MAIN.dvi
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- うきえ ひきぎ
- 4 years ago
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1 01UM1301
2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48 1
3 A 50 A.1 Wannier : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50 A.2 : : : : : : : : : : : : : : 52 B 57 B.1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 B.2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 B.3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 B.3.1 ( ) : : : : : : : : : : : : : : 64 B.3.2 : : : : : : : : : : : : : : : : : : : : : : : : : 66 B.3.3 : : : : : : : : : : : : : : : : 67 B.3.4 : : : : : : : : : : : : : : : : : : : 68 B.3.5 : : : : : : : : : : : : 69 B.3.6 : : : : : : : : : : : 71 B.4 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 C 73 C.1 : : : : : : : : : : : : : : : : : : : : : : : : : 73 C.2 : : : : : : : : : : : : : : : : : 75 D 77 2
4 : 1.2: 1976 Hofstadter [1] 1.3 ( 1.4) 3
![10 4 T nm 1 ο 10T 1:3 [2] 1.3: 1.4: ([2] ) 1.](/docs-images/101/149329570/images/5-0.jpg "2 (1) (2) (1) a H = t X (i;j)2n:n: ^a y i ^a j (1.")
5 10 4 T nm 1 ο 10T 1:3 [2] 1.3: 1.4: ([2] ) 1.2 (1) (2) (1) a H = t X (i;j)2n:n: ^a y i ^a j (1.1) t ^a i i 4
6 E(k) =2tcos(ka) (1.2) t t! 0 k (2) t 6= 0 Mielke Lieb [3, 4, 5] [13] (a) (b) (i) (ii) (iii) (2) 5
7 1.3 Mielke Lieb [3, 4, 5] 1.5 [6] 1.5(a) 1.5(b) fl t H = H hop + H int (1.3) H hop X X t xy c y x;ffc y;ff (1.4) H hop = x;y ff=";# c x;ff x ff t xy = t y x H int H int = X x Un x;" n x;# (1.5) n x;ff x ff H int U U=t fl 1 (1) 1.5(a) t t t H = 6 4 t t t (1.6)
8 1.5: (a) (b) fl t 1.6: (a)s = 1 (b)s =0 E E =(0; 0; ± p 3t) E =0 U=t fl 1 H ex H hop H ex = 4t2 si s j 1 U 4 (1.7) s i i (1.7) 1.6(a) S S =1 S =0 1.6(b) S = 1 (2) 7
9 1.7: U =+1 1.5(b) H = 4 0 t t 7 t 0 t 5 (1.8) t t 0 E E =(2t; t; t) E = t U=t fl 1 U = ffi xy ffi xy = c y x;" cy y;# j0i (1.9) j0i 1.7 t 1.7 U 1.7 8
10 ffl ffl t >0 Φ=ffi 12 ffi 32 + ffi 31 ffi 21 + ffi 23 ffi 13 (1.10) (1.10) Φ=(ffi 12 ffi 21 )+(ffi 23 ffi 32 )+(ffi 31 ffi 13 ) (1.11) (ffi 12 ffi 21 ) ffi 12 ffi 21 = c y 1;" cy 2;# j0i cy 2;" cy 1;# j0i = c y 1;" cy 2;# j0i + cy 1;# cy 2;" j0i (1.12) = j"i 1 j#i 2 + j#i 1 j"i 2 (1.11) S =1 S =1 1.7 ffi 21 ffi 12 H hop (1.12) 1.2 (2) 9
11 ( 1.8) [7, 8] 1.8: ( [7] ) 1.9: 1.8 ( [7] ) 1.4 E B Wannier ffl ffl E B E 0 (1.13) E B =4E 0 (1.14) ffl E B = E ( 0 fi 1) (1.15) E 0 ~ 2 =(2m r a 2 0 ) m r a 0 = (~ 2 ffl 0 )=(m r e 2 ) 0 (A.59) 10
12 1.10: ( [8] ) 0 E B ( ) ( ) 11
13 : 2.1 In 0:72 Ga 0:28 As 10:4nm InAs 72nm 2.2 [8]
![2.2: ( [8] ) 2.3: X X X H = t e ij^ay^a i j + t h ij^b y i^b j + U (r ij )^a y ^a i i^b y j^b j (2.](/docs-images/101/149329570/images/14-0.jpg "1) i;j i;j t e ij th ij U (r ij) ^a i ^b i i i;j [9] U (r ij )= 8 < : U 0 i = j U 1 ( r ij a ) i 6= j (2.")
14 2.2: ( [8] ) 2.3: X X X H = t e ij^ay^a i j + t h ij^b y i^b j + U (r ij )^a y ^a i i^b y j^b j (2.1) i;j i;j t e ij th ij U (r ij) ^a i ^b i i i;j [9] U (r ij )= 8 < : U 0 i = j U 1 ( r ij a ) i 6= j (2.2) a r ij i j r ij =0 i = j
15 2.2 jφii i jχ e i i jχ h i i jφii = X i;j c ij jχ e i ijχ h j i (2.3) c ij E B H E(U 0 ;U 1 ) E B E(U 0 = U 1 =0) E(U 0 6=0;U 1 6=0) (2.4) 2.3 t e t h U 0 U 1 t e t h t e InAs 2.4 [8] 10meV 2.5 t e 6t e 6t e 2.4 t e =1:67meV (2.5) m 0 InAs m e 0:02m 0 m h 0:4m 0 t h t e m e m h m e + m h ο m e (2.6) 14
16 t h t e t e 2.4 t e 3.2(b) t e : [8] 2.5: U 0 U 1 (2.2) U 0 U 1 U 0 U 1 (1) U 0! +1 Polysilane 15
17 U 0 U 1 U 1 =0:75 U 0 (2.7) [9] (2.7) U (r) = e2 4ß" 1 r = e2 4ß"a (2.8) 1 ( r a ) 0:75U 0 e2 4ß"a (2.9) U 0 = 1 e 2 0:75 4ß"a =4:18meV (2.10) InAs " =12:4" 0 a =36nm e(> 0) (2) 2.6 a=2 [6] a 36nm v(r) = ( 1 16 m!a2 cos( ßx a )cos(ßy a ) Λ 2 r< a 2 0 r a 2 (2.11) m! ffi(r) = p 2 2jrj Λ 2 exp (2.12) ßd d =2p ~=m! d 2
18 2.6: fl = ( )=( ) ο 0:6 d d = a =36nm 2.7 U (r) = ZZ d 2 r 1 d 2 r 2 e 2 jffi(r 1 )j 2 jffi(r 2 r)j 2 4ß"jr 1 r 2 j = e2 4ß" 4 2 ZZ ßd 2 e 4jr 1 j 2 d 2 r 1 d 2 d r 2 e 4jr 2 rj 2 d 2 2 jr 1 r 2 j (2.13) r Z 1 jrj = 1 eiq r d 2 q (2.14) 2ß jqj (2.13) U (r) = e2 4ß" = e2 4ß" = e2 4ß" 4 ßd ßd ßd ß 1 2ß 1 2ß ZZZ ZZZ Z d 2 r 1 d 2 r 2 d 2 q eiq(r1 r2) e 4(jr 1j 2 +jr 2 rj 2 ) d jqj 2 d 2 r 1 d 2 r 2 d 2 q eiq(r1 r2 r) jqj d 2 q e iq r q Z d 2 r 1 e iq r 1 e 4r2 1 d 2 e 4(r2 1 +r2 2 ) d 2 Z d 2 r 2 e iq r 2 e 4r2 2 d 2 (2.15) 17
19 r 1 r 2 (2.15) Z U (r) = e2 4ß" 1 2ß Z = e2 4ß" 1 2ß d 2 re iq r e 4r2 d 2 = e q2 d 2 ßd 16 4 e iq r d 2 q q Z +1 Z 2ß 0 d e i q r cos = = =2 =4 Z ß 0 Z 1 1Z 1 Z 1 0 e q2 d 2 8 (2.16) Z 2ß (2.17) dqe d2 8 q2 d e i q r cos d e i q r cos =2ßJ 0 (q r) 0 Z 2ß ß Z e i q r t 1 p 1 t dt e i q r t p1 t 2 dt cos(q r t) p1 t 2 dt i q r cos d e e i q r t p1 t 2 dt (2.18) J 0 (2.18) Z +1 p e ff2 x ß 2 fi J 0 (fix)dx 2 = 2ff e 8ff 2 I 0 ( fi2 8ff ) (2.19) 2 (2.17) 0 p U (r) = e2 4ß" 2ß d e r2 d 2 I 0 ( r2 I 0 d 2 ) (2.20) d = a =36nm U (r) 2.8 (2.20) r =0 U 0 U 0 U(r =0) p = e2 4ß" 2ß a =8:08meV (2.21) InAs " =12:4" 0 U 1 r >a(= 36nm) (2.20) (2.2) U 1 =0:45 U 0 (2.22) 18
20 2.7: ffi a 36nm d 36nm 19
21 2.8: U (r) =U 1 =(r=a) U 1 =0:45 U 0 (1) t t e = t h =1:67meV (2.23) (2) U 0 U 1 (a) (b) U 0 =4:18meV (2.24) U 1 =0:75U 0 (2.25) U 0 =8:08meV (2.26) U 1 =0:45U 0 (2.27) 20
22 jψi jψi = X i;j2( ) c ij jχ e i ijχ h j i (2.28) jχ e i i i jχ h j i j fc ij g 3.7 jψ B i jψ B i = X X i;j2( ) k;l2( ) c ijkl jffi e ijijffi h kli (2.29) jffi e iji S e =0 1 S e =0 jffi e iji = ( 1 p 2 jχ e i ijχ e ji + jχ e jijχ e ii jχ e i ijχ e ii i 6= j i = j S e =1 jffi e iji = n p 1 2 jχ e i ijχ e ji jχ e jijχ e i i (2.30) i 6= j (2.31) jffi h iji S h =0 ( jffi h p 1 2 jχ h i ijχ h j i + jχ h j ijχ h i i i 6= j iji = (2.32) jχ h i ijχ h i i i = j S h =1 jffi h iji = n 1p 2 jχ h i ijχ h j i jχ h j ijχ h i i 21 i 6= j (2.33)
23 2.9: 2 2 S = S e + S h jψ C i jψ C i = X X i;j2( ) k2( ) c ijk jffi e ijijχ h ki (2.34) jffi e iji (2.30) (2.31) S = S e r min U (r min ) 2.9 a p 19a x 2.10(a) p 7a y 2.10(b) p r min = 3a 2.9 p U (r min = 3a) = U 1 p H HITACHI 22
24 2.10: (a)x p 7a (b)y p 3a m (mev) : m m m H H H m m E B m m 150 m
25 2.11: N N N N U 0 =4:18meV U 1 =0:75U
26 2.12: N N N N 25
27 (b) (a) (c) (d) 3.1 t = t e = t h =1 (a)ο(d) 3.2 (a)ο(c) k = 0 K (a)ο(c) 3.1(b) (c) t t 0 t 0 t 0 = t t 0 =0 (b) (c) (a) t =1:67meV U 0 =4:18meV U 1 = 0:75 U 0 3.3(b) t = 1:67meV U 0 =8:08meV U 1 =0:45 U
28 3.1: (a) (b) (c) (d) 3.2: (a) (b) (c) (d) (a) (b) (c) (d) (c) (d) 27
29 3.3: (a) t = 1:67meV U 0 = 4:18meV U 1 = 0:75 U 0 (b) t = 1:67meV U 0 =8:08meV U 1 =0:45 U 0 28
30 3.3 t 0 3.3(a) (b) a 36nm t t e = t h = 1:67meV U 0 =4:18meV U 1 =0:75 U a 36nm a t ( t e = t h ) a E(k) = t r cos k x a + 2 cos k y a +4 q (1+cosk x a)(1 + cos k y a) (3.1) E(k) k x = k y = k p 2 E(k) ' 2t ' 2t s r 2+2cos 4 a2 k 2 2 ka p 2 m (3.2) 1 m = 1 ~ 2 fi fififi k=0 (3.3) (3.2) (3.3) t = 2~2 a 2 m / 1 a 2 (3.4) t a 29
31 3.4: a U 0 a (2.10) (2.21) a a 18nm t t = 1:67meV (18nm=36nm) 2 =6:68meV (3.5) U 0 U 0 = 4:18meV (18nm=36nm) =8:36meV (3.6) a 6nm t = 60:12meV U 0 =25:08meV 3.4 a a =6nm a =36nm 6 20meV 30
32 3.5: s K k 6= 0 s P h (r) =P 0 e r ο (3.7) ο r
33 ο nm : ο 3.4 H = X i;j t e ij^ay i ^a j + X i;j t h ij^b y i^b j + X i;j U (r ij )^a y i ^a i^b y j^b j (3.8) H 0 = T e + T h V U (r ij )= ( U i = j 0 (3.9) T e jffi (e) i jffi (e) i = X i= 32 c i jχ i i (3.10)
34 3.6: jχ i i i c i jffi (e) i 3.6 c i T h jffi (h) i H 0 jψ (0) ii jψ (0) ii = jffi (e) ijffi (h) i (3.11) E (1) E (1) = hhψ (0) jv jψ (0) ii = 1 16 U (3.12) E B E B E (1) = 1 16 U (3.13) H 0 E B = 1 U (3.14) 33
35 (e) 2 (e) 4 (e) 1 (e) 3 3.7: jffi (e) 1 i T e i jffi (e) i i 2 2 i jffi (e) i i (3.10) 3.7 jffi (e) 1 iοjffi (e) 4 i jffi (h) j i H 0 jψ (0) (i;j) ii = jffi(e) i ijffi (h) j i (3.15) H jψ ff ii = 16 X fi=1 a fi ffjφ (0) fi ii + jφ(1) ff ii + jφ (2) ff ii + ::: (3.16) 16X fi=1 a fi ffhhψ (0) fl jv jψ (0) fi ii = E(1) ff 16X fi=1 a fi ffhhψ (0) fl jψ (0) fi ii (ff =1; 2; ; 16) (3.17) 34
36 (3.17) U ::: ::: ::: ::: 6 = E (1) a 1 a 2 a 3. a ο 12ο ::: 4ο 12ο 36 12ο ::: 4ο 12ο 12ο 36 ::: 4ο ο 4ο 4ο ::: a 1 a 2 a 3. a (3.18) ο = ο =1 ο ο (3.18) E ff (1) p E (1) = 2( ) U (3.19) 4(9+7ο) = ο =0 E (1) = 2 U ( = ο =0) (3.20) 12 (3.14) 1 U (3.20) 12 =0 ο =1 E (1) = 1 U ( =0;ο =1) (3.21) 12 = ο =1 E (1) = 3 U ( = ο =1) (3.22) 12 35
37 3.8: (3.22) (3.13) 3.8 E B t t=2 [10]
38 3.9: 3.10: 3.11 U 0 U U 0 37
39 3.11: U :
40 3.6 t ij t ij exp» i 2ße hc Z rj ri A dr (3.23) [11] e h c A r i i :7T :4T t t
41 t =1 N (E 0 )de 9 9 N (E 0 )de t =1 0:1 N (E 0 )de 3.17 ο 3.19 N (E 0 )de ο 3.19 N (E 0 )de N (E 0 )de ±2:7T 0T N (E 0 )de (' ) ±2:7T 0T 40
42 3.13: -3.7T 0T 3.7T 3.14: t t = 1 0 ο 0.5 ο 1-3.7T ο 0T ο 3.7T [6] 41
43 3.15: t t = 1 0 ο 1 ο 2-3.7T ο 0T ο 3.7T [11] 3.16: t t = 1 0 ο 1 0T ο 3.7T [11] 42
44 3.17: 3.18: 43
45 3.19: (2.4 ) S = 1 S =
46 (mev) (S) / / : (S = 1 2 ) (S =0) 1meV 0:1meV S =2 3.20(b) 3.20(a) S = K K S =2 S =0 S =1 S =2 45
47 3.20: (a)s = 0 (b)s = : (a)s = 0 (b)s = 2 46
48 4 4.1 (1) (2) (3) 47
49 meV t a U 0 a t a U 0 a t U 0 U
50 49
51 A A.1 Wannier Wannier [12] ~2 r 2 r 2m r + V (r) ψ(r) =Eψ(r) (A.1) m r V (r) = e2 ffl 0 r (A.2) z V (r) =V (z) = e2 ffl 0 1 jzj+flr (A.3) R fl 0.3 fl ρ = rff (A.4) Wannier ~ 2 ff 2 r 2 ρ 2m + ffe2 ψ(ρ) =Eψ(ρ) r ffl 0 ρ (A.5) 2m re 2 ffl 0 ~ 2 ff = 2 ffa 0 a 0 = ~2 ffl 0 m r e 2 (A.6) (A.7) 50
52 Wannier r 2 ρ ρ 2m re ψ(ρ) = ψ(ρ) (A.8) ~ 2 2 ff E < 0 E > 0 a 2 0ff 2 8m re ~ 2 a E E 0 (A.9) E 0 ~2 2m r a 2 0 = e2 2ffl 0 a 0 = e4 m r 2ffl 2 0 ~2 (A.10) (A.8) Wannier 2 r ρ ψ(ρ) = 1 ρ 4 ψ(ρ) (A.11) r = 2 E0 = ffa 0 E (A.12) = ff (jzj+flr) (A.11) ρ r 2 ψ( ) = 1 4 ψ( ) (A.13) Laplace ffl r 2 ρ = 1 @ρ ^L 2 ρ 2 (A.14) ffl ffl r 2 ρ = @ρ ^L 2 z ρ 2 r 2 (A.15) (A.16) ^L 2 ^L z z ^L 2 = 1 sin @ (A.17) 51
53 ^L z 2 2 (A.18) ^L 2 Y lm ( ; ffi) =l(l +1)Y lm ( ; ffi) (A.19) 1 ^L z p 2ß eimffi = mp2ß 1 eimffi (A.20) jmj» l (A.11) (A.13) ψ ffl ψ(ρ) =f l (ρ) Y lm ( ; ffi) (A.21) ffl ψ(ρ) =f m (ρ) 1 p2ß eimffi (A.22) ffl ψ( ) =f ( ) (A.23) (A.11) (A.13) ffl 1 ρ 2 ffl 1 ρ + ρ 1 l(l +1) f l (ρ) =0 4 @ρ + ρ 1 m2 f 4 ρ 2 m (ρ) f ( ) =0 (A.24) (A.25) (A.26) A.2 ffl 1 @ρ 1 f l (ρ) =0 4 (A.27) 52
54 ffl 1 ρ 2 ρ 1 f m (ρ) = f ( ) =0 (A.28) (A.29) ffl f l (ρ!1)=e ρ 2 (A.30) ffl f m (ρ!1)=e ρ 2 (A.31) ffl 1 ρ 2 ffl l(l ρ2 ρ 2 l @ρ m2 f m (ρ) =0 ρ 2 (A.32) (A.33) ffl f l (ρ! 0) = ρ l (A.34) ffl f m (ρ! 0) = ρ jmj (A.35) ffl f l (ρ) =ρ l e ρ 2 R(ρ) (A.36) ffl f m (ρ) =ρ jmj e ρ 2 R(ρ) (A.37) (A.24) (A.25) (A.26) R 53
55 ffl ffl (2(l +( l 1)R(ρ) =0 (A.38) +( jmj 1 )R(ρ) =0 (A.39) 2 (A.38) (A.39) 2 +(p +1 q = p = ( 2l 2jmj ( l 1 jmj qr(ρ) =0 (A.40) R(ρ) (A.40) R(ρ) = 1X ν=0 fi ν ρ ν (A.41) (A.42) (A.43) ν =0 ρ =0 (A.43) (A.40) 1X ν=2 fi ν ν(ν 1)ρ ν 1 +(p +1) 1X ν=1 fi ν νρ ν 1 ρ ν fi ν+1 = 1X ν=1 fi ν νρ ν + q 1X ν=0 fi ν ρ ν =0 (A.44) ν q (ν +1)(ν + p +1) fi ν (A.45) ν fi ν+1 ο 1 ν fi ν R(ρ) ο e ρ (A.36) (A.37) ρ = +1 ν ν max fi ν>νmax =0 (A.45) fi νmax+1 = q ν max q (ν max + 1)(ν max + p +1) fi ν max 0 (A.46) ν max = q (A.47) n 54
56 ffl n = ν max + l +1 n =1; 2; 3;::: (A.48) ffl n = ν max + jmj+ 1 2 n =0; 1; 2;::: (A.49) (A.12) ffl ffl E n = E 0 n 2 n =1; 2; 3;::: (A.50) E n = E 0 (n )2 n =0; 1; 2;::: (A.51) E B ffl E B = E 0 (A.52) ffl E B =4E 0 (A.53) E 0 (A.10) (A.26) μ2 2 W ;μ ( ) =0 (A.54) μ =+ 1 2 W ;μ( ) Whittaker f (jzj) f (jzj) =N W ; 1 ( 2(jzj+flR) ) (A.55) 2 a 0 N jzj! +1 Whittaker Z W ; 1 ( ) = e 2 1 dt e t (1 + 2 (1 ) 0 t ) (A.56) dw ; 1 ( ) 2 = e 2 d (1 ) Z 1 0 dt e t (1 + t ) ( t ) (A.57) 55
57 z =0 Z 1 dt e t ( )=0 (A.58) + t flR 0ln( )=0 (A.59) 0 a 0 R fi 1 0 fi 1 E B ' E 0 (A.60)
58 B [13] ffl B.1: ffl B.2: ffl B.3: 57
59 B.1 B.1 H H = T AB T BA (B.1) T AB jaj jbj jaj jbj A B jaj > jbj jaj= jbj H = T BA 0 0 T AB (B.2) H 2jBj H (jaj+jbj) 2jBj= (jaj jbj) B.1 jaj= 2 jbj= 1 N (2 1) N = N (B.1) ( ) T AB v = 0 (B.3) jaj v (jaj jbj) (jaj+jbj) ψ ψ = v 1 v 2. v jaj (B.4) A B.4 B.5 B.1 58
60 B.4: B.5: [13] t t =1 E =0 B.2 B.1 B.6 59
61 B.6: 2 6 H = (B.5) E =0; 0; 3 ψ (1) = ψ (2) = (B.6) (B.7) ffl. ( B.7) B.7: H cell t(c y 1 + cy 2 + cy 3 )(c 1 + c 2 + c 3 ) = t (B.8) 60
62 ψ c (c 1 + c 2 + c 3 )ψ c =0 (B.9) ψ (1) c = ψ (2) c = (B.10) (B.11) c i i ffl. B.8 1 H 2cell B.8: H 2cell = t (B.12) ψ 2cell B.8 ψ 2cell = (B.13) H 2cell ψ 2cell 61
63 B.9: ffl. B.9 H 4cell = t (B.14) ψ 4cell B.9 ψ 4cell = (B.15) ψ 4cell H 4cell 62
64 ffl. B.10: B.10 B.11 B.2 B.3 [14] 63
65 B.11: B.2 ( =2 ) 6= 2 E =0 B.3.1 ( ) ffl G(V;E) B.12: G vertex( ) V edge( ) E i vertex v i i j vertex edge e ij =(v i ;v j ) B.12 G(V;E) V = fv 1 ;v 2 ;v 3 ;v 4 g E = f(v 1 ;v 2 ); (v 1 ;v 3 ); (v 3 ;v 2 ); (v 3 ;v 4 )g = fe 12 ;e 13 ;e 32 ;e 34 g (B.16) (B.17) 64
66 edge vertex edge (multiple edges) vertex edge vertex edge (loop) B.13 B.13: ffl edge vertex B.14 edge vertex B.14: ffl vertex ( vertex ) ffl G A(G) B(G) A(G) jv j jv j ( 1 e ij 2 E(G) a ij = 0 e ij =2 E(G) (B.18) 65
67 jv j V vertex jej edge B(G) b ij = 8 >< >: 1 (v i ( i) e kl ( j) i = k ) 1 (v i ( i) e kl ( j) i = l ) 0 ( ) jv j jej (B.19) B.3.2 (1) G(V;E) edge vertex v L ( B.15) B.15: (G) vertex v L (2) v L G edge G vertex v L edge e L ( B.16) G L G (V L ;E L ) B.16: (G) (L G ) B.15 B.16 G L G B.3 66
68 B.3.3 (1) G A(G) =D(G) B(G) B(G) t [14] D(G) ( (vertexv i vertex ) d ij = i = j 0 i 6= j (B.20) (B.21) G vertex z I jv j jv j jv j D(G) =zi jv j (B.22) (B.20) G A(G) =zi jv j B(G) B(G) t (B.23) (B.20) (BB t ) ij P m l=1 b ilb jl i = j G vertex v i vertex (v i ;v j ) G edge 1 (v i ;v j ) G edge i 6= j 0 (2) G A(L(G)) = B(G) t B(G) 2I jej (B.24) [3] edge (3) G B(G) jej jv j G B(G) G z c B(G)z c = 0 (B.25) 67
69 B.17: z c [14] z c z c = 8 >< >: 1 e i 2 e i 1 e i 2 e i 0 e i =2 B.17 G edge V (B.26) V = fe 1 ;e 2 ;e 3 ;e 4 ;e 5 ;e 6 g (B.27) z c z c = f e 1 1; e 2 1; e 3 1; e 4 1; e 5 1; e 6 0g (B.28) B.3.4 H = X i;j t ij^a y i ^a j (B.29) t 1 G H A(G) 68
70 G L G (L G ) A(L G ) A(L G )Ψ L jej = E LΨ L jej (B.30) (B.24) B(G) t B(G)Ψ L jej 2Ψ L jej = E LΨ L jej (B.31) Ψ L jej (G) (B.25) 2Ψ L jej = E LΨ L jej (B.32) E L = 2 Ψ L jej ( ) ( B.18) E = 2 B.18: C 1 C 2 C 3 B.3.5 B.3.5 z c = f e 12 0 ; e 14 0 ; e 16 0 ; e 32 +1; e 34 1; e 38 0 ; e 52 1; e 56 +1; e 58 0 ; e 74 +1; e 76 1; e 78 0 g (B.33) edge vertex Ψ L jej = fe 12 0 ; e 14 0 ; e 16 0 ; e 32 +1; e 34 1; e 38 0 ; e 52 1; e 56 +1; e 58 0 ; e 74 +1; e 76 1; e 78 0 g (B.34) B.3.5 B.21(b) E = 2t 69
71 B.19: B.20: 70
72 B.3.6 (G) A(G)Ψ G jv j = E GΨ G jv j (B.35) (B.23) zψ G jv j B(G) B(G) t Ψ G jv j = E GΨ G jv j (B.36) (L G ) B(G) t B(G)Ψ L jej 2Ψ L jej = E LΨ L jej (B.37) B(G) t B(G) B(G) B(G) t (B.36) (B.37) E L = z 2 E G (B.38) G G E G E = 0 G L G E L = z 2+E G (B.39) B.21(a)(b) (B.39) B.4 (1) : : E =0 E =0 (2) : (3) : (G) (L G ) : E = 2t L G G 71
73 B.21: (a) (b) 72
74 C C.1 H H = h 11 h 12 h 13 h 14 ::: h 1n h 21 h 22 h 23 h 24 ::: h 2n h 31 h 32 h 33 h 34 ::: h 3n h 41 h 42 h 43 h 44 ::: h 4n h n1 h n2 h n3 h n4 ::: h nn T T = ff 1 fi 1 0 fi 1 ff 2 fi 2 fi 2 ff 3 fi fi n 2 ff n 1 fi n 1 0 fi n 1 ff n (C.1) (C.2) T H Q T = Q 1 HQ (C.3) (C.3) H h q 1 q 2 ::: q n i = h q 1 q 2 ::: q n i T (C.4) 73
75 q k Q k Hq 1 = ff 1 q 1 + fi 1 q 2 Hq 2 = fi 1 q 1 + ff 2 q 2 + fi 2 q 3. Hq k = fi k 1 q k 1 + ff k q k + fi k q k+1 (C.5). Hq n = fi n 1 q n 1 + ff n q n (C.5) q k (q k ; q k )=1 (C.6) (q k ; q k+1 )=0 (C.7) (q k ; q k 1 )=0 (C.8) ff k ff k =(q k ;Hq k ) k =1; 2;:::;n (C.9) (, ) (C.5) fi k q k+1 = Hq k fi k 1 q k 1 ff k q k (C.10) r k Hq k fi k 1 q k 1 ff k q k k =1; 2;:::;n 1 (C.11) fi 0 0 (C.12) fi 2 k (q k+1; q k+1 )=(r k ; r k ) (C.13) fi k p (r k ; r k ) k =1; 2;:::;n 1 (C.14) fi k = (C.10) (C.14) q k+1 q k+1 = r k fi k = p r k (r k ; r k ) (C.15) q 1 (C.9) ff 1 ff 1 (C.11) (C.12) r 1 (C.14) fi 1 q 2 (C.15) (C.9) ff 2 n T ff k fi k 74
76 C.2 T m H m m m T m T m = ff 1 fi 1 0 fi 1 ff 2 fi 2 fi 2 ff 3 fi fi m 2 ff m 1 fi m 1 0 fi m 1 ff m (C.16) m n q 1 ; q 2 ;:::;q m n m Q m Q m =[q 1 ; q 2 ;:::;q m ] (C.17) H T m HQ m = Q m T m + fi m q m+1 e T m (C.18) e T m m e T m =(0; 0;:::;0; 1) (C.19) (C.18) m fi m q m+1 n m (C.5) k = m Hq m = fi m 1 q m 1 + ff m q m + fi m q m+1 (C.20) m T m T m ~y = ~ ~y (C.21) ~y (C.18) HQ m ~y = Q m T m ~y + fi m q m+1 e T m ~y (C.22) ~x = Q m ~y (C.23) 75
77 j~y m j H ~x = ~ Q m ~y + fi m q m+1 ~y m = ~ ~x + fi m ~y m q m+1 (C.24) H ~x w ~ ~x (C.25) ~ ~x H j~y m j 76
78 D ^H 0 E n (0) N n jψ n;ffi (0) (ff =1; 2; ;N n ) ^H 0 jψ (0) n;ffi = E (0) n jψ (0) n;ffi (ff =1; 2; ;N n ) (D.1) jψ n;ffi (0) ^H 0 ^V ^H 0 E n (0) ^H = ^H 0 + ^V E n;ff jψ n;ff i ^Hjψ n;ff i = E n;ff jψ n;ff i (D.2) E n;ff jψ n;ff i E n;ff = E (0) n jψ n;ff i = XN n fi=1 + E(1) n;ff + E(2) n;ff + a fi n;ffjψ (0) n;fi i + jψ(1) n;ffi + (D.3) (D.4) (D.2) ^H 0 jψ (1) n;ffi + ^V XN n fi=1 a fi n;ffjψ (0) i n;fi = E(0) n jψ n;ffi (1) + E (1) n;ff XN n fi=1 a fi n;ffjψ (0) n;fi i (D.3) (D.4) (D.5) hψ (0) m;flj X hψ m;flj (0) ^H N n 0 jψ n;ffi+ (1) fi=1 a fi n;ffhψ m;flj (0) ^V jψ (0) i n;fi = E(0) n hψ m;fljψ (0) n;ffi+e (1) (1) n;ff XN n fi=1 a fi n;ffhψ (0) m;fljψ (0) n;fi i (D.6) m = n =0 ^H (0) E (0) 0 E (1) 0;ff XN 0 fi=1 a fi 0;ffhψ (0) 0;flj ^V jψ (0) 0;fi i = E(1) 0;ff XN 0 fi=1 a fi 0;ffhψ (0) 0;fljψ (0) 0;fi i (ff =1; 2; ;N 0) (D.7) 77
79 [1] D. R. Hofstadter, Phys. Rev. B, 14, (1976) 2239 [2] C. Albrecht, J. H. Smet, K. von Klizing, D. Weiss, V. Umansky, and H. Schweizer, Phys. Rev. Lett. 86, (2001) 147 [3] A. Mielke, J. Phys. A: Math. Gen. 24 (1991) L73; ibid. 24 (1991) 3311; ibid. 25 (1992) [4] E. H. Lieb, Phys. Rev. Lett. 62 (1989) 1201; ibid [5] A. Mielke and H. Tasaki, Commun. Math. Phys. 158 (1993) 341 [6] vol.36 No.10 (2001) [7] K. Shiraishi, H. Tamura and H. Takayanagi, Appl. Phys.Lett. 78 (2001) 3702 [8] K. Shiraishi, H. Tamura and H. Takayanagi, unpubished [9] K. Ishida, Phys. Rev. B, 49 (1994) 5541 [10] K. Kusakabe and H. Aoki, Phys. Rev. Lett. 72 (1994) 144 [11] 1993 [12] Quantum thory of the optical and electronic properties of semiconductors (Third edition), Hartmut Hang and Stephan W. Koch, world Scientific. [13] 1998 [14] B
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
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
) ] [ 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
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
X線-m.dvi
X Λ 1 X 1 O Y Z X Z ν X O r Y ' P I('; r) =I e 4 m c 4 1 r sin ' (1.1) I X 1sec 1cm e = 4:8 1 1 e.s.u. m = :1 1 8 g c =3: 1 1 cm/sec X sin '! 1 ß Z ß Z sin 'd! = 1 ß ß 1 sin χ cos! d! = 1+cos χ (1.) e
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di-problem.dvi
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di-problem.dvi
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1 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
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
![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](/thumbs/103/158001152.jpg "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")
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..
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
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,
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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.........................
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
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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
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 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 z
ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y
01 4 17 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy + r (, y) z = p + qy + r 1 y = + + 1 y = y = + 1 6 + + 1 ( = + 1 ) + 7 4 16 y y y + = O O O y = y
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Report10.dvi
[76 ] Yuji Chinone - t t t = t t t = fl B = ce () - Δθ u u ΔS /γ /γ observer = fl t t t t = = =fl B = ce - Eq.() t ο t v ο fl ce () c v fl fl - S = r = r fl = v ce S =c t t t ο t S c = ce ce v c = ce v
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)
(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
x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
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013 4 16 5 54 (03-5465-7040) nkiyono@mail.ecc.u-okyo.ac.jp hp://lecure.ecc.u-okyo.ac.jp/~nkiyono/inde.hml 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy
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
.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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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
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35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m
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SO(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