2 = P n 1/2 Q n t (3.12) Q n+1 = Q n + P n+1/2 t (3.13) 3.13 n n 1 Q n = Q n 1 + P n 1/2 t (3.14) P n+1/2,p n 1/2 Q n+1,q n,q n 1 Q n+1 ( 2 ( t) 2) Q n + Q n 1 = 0 (3.15) Q n = Aλ n λ λ 2 ( 2 ( t) 2) λ + 1 = 0 (3.16) λ 1,λ 2 Q n = Aλ n 1 + Bλ n 2 (3.17)
19 3 Verlet 16 (1) cos α =1 1 2 ( t)2 λ 1,2 = e ±iα t α = t ( t) ( t)5 + t n P n P n = 1 2 ( Pn+1/2 + P n 1/2 ) = Q n+1 Q n 1 2 t (3.18) (2) Q 0,P 0 Q n,p n ( ) t Q n = Q 0 cos nα + P 0 sin nα (3.19) sin α ( ) sin α P n = P 0 cos nα Q 0 sin nα (3.20) t (3) [ S n = 1 Pn ( sin α t ) 2 Q 2 n ] (3.21) ( ) 2 sin α =1 1 t 4 ( t)2 S n E n 1 8 ( t)2 Q 2 n ( t)2 3.7 Verlet t = 2π 2π t = Q, P, E δq, δp, δe, δx = (δq) 2 +)δp) 2
20 3 Verlet t = 2π δq, δp, δe, δx t δe t ( t) k δq, δp ( t) t = 2π 4 Verlet t =2 t = 2π 4 (Q n,p n ) Q Q n Q(t) =Q 0 cos(αt)+ α α = 2 ( ) π cos 1 1 π2 8 P 0 1 π2 16 sin(αt) = ˆQ(t) =Q 0 cos t + P 0 sin t 2 Verlet E t
21 18 4 Verlet MD MD (Rahman Stillinger m d2 r = F (r) (4.1) dt2 t r(t) r (1),r (2),r (3),r (4),r (5) X k = 1 k! ( t)k r (k), (k =0, 1, 2, 3, 4, 5) (4.2) t X k (t) t + t X k (t + t) X 0(t + t) =X 0 (t)+x 1 (t)+x 2 (t)+x 3 (t)+x 4 (t)+x 5 (t) (4.3) X 1 (t + t) =X 1(t)+2X 2 (t)+3x 3 (t)+4x 4 (t)+5x 5 (t) (4.4) X 2(t + t) =X 2 (t)+3x 3 (t)+6x 4 (t)+10x 5 (t) (4.5) X 3(t + t) =X 3 (t)+4x 4 (t)+10x 5 (t) (4.6) X 4(t + t) =X 4 (t)+5x 5 (t) (4.7) X 5(t + t) =X 5 (t) (4.8)
22 4 19 X 0(t + ) F (X 0(t + )) X 2(t + ) δ = 1 F (X 0 (t + )) ( t) 2 X 2 m 2(t + ) (4.9) 0 ( X k (t + ) δ X k (t + )=X k(t + )+C k δ (4.10) C k C k Nordsieck Gear C k Gear C 0 = 3 20,C 1 = ,C 2 =1,C 3 = 11 18,C 4 = 1 6,C 5 = 1 60 (4.11) 4.1 d 2 Q = Q (4.12) dt2 t, Q 0,P 0 = Q 0 X k (n t) t =0 X 0 (0) = Q 0, X 1 (0) = P 0 t, X 2 (0) = 1 2 Q 0( t) 2,X 3 (0) = 1 6 P 0( t) 3 X 4 (0) = 1 24 Q 0( t) 4, X 5 (0) = P 0( t) 5 (4.13) Q 0 =1.0, P 0 =1.0, t = ( ) X 0 (n t), X 1(n t),e n X 0 t (n t), X 1(n t) t
23 4 20 X 0 (n t), X ( 1(n t),e n X 0 (n t), X ) 1(n t) t t ˆQ n, ˆPn, Ê n , 0.1, 0.05, 0.04, , 0.02, 0.01, , 0.005, t=1.0 Q, P, E ln E =ln E n+1 E n
24 4 21 t Verlet d 2 Q dt 2 = Q 2adQ dt (4.14) Q Q = P E = 1 2 (P 2 + Q 2 ) t =0 Q(0) = Q 0,Q (1) (0) = P 0 Q (2) (0) = Q 0 2aP 0 Q (3) (0) = 2aQ 0 +(4a 2 1)P 0 Q (4) (0) = (4a 2 1)Q 0 +4a(1 2a 2 )P 0 Q (5) (0) = 4a(2a 2 1)Q 0 +(1 12a 2 +16a 4 )P 0 (4.15) [ Q(t) =e at Q 0 cos( 1 a 2 t)+ P ] 0 + aq 0 sin( 1 a 2 t) 1 a 2 [ P (t) = Q(t) =e at P 0 cos( 1 a 2 t) Q ] 0 + ap 0 sin( 1 a 2 t) 1 a 2 (4.16) (4.17)
25 4 22 a = E = 1 2 (P 2 + Q 2 ) de dt = P dp dt + QdQ dt = P ( Q 2aP )+QP = 2aP 2 (4.18) E(t) E(0) = 2a E(0) t 0 P (t ) 2 dt (4.19) t E(0) = E(t)+2a P (t ) 2 dt (4.20) 0 t A(t )dt t 0 0 [ ] 1 A(t )dt = t 2 A(0) + A( t)+a(2 t)+ + A((n 1) t)+1 2 A(n t) ( t) 2 (4.21)
26 23 5 [ ( ) 12 ( ) ] 6 σ σ φ LJ (r ij )=4ɛ r ij r ij (5.1) r ij = r i r j i j H = m 2 ṙ2 1 + m 2 ṙ2 2 + φ(r 12 ) (5.2) m r 1 = φ(r 12) r 1 = φ(r 12) r 12 r 12 r 1 = φ(r 12) r 12 r 1 r 2 r 12 (5.3) m r 2 = φ(r 12) r 2 = φ(r 12) r 12 r 12 r 2 = φ(r 12) r 12 r 2 r 1 r 12 (5.4) m =39.9m u = kg ɛ = 120kK = J σ =3.4A = m m u m, ɛ, σ MKSA σ m ɛ
27 5 24 ɛ, σ 1 A g/mol MPa, GPa 5.1 Verlet 2σ 5.2 τ = mσ 2 ɛ s τ 0.02 ps 1000 m(v 1 + v 2 ) 0 t t
28
29 26 6 r c t fcc r c g(r) 0 r 1 r 2 1(r r 2 ) t r c
30 6 27 r c 0 φ sh (r) =φ(r) φ(r c ) (6.1) r c n c E = E c φ(r c)nn c +2πNρ r c φ(r)r 2 dr (6.2) fcc 3. T eq 1 2 m iv 2 i = 3N 2 kt eq (6.3) i
31 6 28 i 1 2 m iv 2 i = 3N 2 kt (6.4) T T v i = sv i (6.5) s s 3N 2 kt = i 1 2 m iv i 2 = s 2 i 1 2 m iv 2 i = s2 3N 2 kt (6.6) s = 3N kt 2 1 i m 2 iv 2 i (6.7) T T MD
32 6 29 A 0 A i A 0 (A i A 0 ) 2 A i = A 0 + A i A 0 = A M M (A i A 0 ) (6.8) i=1 (δa i ) 2 = (A i A i ) 2 = ( A i A 0 ( A i A 0 ) ) 2 = (A i A 0 ) 2 ( A i A 0 ) 2 = 1 M (A i A 0 ) 2 ( A i A 0 ) 2 M i=1 (6.9) M
33 6 30 / 6.1 Verlet 1) 5.4 FORTRAN
34 6 31 N = 256 ( ) fcc L x,l y,l z 2) 3) 4) K(v) = i 1 2 m iv 2 i (6.10) U(r) = i<j φ(r ij ) (6.11) E = K(v)+U(r) (6.12) P = 1 3V [ m i v 2 i i i<j ] dφ(r ij ) r ij dr ij (6.13) T = 2 K (6.14) 3Nk
35 6 32 k C V = 2 3N (δk)2 K 2 (6.15) 5) g(r) g(r) MD x L x alx alxh (3,N) x r c rcut2 do 2000 i=1,n-1 do 1000 j=i+1,n xx=x(1,i)-x(1,j) if(xx.gt.alxh) xx=xx-alx if(xx.lt.-alxh) xx=xx+alh Y Z rr=xx*xx+yy*yy+zz*zz if(rr.gt.rcut2) go to 1000 r=dsqrt(rr) g(r) 1000 continue 2000 continue g(r) DO DO
36 6 33 6) r i (0) δ(t) = 1 r i (t) r i (0) 2 (6.16) N i 6Dt x, y, z r i (0) r i (t) r i (0) if(x(1,i).gt.alx) then x(1,i)=x(1,i)-alx x0(1,i)=x0(1,i)-alx endif if(x(1,i).lt.0.0d0) then x(1,i)=x(1,i)+alx x0(1,i)=x0(1,i)+alx endif N = 108 1) fcc ( 84K ɛ/k 0.7 t 0.02ps 0.01 ) 100 t
37 6 34 2) 3) N = M m M/m M A M m A m A M = A m (6.17) (δa M ) 2 = m M (δa m) 2 (6.18) M m A M ± M (δa m) 2 (6.19) 1 A M ± M (δa)2 (6.20)
38 6 35 A M ± (δa) 2 (6.21) (6.20) D δ(t) δ(t) 6D δ(t) lim t 6t N 1 N(N 1) 2 N r c r c N 100 N 2 N r c r t r t ntable(j,i) i j ntable(j,i) i nnum(i) r t rtab2
39 6 36 do 2000 i=1,n-1 nn=0 do 1000 j=i+1,n xx=x(1,i)-x(1,j) if(xx.gt.alxh) xx=xx-alx if(xx.lt.-alxh) xx=xx+alx y z rr=xx*xx+yy*yy+zz*zz if(rr.gt.rtab2) go to 1000 nn=nn+1 ntable(nn,i)=j 1000 continue nnum(i)=nn 2000 continue do 4000 i=1,n-1 nmax=nnum(i) if(nmax.eq.0) go to 4000 do 3000 nn=1,nmax j=ntable(nn,i) xx=x(1,i)-x(1,j) if(xx.gt.alxh) xx=xx-alx if(xx.lt.-alxh) xx=xx+alx y z rr=xx*xx+yy*yy+zz*zz if(rr.gt.rcut2) go to continue 4000 continue
40 6 37 ntable(j,i) j r t j>i i ntable(i) i-1 npoint(i) nn=0 do 2000 i=1,n-1 npoint(i)=nn do 1000 j=i+1,n rr=xx*xx+yy*yy+zz*zz if(rr.gt.rtab2) go to 1000 nn=nn+1 ntab1(nn)=j 1000 continue 2000 continue npoint(n)=nn do 4000 i=1,n-1 nstart=npoint(i)+1 nend=npoint(i+1) if(nened-nstart.lt.0) go to 4000 do 3000 nn=nstart,nend j=ntab1(nn) 3000 continue 4000 continue
41 6 38 npoint(i) i-1 i ntab1(i) i=npoint(i)+1 npoint(i+1) x, y, z m m N = 256 N = 108 N = 108, N= 256 CPU CPU CPU (
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2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
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
SFGÇÃÉXÉyÉNÉgÉãå`.pdf
SFG 1 SFG SFG I SFG (ω) χ SFG (ω). SFG χ χ SFG (ω) = χ NR e iϕ +. ω ω + iγ SFG φ = ±π/, χ φ = ±π 3 χ SFG χ SFG = χ NR + χ (ω ω ) + Γ + χ NR χ (ω ω ) (ω ω ) + Γ cosϕ χ NR χ Γ (ω ω ) + Γ sinϕ. 3 (θ) 180
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B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
sec13.dvi
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:
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.........................
C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0
![C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0](/thumbs/92/108844778.jpg "C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0")
1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :
(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)
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
dynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
V(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)
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
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
![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](/thumbs/101/149159646.jpg "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")
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
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......................
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
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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
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
3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C
![3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C](/thumbs/94/119172213.jpg "3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C")
3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C nn (r r ) = C nn (R(r r )) [2 ] 2 g(r, r ) ˆn(r) ˆn(r
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
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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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29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
2000年度『数学展望 I』講義録
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Hanbury-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............................................
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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
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
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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
p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
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t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
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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 dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
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(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
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構造と連続体の力学基礎
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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 ]
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
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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 )
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) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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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
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0
(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
2 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
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No 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
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
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4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20)
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
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量子力学 問題
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