C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq
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1 ( ) F K 0 (q 1,,q N,p 1,,p N ) (Q 1,,Q N,P 1,,P N ) Q i Q i (q, p). (1) P i P i (q, p), (2) (p i dq i P i dq i )df. (3) [ ] Q αq + βp, P γq + δp α, β, γ, δ [ ] PdQ pdq (γq + δp)(αdq + βdp) pdq γαqdq + δβpdp + δαpdq + γβqdp pdq d( 1 2 γαq δβp2 + γβqp)+(δα γβ 1)pdq δα γβ 1 (3) C (p i dq i P i dq i ) 0 (4) C 1
2 C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq i P i dq i (6) C (u, v) C 0 D 0 ( Av (A u (u, v)du + A v (u, v)dv) C D u A ) u dudv. (7) v C D C (u, v) C (u, v) C 0 C (u, v) D 0 C 2
3 D (q i,p i ) (u, v) (6) (7) C p i dq i C 0 ( qi p i u du + q ) i v dv (8) A u (u, v) p i q i u, A v(u, v) p i q i v. (9) (7) A v u A u v N ( pi q i u v p ) i q i v u (p i,q i ) p (u, v) i p i u v p i u q i u q i v (p i,q i ) (u, v), (10) q i v p i q i v u. (11) (10) p i dq i C D dp i dq i (12) dp i dq i D D 0 (p i,q i ) dudv (13) (u, v) (1)(2) D D D (Q, P ) (q, p) D (q, p) (u, v) D (Q, P ) (u, v) D D dp i dq i D 0 (P i,q i ) dudv. (14) (u, v) (3) D dp i dq i D dp i dq i (15) D D (1)(2) 3
4 5.2 (15) (13)(14) (p i,q i ) (u, v) dudv D 0 D 0 (P i,q i ) (u, v) dudv. u v {u, v} q,p ( qi p i u v p i u ) q i v (15) (16) {u, v} q,p {u, v} Q,P (17) q, p u, v q i p j [ ] {q i,p j } q,p {q i,q j } 0, {p i,p j } 0, {q i,p j } δ ij (18) ( qk p k p ) k q k q i p j q i p j δ ik δ jk δ ij 2N x 1 q 1 X 1 Q 1 : : [ ] x, X 0 1 N, J 1 N 0 x 2N q N p 1 : p N X 2N 1 N N N x T, X T ( qi p i {u, v} q,p u v p ) i q i x T J x (19) u v u v ( Qi P i {u, v} Q,P u v P ) i Q i X T J X (20) u v u v 4 Q N P 1 : P N
5 x i X i X i (x 1,,x 2N ) (21) X i u j X i x j x j u j M ij x j u M M ij X i x j (22) X v M x v, T X u x T M T. (23) u (20) (19) M (21) (21) M T JM J. (24) M det J (det M) 2 1 det M ±1 (Liouville ) dq 1 dq N dp 1 dp N ±dq 1 dq N dp 1 dp N. det M 1 (24) M 1 M T J JM 1 5
6 J J 2 1 (25) JM T M 1 J M MJM T J (26) 5.3 q k,p k (k 1, 2,,N) (Poisson Bracket) ( u v [u, v] v ) u q k p k q k p k (27) [ ] x 1 q 1 X 1 Q 1 : : q N, p 1 Q N P 1 : : (28) x 2N p N X 2N P N [u, v] q,p ( u v v ) u q k p k q k p k 2 i,j1 u v J ij ( u) T J ( v) (29) x i x j i v v x i j v X j X j x i j M ji v X j (30) v M T v (31) ( v) T ( v) T M (32) 6
7 (31) [u, v] q,p ( u) T J ( v) ( u) T MJM T ( v) ( u) T J ( v) [u, v] Q,P (33) M (26) [q i,p j ] [q i,q j ] [p i,p j ] ( qi p j p ) j q i δ ik δ jk δ ij (34) q k p k q k p k ( qi q j q ) j q i 0 (35) q k p k q k p k ( pi p j p ) j p i 0 (36) q k p k q k p k [q i,q j ] 0 [p i,p j ] 0 (37) [q i,p j ] δ ij [u, v] [v,u] (38) [αu + βv,w]α[u, w]+β[v, w] (α, β ) (39) [uv, w] u[v, w]+[u, w]v (40) [q i,f(q, p)] F, [p i,g(q, p)] G p i q i (41) [u, [v, w]] + [v, [w, u]] + [w, [u, v]] 0 (Jacobi ) (42) (38)(39)(41) (40) (uv) q k u q k v + u v q k, (uv) p k u p k v + u v p k 7
8 J ij (31) u x i u, i 2 u x i x j u, ij (43) [v, w] 2 k,l1 [u, [v, w]] + [v, [w, u]] [u, [v, w]] [v, [u, w]] v, k J w, l (44) ij ij ij u, i J ij ([v, w]), j v, i J ij ([u, w]), j ij J ij J (u, i (v, k w, l ), j v, i (u, k w, l ), j ) J ij J (u, i v, k v, i u, k )w, lj + ij J ij J (u, i v, kj v, i u, kj )w, l (45) ([v, w]), j J (v, k w, l ), j J (v, kj w, l +v, k w, lj ) (45) u, i v, k v, i u, k i k J ij J i, k ij 1 2 (J ijj J kj J il )(u, i v, k v, i u, k )w, lj J ij J J kj J il l j w, lj l, j l, j (45) J ij J (u, i v, kj v, i u, kj )w, l ij i j J ij J ij J (u, i v, kj +v, j u, ki )w, l ij ij J ij J (u, i v, j ), k w, l J [u, v], k w, l [[u, v],w] [w, [u, v]] 8
9 [ 1] L x yp z zp y,l y zp x xp z,l z xp y yp x Poisson Blacket [x, L z ], [y, L z ], [z, L z ] [p x,l z ], [p y,l z ], [p z,l z ] [L x,l y ], [L y,l z ], [L z,l x ] 5.4 (26) (24) q, p Q, P Q Q(q, p), P P (q, p) MJM T J 2N 2N (i, j) 2 k,l1 X i X j J x k l J ij (46) i, j 1 N N +1 2N k k 1,,N k N +1,, 2N ( Qi Q j Q ) i Q j 0 q k p k p k q k ( Qi P j Q ) i P j δ ij (47) q k p k p k q k ( Pi P j P ) i P j 0 q k p k p k q k ( Pi Q j P ) i Q j δ ij q k p k p k q k q, p Q, P Q Q(q, p), P P (q, p) [Q i,p j ] 0, (48) [Q i,p j ] δ ij (49) [P i.p j ] 0 (50) (37) 9
10 [ 2] (N1) q, p Q, P Q Q(q, p), P P (q, p) MJM T J M [ Q q P q Q p P p ] [ J [Q, Q] 0, [P, P] 0, [Q, P ]1 ] 5.5 dq i dp i H (51) p i H q i (41) dq i dp i [q i,h] (52) [p i,h] q p q i,p i F (q, p) (51) df (q, p) ( F dq i q i i + F ) dp i p i ( F H F ) H q i i p i p i q i [F (q, p),h] (52) df (q, p) [F (q, p),h] (53) 10
11 q, p F (q, p) F F F (q, p) G(q, p) [F, G] F (q, p), G(q, p) [F, H] 0, [G, H] 0 (42) [H, [F, G]] [G, [F, H]] [F, [G, H]] 0 [F, G] 5.6 G(q, p, t) (41) G(q, p, t) G(q, p, t) q i ɛ, p i ɛ (54) p i q i q i ɛ [q i,g], p i ɛ [p i,g] (55) q i,p i F (q, p) F (q, p) ( ) F F q i + p i q i i p i ɛ ( G F G ) F p i i q i q i p i ɛ [F, G] (56) G(q, p, t) (56) G (53) H(q, p) ɛ [H, G] (57) dg(q, p) [H, G] 0 (58) [G(q, p),h] 0 (59) G [ 3]( ) 11
12 1. (P total i p i) q i ɛ [q i,p total ] p i ɛ [p i,p total ] 2. n (θ ε) x i ɛ [x i, n L] p i ɛ [p i, n L] [ 4]( ) r x 2 + y 2 + z 2 L i (i x, y, z) p 2 p 2 x + p2 y + p2 z L i (i x, y, z) H 1 2m p2 + V (r) L i (i x, y, z) 5.7 A, B Â, ˆB [A, B] 1 i (Â ˆB ˆBÂ). (60) [q i,q j ]0 1 i (ˆq iˆq j ˆq j ˆq i )0, [q i,p j ]δ ij 1 i (ˆq iˆp j ˆp j ˆq i )δ ij, (61) [p i,p j ]0 1 i (ˆp iˆp j ˆp j ˆp i )0. (53) df (ˆq, ˆp) 1 (F (ˆq, ˆp)H(ˆq, ˆp) H(ˆq, ˆp)F (ˆq, ˆp)) (62) i (61) (62) 12
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..........................
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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.......................
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
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n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
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24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
SO(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 )
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 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
弾性定数の対称性について
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D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
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6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
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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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19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352
1 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................
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
1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915
1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X
4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
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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
Gmech08.dvi
63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
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2005 1 1991 1996 5 i 1 12 *1 *2 (1991) (1992) (2002) (1991) (1992) (2002) 13 (1991) (1992) (2002) *1 (2003) *2 (1997) 1 2 13 *3 *4 200 1 14 2 250m :64.3km 457mm :76.4km 200 1 548mm 16 9 12 589 13 8 50m
Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4
[2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e
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5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
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83 ( Intrinsic ( (1 V v i V {e 1,, e n } V v V v = v 1 e 1 + + v n e n = v i e i V V V V w i V {f 1,, f n } V w 1 V w = w 1 f 1 + + w n f n = w i f i V V V {e 1,, e n } V {e 1,, e n } e 1 (e 1 e n e n
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,
meiji_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
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φ 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
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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構造と連続体の力学基礎
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(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
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量子力学Ａ
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