Note5.dvi

12 2011 7 4 2.2.2 Feynman ( ) S M N S M + N S Ai Ao t ij (i Ai, j Ao) N M G = 2e2 t ij 2 (8.28) h i μ 1 μ 2 J 12 J 12 / μ 2 μ 1 (8.28) S S (8.28) (8.28) 2 ( ) (collapse) j 12-1

2.3 2.3.1 Onsager S B S(B) = t S( B) (S mn (B) =S nm ( B)) (8.29) Schrödinger Schrödinger [ (i + ea) 2 2m + V ] ψ = Eψ (8.30) A A [ ] (i + ea) 2 + V ψ = Eψ 2m {ψ ( B)} = {ψ(b)} (8.31) ψ(b) ψ ( B) ({ } ) ψ(b) Schrödinger (8.30) Sc{a b}(a S b ) Sc{a(B) b(b)} {ψ(b)}, (8.32) i.e., b(b) =S(B)a(B) (8.33) (8.33) b (B) = S(B)a (B). (8.34) exp(±ikr) *1 Sc(b (B) a (B)) {ψ (B)} (8.35) B B Sc{b ( B) a ( B)} {ψ ( B)} = {ψ(b)} (8.36) i.e. a ( B) =S(B)b ( B) (8.37) (8.37) b (B) =S 1 ( B)a (B) (8.38) *1 (8.30) Schrödinger iωt ikr 12-2

(8.34) S(B) =S 1 ( B) =S ( B) ( unitarity SS = S S = I) S(B) = t S( B) (8.39) (ρ xx ) ( ) ρ xx (B) =ρ xx ( B) (8.40) 2.3.2 Landauer-Büttiker p q 8.16 p μ p = ev p p J p p p p T p q J p = 2e h [T q p μ p T p q μ q ] (8.41) q T pq T p q (p q), T pp q p T q p T J = t (J 1,J 2, ) µ = t (μ 1,μ 2, )( ) J = 2e h T µ V q = μ q e, G pq 2e 2 ht p q J p = q [G qp V p G pq V q ] (8.42) J q = 0 (8.43) q 12-3

B Onsager [G qp G pq ] = 0 (8.44) q G qp (B) =G pq ( B) (8.45) S-matrix Onsager 4 V 4 =0 J 1 J 2 = G 12 + G 13 + G 14 G 12 G 13 G 21 G 21 + G 23 + G 24 G 23 V 1 V 2 (8.46) J 3 G 31 G 32 G 31 + G 32 + G 34 V 3 (Casimir) J 1 = J 3, J 2 = J 4 (8.47) J 2 =0 1 3 2 4 V ij V i V j ( J1 ) ( )( ) α11 α = 12 V13 J 2 α 22 V 24 α 21 (8.48) q ev q q J q 1 1 ev 1 J 1 J 2 2 2 ev 2 J p p p ev p 8.16 LB 12-4

α 11 =2G q [ T 11 S 1 (T 14 + T 12 )(T 41 + T 21 )] (8.49a) α 12 =2G q S 1 (T 12 T 34 T 14 T 32 ) (8.49b) α 21 =2G q S 1 (T 21 T 43 T 23 T 41 ) (8.49c) α 22 =2G q [ T 22 S 1 (T 21 T 23 )(T 32 + T 12 )] (8.49d) S = T 12 + T 14 + T 32 + T 34 = T 21 + T 41 + T 23 + T 43 (8.50) (8.48) (8.47) V 1 V 2 V 3 2.3.3 2 Onsager (8.40) Landauer- Büttiker 4 (8.45) (8.48) α 11 (B) =α 11 ( B), α 22 (B) =α 22 ( B), α 12 (B) =α 21 ( B) (8.51) 13: 24: LB R 13,24 R 13,24 = V 2 V 4 J 1 = α 21 α 11 α 22 α 12 α 21 (8.52) Onsager R (8.40) R 24,13 = α 12 α 11 α 22 α 12 α 21 (8.53) (8.51) T km T ln T kn T lm R mn,kl = R q, D R 2 D q(α 11 α 22 α 12 α 21 )S (8.54) R mn,kl (B) = R kl,mn ( B) (8.55) B B 12-5

2.4 Aharonov-Bohm 1 a b 2 S Aharonov-Bohm(AB) AB 1 2 ψ 1,2 ψ = ψ 1 + ψ 2 ψ 2 = ψ 1 2 + ψ 2 2 +2 ψ 1 ψ 2 cos ϕ (8.56) ϕ? p p + ea (k + e A) exp(ik x) θ AB = b a k dx Δθ AB = e b a A dx (8.57) [ Δθ = e b A ds (1) a b a ] A ds (2) = e A ds = e loop BdS =2π Φ Φ 0, Φ 0 h e. (8.58) Φ 0 2π (8.56) Φ 0 S AB S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (8.59) 2 1/2 1/2 *2 AB ( ) 0 e iθ AB S AB = e iθ, θ 2π φ = e φ (φ ) (8.60) AB 0 φ 0 S S ( ) 0 e iθ 0 S w = e iθ0 0 (8.61) *2 12-6

a 1 =1 b 1 S t b 2 a 2 b 3 S AB S w a 4 b 4 a 5 S t b 6 a 6 =0 2 (a) a 3 b 5 0 (c) 0 / 0 1 2.8 0 2.2 1.6 0 / 0 1.0 t 2 (b) 2 0 (d) 0 =0.4 0 / 0 1 8.17 (a)ab S (b) (8.62) AB t 2 (θ 0 ) φ/φ 0 (c)(b) (d) φ/φ 0 φ 0 AB θ 0 =1.6 π θ 0 θ AB (8.29) S t = 4sinθ 0 1+e iθ AB (e iθ AB + e iθ 0 3e iθ 0) (8.62) ((8.28) )T = t 2 φ 8.17(b) φ 0 AB θ 0 2π t 2 φ =0 S (8.60) Onsager (8.62) φ 0 θ 0 π 12-7

φ 0 /2 θ 0 φ 0 0 π AB (phase rigidity) AB AB AB *3 (8.56) (8.60) S (8.62) 8.17(d) AB 2DEG AB 10 2.5 Green S S Keldysh 2.5.1 Green Green S T Green D GR G A G R 1 = E H + iη G A 1 = E H iη (η +0), (8.63a) (η +0) (8.63b) 1 (x) G R (E H + iη)g R (x, x )=δ(x x ) (8.64) x x Ψ α x Ψ α = ψ α (x) G R (x, x )= x α Ψ α Ψ α E H + iη x = α ψ α (x)ψ α (x ) E H + iη (8.65) *3 Onsager 12-8

A m A m y p y q x p 0 0 x q 8.18 S Green 2.5.2 1 D (D.10) x y y ϕ n (y) (8.65) G R (x, y; x,y ) G R (x, y; x,y )= n ( i ) ϕ n (y)ϕ n v (y )exp[ik n x x ]. (8.66) n 2.5.3 S 8.18 x p =0 x q =0 G R (y q ; y p )= (δ nm A m + s vm nm A + m v )χ n(y q ) (8.67) n n q,m p A ± m s nm S v i (8.66) A ± m = χ m(y p ) i v m G R (y q ; y p )= n q,m p i v n v m χ n (y q )(δ nm + s nm )χ m (y p ) (8.68) s nm = δnm + i v n v m dy q dy p χ n (y q )G R (y q ; y p )χ m (y p ) (8.69) S [1]T.Ando,A.B.Fowler,andF.Stern,Rev.Mod.Phys.54, 437 (1982). 12-9

[2] S. Datta, ElectronTransport in Mesoscopic Systems (Cambridge Univ. Press, 1995). [3] ( 2002) [4] K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). [5] S. Kawaji and J. Wakabayashi, in Physics in High Magnetic Fields, eds.s.chikazumi and N. Miura (Springer, Berlin 1981). [6] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). [7] (, 1995) [8] (, 1998) D Green S R Green ( )Ĝ R = ĜS Ĝ = D 1 when DR = S (D.1) S Schrödinger [E H ]ψ = S (D.2) Green Ĝ =[E H ] 1 (D.3) Ĝ ( 1 ) (x, x ) G(x, x ) (D.3) [E H ]G(x, x )=δ(x x ) (D.4) (D.2) G(x, x ) x x 1 E E H = E + 2 2 2m x 2 (D.5) G x k x G(x, x )=A ± exp[±ik(x x )], x x (k = 2mE/ ), (D.6) x = x G(x x +0,x )=G(x x 0,x ). (D.7) 12-10

(D.4) (D.5) G/ x (D.7) x = x G E [ ] G = 2m x x δ(x 2 x ) x [ ] [ ] G G = 2m x x 2. (D.8) (D.7) (D.8) (D.6) x x +0 x x 0 A + = A, ik[a + + A ]=2m/ 2, A + = A = 1 m ik 2, (D.9) G R (x, x )= i v exp [ik x x ], v k m, (D.10) (retarded solution) (advanced solution) G A (x, x )= i v exp [ ik x x ], η [E + 2 2 ] 2m x + iη G R (x, x )=δ(x x ) 2 (D.11) (D.12) k k = k [ 1+ iη 2E ( ) ( ) ]. G R 1 = E H + iη G A 1 = E H iη (η +0), (D.13a) (η +0) (D.13b) 12-11

E α Casimir α ( (8.48)) T (8.50) α 11 α 21 Casimir V 2 = V 4 =0 h 2e 2 J 1 =(T 21 + T 31 + T 41 )V 1 T 13 V 3 h 2e 2 J 3 = T 31 V 1 +(T 13 + T 23 + T 43 )V 3 Casimir J 1 = J 3 J 1 + J 3 =(T 21 + T 41 )V 1 +(T 23 + T 43 V 3=0 V 3 = T 21 + T 41 T 23 + T 43 V 1. J 2 = J 4 V 1 =1 (2G q ) 1 J 2 = T 2 1+T 23 T 21 + T 41 T 23 + T 43 = T 23T 41 T 21 T 43 T 23 + T 43 V 13 =1+ T 21 + T 41 = T 21 + T 41 + T 23 + T 43 T 23 + T 43 T 23 + T ( ) 43 T23 T 41 T 21 T 43 α 21 = 2G q =2G q S 1 (T 21 T 43 T 23 T 41 ). T 21 + T 41 + T 23 + T 43 (2G q ) 1 J 1 = T 21 + T 31 + T 41 + T 13 T 21 + T 41 T 23 + T 43 α 11 =2G q [ (T21 + T 31 + T 41 )(T 23 + T 43 S + T ] 13(T 21 + T 41 ) S =2G q (T 12 + T 13 + T 14 S 1 (T 14 + T 12 )(T 21 + T 41 )) =2G q ( T 11 S 1 (T 14 + T 12 )(T 21 + T 41 )). 12-12