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1 Ehrenberg-Siday-Bohm-Aharonov 1. Aharonov Bohm 1) 0 A 0 A A = 0 Z ϕ = e A(r) dr C R C e I ϕ 1 ϕ 2 = e A dr = eφ H Φ Φ 1

2 Aharonov-Bohm Aharonov Bohm 10 Ehrenberg Siday 2) Ehrenberg-Siday-Bohm-Aharonov ESBA( 3). Schrödinger Φ Aharonov Bohm Φ 2π/e Φ 2π/e R B Φ = πr 2 B z z (r, θ, z) x = r cos θ y = r sin θ A = rot B A θ, r, z ( Φ r/(2πr 2 ) (r < R) A θ = Φ/(2πr) (r > R) A r = A z = 0 (1) A θ = A x sin θ + A y cos θ A r = A x cos θ + A y sin θ Schrödinger H = 1 2µ ( + iea)2 (2) µ h = c = 1 Schrödinger H = E E (1) " 1 µ # 2 2µ r = E 2 β = eφ 2π 2

3 β A θ = Φ/(2πr) = β/(er) (3) (r = R) = 0 Schrödinger A A + χ, exp( ieχ) (4) χ (4) χ = (β/e)θ = e iβθ (5) (3) β 1 2µ 2 = E (6) r > R χ = (β/e)θ z z (5) (6) Schrödinger ESBA θ r (θ + 2π) = (θ) (5) (θ + 2π) = e 2πiβ (θ) (7) z β e 2πiβ θ θ 0 θ < 2π (7) x θ = 0 lim θ 2π (θ) = e 2πiβ lim θ 0 (θ), lim (θ) = e 2πiβ lim (θ) (8) x e 2πiβ x 0 θ < 2π 3

4 θ r (r = R) = 0 θ (7) (8) (6) β β Φ ESBA (7) β 3. Φ Φ θ Faraday θ θ z E θ B z ee θ + ev r B z θ v r r z ρ(t) L dl dt = ρee θ(ρ) + ρev r B z (ρ) (9) z Stokes ρee θ (ρ) = e 2π I r=ρ E dr = e 2π Z Z r ρ rote ds = e Z ds (10) 2π r H r=ρ ρ RR r ρ ρ z r φ(r) d dt φ(ρ) + r φ(ρ) ρ φ(ρ) 4

5 v r = dρ/dt φ(ρ) = @ρ φ(ρ) = 2πρB z(ρ) (12) (9) (12) dl dt = e d φ(ρ) (13) 2π dt t 0 t 1 r < R Φ t 1 Φ t 0 L 0 (13) t 0 t > t 1 L = L 0 + (e/2π)φ(ρ) ρ t t > t 1 φ(ρ) = φ(r) = Φ β L = L 0 + β (14) Φ β β L 0 r p = i p canonical momentum p + ea mechanical momentum r p r (p + ea) x-y z θ + era θ r > R (θ + 2π) = (θ) (θ + 2π) = (θ) Schrödinger (θ + 2π) = (θ) (5) (θ + 2π) = (θ) (θ + 2π) = (θ) 5

6 Schrödinger = H H (2) L = + era θ Schrödinger d D, E + era θ = h, era D h θ i +, i + iera θ E = A θ E θ (e/2)(v r rb z + rb z v r ) B z + A θ r r v r = i r + iea r d dt h, L i = eh, re θ i + e 2 h, (v rrb z + rb z v r ) i (15) hφ, L i (15) v r rb z (9) t 0 t 1 Φ t 0 t 1 A θ = Φ/(2πr) A r = A z = 0 θ t 0 θ e imθ m m m A θ H θ [H, = 0 m r θ e imθ t 1 Φ m A θ = Φ/(2πr) m + era θ = m + β Φ β θ Φ t > t 1 6

7 t > t 1 +β β t > t 1 β 2π/e A θ = Φ/(2πr) β (5) (θ + 2π) = (θ) m + β θ e i(m+β)θ β (7) β 4. Schrödinger x-y z Bessel J N β k,m (r, θ) = N m+β (kr)j m+β (kr) J m+β (kr)n m+β (kr) e imθ (16) m k Schrödinger (3) β k,m (R, θ) = 0 E = k2 /(2µ) 7

8 m A θ = β/(er) m + β Aharonov Bohm t 0 t 0 t 1 Φ t 1 Φ t k,m 0 = Z 1 0 dk X m C k,m 0 k,m (C k,m (17) t 0 t 1 t = Z 1 0 dk X m D k,m β k,m (D k,m (18) (17) (18) β k,m 0 k,m t 0 t 1 A θ = β/(er) t 0 t 1 +β (5) t 1 1 = Z 1 0 dk X m D k,m β k,m, β k,m = β k,m eiβθ (19) β k,m θ ei(m+β)θ m + β β k,m 1 1 (θ+2π) = e 2πiβ 1 (θ) θ 0 t 1 Schrödinger = (1/2µ) 2 β k,m k2 /(2µ) t > t 1 (19) 1 β k,m β k,m exp[ ik2 (t t 1 )/(2µ)] = Z 1 0 dk X m D k,m β k,m exp[ ik2 (t t 1 )/(2µ)] (20) 8

9 ESBA t t 1 1 t 0 t 1 t 1 1 +β β 1 1 (θ + 2π) = e 2πiβ 1 (θ) t 0 t 1 θ 0 θ < 2π 1 θ x (8) r r = R 1 θ = π x v 1 p 1 = v 1 /µ x v 1 θ σ(θ) p 1 C exp(ip 1 x) C exp(ip 1 x) x β β k,m x (1) p 1 C exp(ip 1 x iβθ) β (19) D k,m 1 v 1 (20) v 1 β x β β β ESBA 9

10 (7) β 1 Φ Φ + 2jπ/e j Φ 2π/e (16) β+j j k,m = β β+j k,m+j { k,m } { β k,m } +j + β +β β 6. ESBA Φ t 1 (16) Φ ESBA 10

11 7. Φ θ +β ESBA β Φ β (14) L 0 eφ/(2π) (7) ESBA ESBA m 1, m 2, m 3,... m i m i m j m i +α 0 α < 1 α i α α α α α β = eφ/2π ESBA 11

12 1) Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959), ) W. Ehrenberg and R. E. Siday, Proc. Roy. Soc. London, B62 (1949), 8. 3) John Bell H.M.) J.B. Ehrenberg Siday Ehrenberg-Siday- Bohm-Aharonov ESBA 4). Tonomura et al., Phys. Rev. Lett. 56 (1986),

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