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- あやか いなおか
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2 i E B Maxwell Maxwell Newton Newton Schrödinger Newton Maxwell Kepler Maxwell Maxwell B H B
3 ii Newton
4 i Coulomb E φ E n E φ φ φ φ Gauss Gauss ( ) Gauss ( ) Poisson Laplace Laplace Poisson
5 ii x y N ( ) J
6 iii C R RC = ε κ Kirchhoff Biot-Savart Biot-Savart Ampere Biot-Savart m 1 m Faraday (1) (2)
7 iv Maxwell : RC : LC ɛ µ (k, λ) Lorentz Maxwell Lorentz Hamiltonian Zeeman Lamour A 81 A A A A A.2 : A A A B Lagrange 85 B.1 Lagrange B.1.1 Newton
8 v B.1.2 Lagrange B B B B.3 Lagrange B.3.1 Lagrangian B.3.2 Lagrange B.3.3 Schrödinger B.3.4 Hamiltonian B.4 QED+ Lagrangian B B.4.2 Lagrangian C Kepler 97 C C.2 Kepler C.2.1 Kepler ( ) C C C.3.2 GPS C D 109 D D D D D D D D D D D
9 vi E 115 E E E E E.2.1 ( ) E E E E E.4.1 δ E E E E E E.5.1 (x, y, z) E.5.2 (r, ϕ, z) E.5.3 (r, θ, ϕ) E E E.6.1 Euler E E E E.9 Taylor E E
10 1 1.1 E e E F = ee Maxwell Maxwell Maxwell E = ρ ε 0 (Gauss ) (1.1a) B = 0 E = B t B = µ 0 j + 1 c 2 E t ( ) (1.1b) (Faraday ) (1.1c) (Ampere Maxwell ) (1.1d) ρ j ε 0 µ 0 E B E = E(t, r), B = B(t, r)
11 2 1.2 Coulomb Coulomb q, Q F F = qq r (1.2) 4πε 0 r 3 q Q r Coulomb q, Q V (r) V (r) = qq 1 4πε 0 r (1.3) q, Q 1.3 e R e q, Q F F = e 4πε 0 ( ) q R R 2 R + Q (R r) r R 2 r R q, Q Maxwell (1.4)
12 3 2.1 E q Q r Q F = qq 4πε 0 r r 3 (2.1) φ Q q E q r (2.2) 4πε 0 r 3 E q r Q F r ψ(r) E q 1 r 1 q 2 r 2 r E = 1 { } q 1 (r r 1 ) 4πε 0 r r 1 2 r r 1 + q 2 (r r 2 ) (2.3) r r 2 2 r r 2
13 n E q 1,, q n r 1,, r n r E = 1 4πε 0 n i=1 q i (r r i ) r r i 2 r r i (2.4) 2.2 φ φ φ φ E E = φ (2.5) E φ E = 0 (2.6) E = φ Maxwell Faraday (2.6) (2.5) φ φ q r φ φ(r) = q 1 (2.7) 4πε 0 r E E = φ E(r) = q r (2.8) 4πε 0 r 3
14 2.2. φ 5 (2.2) φ q (0, 0, d) q (0, 0, d) φ(r) = 1 q 4πε 0 x 2 + y 2 + (z d) + q (2.9) 2 x 2 + y 2 + (z + d) 2 ( r >> d) x (1 + x) α 1 + αx α(1 α)x2 + (2.10) φ(r) = q ( 1 2zd ) 1 4πε 0 r r + d2 2 (1 + 2zd ) 1 2 r 2 r + d qdz 2 r 2 4πε 0 r 3 (2.11) p 2qd φ φ(r) = 1 pz (2.12) 4πε 0 r 3 p q q d p = qd p r φ φ(r) = 1 [ ] q 4πε 0 r + 1d + q r 1d 2 2 φ(r) = q 4πε 0 r ( 1 d r r 2 ) 1 + d2 2 (1 + d r ) 1 + d2 2 r 2 r 2 r 2
15 6 (2.9) ( r >> d ) φ(r) 1 4πε 0 p r r 3 (2.13) p p p E = φ E(r) = 1 [ ] p 3r(p r) 4πε 0 r3 r 5 (2.14) φ a Q ρ ρ = 3 Q r = (x, y, z ) 4πa 3 dv dv = dx dy dz δq δq = ρdx dy dz r δφ δφ(r) = 1 δq (2.15) 4πε 0 r r φ(r) = 1 ρdx dy dz (2.16) 4πε 0 r r dx dy dz = r 2 dr sin θdθdϕ t = cos θ φ φ(r) = ρ a 1 2π r 2 dr 4πε dt r 2 + r 2 2r rt (2.17)
16 2.2. φ 7 φ(r) = 1 Q 4πε 0 r ( 1 Q 3 ) r2 4πε 0 a 2 2a 2 r > a (2.18) r < a
17
18 9 Gauss 3.1 Gauss ( ) Gauss E = ρ ε 0 (3.1) E E = q r (2.2) 4πε 0 r 3 Gauss E = q ( ) r = q [ x 4πε 0 r 3 4πε 0 x r + y 3 y r + ] z = 0 (3.2) 3 z r 3 (3.2) a I = Ed 3 r = q ( ) r d 3 r = q ( ) r ds r <a 4πε 0 r <a r 3 4πε 0 r =a r 3 = q ( ) 1 a 2 sin θdθdϕ = q (3.3) 4πε 0 r =a a 2 ε 0 Gauss ρ(r)d 3 r = q ρ(r) ρ(r) = qδ(r) (3.4) E 2 1 r = 4πδ(r) (3.5)
19 10 Gauss 3.2 Gauss ( ) Gauss Gauss A Ad 3 r = A n ds (3.6) V S A n A Gauss Gauss V Ed 3 r = S E n ds = 1 ε 0 V ρ(r)d 3 r (3.7) (3.7) E r q r r Gauss r <r E(r )d 3 r = r =r r =r E r r 2 dω = 1 ε 0 V ρ(r )d 3 r = q ε 0 (3.8) E r r 2 dω = E r r 2 4π (3.9) E r = 1 q E = 1 q r 4πε 0 r 2 4πε 0 r 2 r (3.10)
20 3.2. Gauss ( ) a Q ρ = 3Q r 4πa 3 Gauss r a (1) r > a Gauss r r <r E(r )d 3 r = E r = r =r E r r 2 dω = 1 ε 0 V ρ(r )d 3 r = Q ε 0 (3.11) Q 1 E = Q r (3.12) 4πε 0 r 2 4πε 0 r 3 (2) r < a r r <r E r = 4πr3 ρ 3ε 0 1 4πr 2 = ρ(r )d 3 r = 4πr3 ρ 3ε 0 (3.13) ρ 3ε 0 r = Q r (3.14) 4πε 0 a 3 (2.18) E r = φ(r) r = Q 4πε 0 r a 3
21 12 Gauss σ S Q Q = σs a x, y z Gauss E n ds = 1 ρ(r)d 3 r (3.15) ε 0 S 2E z a 2 E z = V σa 2 ε 0 σ 2ε 0 (3.16) Q Q S, d (3.16) E = σ ε 0 V V = φ 2 φ 1 = Edz = Ed V = Ed = σ d = d Q (3.17) ε 0 Sε 0 V Q = CV C C = Sε 0 d
22 3.2. Gauss ( ) a Q (1) r > a Gauss r E(r )d 3 r = E r r 2 dω = 1 ρ(r )d 3 r = Q (3.18) ε 0 ε 0 r <r E r = r =r V Q 1 E = Q r (3.19) 4πε 0 r 2 4πε 0 r 3 (2) r < a E r = 0 (3.20) 3.2.5
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24 15 Poisson Gauss E = ρ ε 0 Faraday E = 0 E = φ Poisson 2 φ = ρ ε 0 (4.0) 4.1 Laplace Poisson 2 φ(r) = 0 (4.1) Laplace Laplace Laplace 2 2 = 1 r 2 r r2 r + 1 r 2 sin θ θ sin θ θ + 1 r 2 sin 2 θ 2 ϕ 2 (4.2) Laplace [ 1 r 2 r r2 r + 1 r 2 sin θ θ sin θ θ ] r 2 sin 2 φ(r) = 0 (4.3) θ ϕ 2 ϕ ( ) ( ) φ(r, θ) = a 0 + b 0 r + a 1 r + b 1 r 2 cos θ (4.4)
25 16 Poisson 4.2 Poisson Poisson Poisson 2 φ(r) = ρ(r) (4.5) ε q ρ(r) = qδ(r) (4.6) 2 1 r = 4πδ(r) (4.7) (4.5) φ(r) = q 1 4πε 0 r Poisson (4.8) σ φ r = (0, 0, z) φ(r) = 1 σ dx 4πε 0 x dy (4.9) 2 + y 2 + z 2 r = (0, 0, z) x, y φ(r) = σ 4πε 0 2π 0 dθ Λ 0 r dr 1 r 2 + z 2 (4.10)
26 4.2. Poisson 17 r Λ t = r 2 φ(r) = σ (Λ z ) (4.11) 2ε 0 z E z (r) = φ z = θ(z) ( ) 1 z < 0 θ(z) = 1 z > 0 σ 2ε 0 θ(z) (4.12) (4.5) φ 2 1 r r = 4πδ(r r ) (4.13) φ(r) = 1 4πε 0 ρ(r ) r r d3 r (4.14) Poisson 2 r 2 φ(r) = 1 ρ(r ) 2 1 4πε 0 r r d3 r 2 φ(r) = 1 ρ(r ){ 4πδ(r r )}d 3 r = ρ(r) (4.15) 4πε 0 ε 0 Poisson (4.10)
27 18 Poisson x y z d q r φ Poisson 2 qδ(r d) φ(r) = (4.16) ε 0 d = (0, 0, d) φ(r) = q 1 4πε 0 r d (4.17) Poisson x y Poisson Poisson Laplace Poisson φ(r) = q 1 (4.18) 4πε 0 r + d 2 φ(r) = q 2 1 4πε 0 r + d = q δ(r + d) (4.19) ε 0 d r φ(r) = 1 [ q 4πε 0 r d + q ] (4.20) r + d Poisson q d
28 4.2. Poisson x y φ(r) = q 1 4πε 0 x 2 + y 2 + (z d) 1 (4.21) 2 x 2 + y 2 + (z + d) 2 E x = φ(r) x = qx { (x 2 + y 2 + (z d) ) 3 2 ( x 2 + y 2 + (z + d) ) } 3 2 4πε 0 (4.22) z = 0 E y E z E z (x, y, 0) = φ(r) z = qd ( x 2 + y 2 + d 2) 3 2 2πε 0 (4.23) σ Gauss σ = ε 0 E z σ = qd ( x 2 + y 2 + d 2) 3 2 2π (4.24) Q = σdxdy = qd dxdy = qd rdrdθ = q 2π (x 2 + y 2 + d 2 ) 3 2 2π (r 2 + d 2 ) 3 2 (4.25) z q p z d p
29 20 Poisson z p z p z = d φ φ(r) = 1 { p (r d) + 4πε 0 r d 3 } p (r + d) r + d 3 (4.26) z p z p z = d φ φ(r) = 1 { p (r d) 4πε 0 r d 3 } p (r + d) r + d 3 (4.27) p d = 0
30 q 1 r 1 q 2 r 2 Coulomb U U = 1 q 1 q 2 (5.1) 4πε 0 r 1 r 2 q 1, q 2, q 3 r 1, r 2, r 3 U = 1 ( q1 q 2 4πε 0 r 1 r 2 + q 2q 3 r 2 r 3 + q ) 3q 1 r 3 r 1 N U = 1 8πε 0 N i,j=1 q i q j r i r j (5.2) (5.3) 5.2 N q i φ i = 1 4πε 0 N j=1 q j r i r j (5.4)
31 22 N U = 1 N q i φ i (5.5) 2 j=1 5.3 d 3 r, q i ρ(r), φ i φ(r) (5.6) i (5.5) U = 1 ρ(r)φ(r)d 3 r (5.7) 2 Poisson 2 φ(r) = ρ(r) ε 0 ρ(r) φ(r) U = ε 0 ( 2 φ(r))φ(r)d 3 r (5.8) 2 ( φ(r)φ(r)) = ( 2 φ(r))φ(r) + φ(r) φ(r) (5.9) U = ε 0 φ(r) φ(r)d 3 r ε ( φ(r)φ(r))d 3 r (5.10) R ( φ(r)φ(r))d 3 r = ( φ(r)φ(r)) n ds 1 R 0 (5.11) r=r R 1 U U = ε 0 φ(r) φ(r)d 3 r = ε 0 E(r) 2 d 3 r (5.12) 2 2
32 q E = q r 4πε 0 r 3 U U = ε 0 2 E(r) 2 d 3 r = ε 0 2 ( q 4πε 0 ) 2 4π 1 r 4 r2 dr = q2 8πε 0 1 r 0 (5.13) r a Q E r = 1 q U = ε 0 2 4πε 0 r 2 E(r) 2 d 3 r = ε 0 2 ( Q 4πε 0 ) 2 4π a 1 r 4 r2 dr = Q2 8πε 0 a (5.14) U = 1 2 ρ(r)φ(r)d 3 r = 1 2 Q 1 Q 4πε 0 a = 1 2 Q 2 4πε 0 a (5.15) a Q Q r 4πε 0 r < a a 3 E r = (5.16) r > a Q 4πε 0 1 r 2
33 24 U = ε 0 2 [ ( ) Q 2 a ( ) Q 2 r 2 (4πr 2 )dr + a 3 4πε 0 0 4πε 0 a Q 2 U = 3 5 4πε 0 a ] 1 r 4 (4πr2 )dr (5.17) Coulomb
34 25 S d V C Q = CV (6.1) C Q 6.1 Q S σ σ = Q S (6.2) Q x y z E z E z = σ z (6.3) 2ε 0 z Q x y d E = σ ε 0 (6.4)
35 V φ + φ V = C C = ε 0S d Edz = Ed (6.5) V = Ed = σ ε 0 d = Q ε 0 S d (6.6) (6.7) 6.3 U U = ε 0 2 E 2 d 3 r = ε 0 2 ( σ ε0 ) 2 1 ε 0 S Sd = 2 d V 2 (6.8) U = 1CV t Q σ σ U = ε 0 2 ( σ ε0 ) 2 1 ε 0 S S(d t) = 2 d V 2 ( ) d t d (6.9)
36 E (Polarization) P P = np d (7.1) p d n r n = n(r) E P E P = χ e ε 0 E (7.2) χ e 7.2 ρ p ρ p = P (7.3)
37 28 Poisson ρ r ε 0 E = ρ r + ρ p (7.4) D Poisson D D = ε 0 E + P (7.5) D = ρ r (7.6) D = ε 0 E + P = ε 0 E + χ e ε 0 E = εe (7.7) ε ε = ε 0 (1 + χ e ) (7.8) ε 7.3 p d p d r φ d (r) φ d (r) = 1 4πε 0 p d r r 3 (7.9) p d R r φ d (r) φ d (r) = 1 4πε 0 p d (r R) r R 3 (7.10) n(r) φ d (r) φ d (r) = 1 4πε 0 n(r) p d (r R) r R 3 d 3 R (7.11)
38 r R r R = 1 3 R r R (7.12) φ d (r) φ d (r) = 1 4πε 0 P R 1 r R d3 R (7.13) P = n(r)p d ( ) 1 1 R P = P R r R r R + ( 1 R P ) r R (7.11) φ d (r) = 1 [ ( ) 1 R P d 3 R 4πε 0 r R ] 1 ( R P ) r R d3 R (7.14) Gauss V R ( P ) ( ) 1 d 3 1 R = P ds 0 r R S r R n ρ p = P φ d (r) ρ p φ d (r) = r R d3 R (7.15) ρ p Gauss ε 0 E = ρ r + ρ p (7.16)
39 D = εe Gauss D = ρ r (7.17) ρ r ρ r Gauss ρ r D = D n ds = D n (1) S D n (2) S = 0 (7.18) V S D (1) n = D (2) n ε 1 E (1) n = ε 2 E (2) n (7.19) n Faraday E = 0 (7.20) a Stokes E ds = E ds = (E (1) t E (2) t )a = 0 (7.21) S s E t x y E (1) t = E (2) t E (1) x = E (2) x, E (1) y = E (2) y (7.22)
40 U = 1 2 ρ r (r)φ(r)d 3 r (7.23) Gauss D = ρ r (r) U = 1 2 D(r)φ(r)d 3 r (7.24) U = 1 2 D(r) E(r)d 3 r (7.25) D = εe U = ε E(r) 2 d 3 r (7.26) 2
41
42 q ( m ) E F F = qe = q φ = U (8.1) U F m Newton m r = F (8.2) 8.2 r p = qd E φ U r >> d U(r) = qφ (r 1 ) 2 d + qφ (r + 1 ) 2 d = q φ d = p E (8.3) F = U(r) = (p E) (8.4)
43 34 p 8.3 Q Q 2 d x y U U = ε 0 E 2 d 3 r = ε ( ) 0 σ 2 1 ε 0 S Sd = 2 2 ε0 2 d V 2 (8.5) Q z U(z) = ε ( ) 0 σ 2 Q 2 zs = 2 ε0 2ε 0 S z (8.6) F F z = U z = Q2 2ε 0 S z x y (8.7) 8.4 q m Lagrangian L = 1 2 mṙ2 + q (ṙ A φ) (8.8)
44 A B = 0 B = A Lagrange L x = d L dt ẋ, L y = d L dt ẏ, L z = d L dt ż (8.9) m r = qṙ B + qe (8.10) Lorentz E F Lagrangian Hamiltonian Hamiltonian H H = 1 2m (p qa)2 + qφ (8.11) q Schrödinger [ ] 1 2m ( i h qa)2 + qφ ψ(r) = Eψ(r) (8.12) E B A φ
45
46 ( ) ρ Q = ρ(r)d 3 r J J = dq (9.1) dt j J J = j ds (9.2) V dq dt = j ds (9.3) Q = V ρ(r)d3 r Gauss ρ V t d3 r = j d 3 r (9.4) V V ρ t + j = 0 (9.5)
47 j E j = κe (9.6) κ t t J J J = j n ds = j ds (9.7) J = j n ds = κ E n ds (9.8) φ 1 φ 2 = V = RJ (9.9) R C R Maxwell D = ρ (9.10) Q Q = ρd 3 r = D n ds = ε E n ds (9.11)
48 C C = Q V = ε E n ds (9.12) V J J = κ E n ds (9.13) R 1 R = J V = κ V E n ds (9.14) RC = ε (9.15) κ RC = ε κ RC = ε κ C V S, d ε E V Q V = Ed = Qd εs, C = εs (9.16) d κ V j = κe V = Ed = Jd Sκ, R = d Sκ (9.17) RC = ε κ ε κ ε κ
49 Kirchhoff Kirchhoff J i = 0 (9.18) i Kirchhoff R i J i = V i (9.19) i i
50 Biot-Savart Ampere B = µ 0 j (10.1) Ampere Biot-Savart Biot-Savart B = µ 0J 4π dr (r r ) r r 3 (10.2) J j Jdr = j(r )d 3 r (10.3) Biot-Savart B = µ 0 j(r )d 3 r (r r ) (10.4) 4π r r Biot-Savart Ampere B = µ 0 4π j(r )d 3 r 1 r r = µ 0 4π a (b c) = (a c)b (a b)c) j(r )d 3 r 2 1 r r
51 r r = 4πδ(r r ) B = µ 0 j(r )d 3 r ( 4πδ(r r )) = µ 0 j (10.5) 4π Ampere Ampere Biot-Savart z J P (x, y, z) r = (0, 0, z ) dr (r r ) = ( ydz, xdz, 0) B x = µ 0J 4π B y = µ 0J 4π ydz (x 2 + y 2 + z 2 ) 3 2 xdz (x 2 + y 2 + z 2 ) 3 2 = µ 0J 2π = µ 0J 2π y x 2 + y 2 (10.6) x x 2 + y 2 (10.7) x = r cos θ, y = r sin θ B = (B x cos θ + B y sin θ)e r + ( B x sin θ + B y cos θ)e θ = µ 0J 2πr e θ (10.8) a J x y P (x, y, z) r = (a cos θ, a sin θ, 0) dr = ( a sin θdθ, a cos θdθ, 0) r r = (x a cos θ, y a sin θ, z) dr (r r ) = za cos θdθe x +za sin θdθe y (ya sin θ+xa cos θ a 2 )dθe z (10.9)
52 10.1. Biot-Savart 43 B = µ 0J 4π 2π 0 dθ za cos θe x + za sin θe y (ya sin θ + xa cos θ a 2 )e z (r 2 2ax cos θ 2ay sin θ + a 2 ) 3 2 (10.10) z x = 0, y = 0, r = z B = µ 0J 2 a 2 e (z 2 + a 2 ) 3 z (10.11) 2 (r >> a) r >> a [ ] (r 2 2ax cos θ 2ay sin θ + a 2 ) 3 1 3ax cos θ 3ay sin θ a2 r 3 r 2 r 2 2r + 2 (10.12) (10.10) B = µ 0m 4π [ 3zx r e 5 x + 3zy r e 5 y + 3z2 r 2 r 5 e z ], m = Jπa 2 (10.13) m = Jπa 2 m = (0, 0, m) B = µ ( ) 0 m r 4π r 3 (10.14) E = 1 ( ) p r (10.15) 4πε 0 r 3 φ m φ m = µ ( ) 0 m r (10.16) 4π r 3
53 44 φ m 10.2 Ampere Ampere B = µ 0 j (10.17) Stokes B ds = B dr (10.18) µ 0 S S C j ds = µ 0 J = C B dr (10.19) J Ampere Gauss Ampere z J z r Ampere ds A ds A ds dr C
54 r µ 0 J = C dr = re θ dθ (10.20) 2π B dr = r B e θ dθ = r2πb θ (10.21) 0 B = µ 0J 2πr e θ (10.22) Biot-Savart x y y K Ampere x y l Stokes B dr = 2Bl = µ 0 j ds = µ 0 Kl (10.23) C B = µ 0K 2 S z z e x (10.24) 10.3 v Lorentz Lorentz F = ev B J Jdr = ev n e < v >= nevd 3 r = jd 3 r = Jdr (10.25)
55 46 J df = Jdr B z J 1 x d a J 2 x z J 1 x z B 1 r = (x, 0, z) B 1 = µ 0J 1 2π x e y (10.26) F = J 2 dr B 1 (10.27) r = (d + a cos θ)e x + a sin θe y (10.28) dr = a sin θdθe x + a cos θdθe y (10.29) µ 0 J 1 F = J 2 ( a sin θdθe x + a cos θdθe y ) 2π(d + a cos θ) e y (10.30) F = J 2J 1 µ 0 a ( sin θ 2π d + a cos θ e z + cos θ ) d + a cos θ e x dθ (10.31)
56 (1) 0 < θ < π (2) π < θ < 2π F = J 2 J 1 µ 0 ln ( 1 F = J 2J 1 µ 0 2π ln ( ) d + a e z d a ) d e d2 a 2 x + J ( ) 2J 1 µ 0 d + a ln e z 2π d a (10.32a) (10.32b) 10.4 A B = 0 B = A (10.33) B = 0 1 B 3 A 3 1 Coulomb A = 0 (10.34) B = A A A = A + χ A B = A A χ
57 48 Maxwell Biot-Savart Ampere Biot-Savart Ampere ( A) = µ 0 j (10.35) ( A) = ( A) ( )A (Coulomb ) A = 0 2 A = µ 0 j (10.36) A(r) = µ 0 4π j(r ) r r d3 r (10.37) B B = A B = A = µ 0J 4π dr (r r ) r r 3 (10.38) Biot-Savart Jdr = j(r )d 3 r
58 U = 1 j Ad 3 r (11.1) 2 H H = eṙ A eṙ = jd 3 r H = eṙ A j Ad 3 r Ampere B = µj U = 1 2µ 0 B Ad 3 r (11.2) U = 1 B 2 d 3 r (11.3) 2µ
59 j A A(r) = µ 0J 4π dr r r (11.4) a J x y ( r >> a) 1 r r = 1 r + r r r 3 + (11.5) r = a cos θe x + a sin θe y, dr = a sin θdθe x + a cos θdθe y (11.6) A(r) = µ 0J 4πr 3 [ a 2 yπe x + a 2 xπe y ] (11.7) m = Jπa 2 e z (11.8) m r = Jπa 2 (xe y ye x ) (11.9) A(r) = µ 0 m r (11.10) 4π r 3 B = A B = A = µ [ ] 0 m 3r(m r) (11.11a) 4π r3 r 5 B = µ ( ) 0 m r 4π (11.11b) r 3 φ m φ m = µ ( ) 0 m r (11.12) 4π r 3
60 A(r) = µ 0 m r 4π r 3 n A(r) = µ 0 n(r )m (r r ) d 3 r (11.13) 4π r r 3 M = nm A(r) = µ 0 4π M(r ) 1 r r d3 r (11.14) A(r) = µ 0 M(r ) d 3 r (11.15) 4π r r j M = M(r) (11.16) A(r) = µ 0 jm (r ) 4π r r d3 r (11.17) Ampere B = µ 0 (j + j M ) (11.18) j M = M(r) ( ) B M = j (11.19) µ 0
61 52 H H B µ 0 M (11.20) Ampere H = j (11.21) B = µ 0 (H + M) = µ 0 (1 + χ m )H = µh (11.22) µ χ m χ m 1. χ m > 0 : (paramagnetic) B 2. χ m < 0 : (diamagnetic) B 3. B H B H E D B H (11.20) m m = 1 2 J r dr (11.23) Jdr = jd 3 r = ev (11.24) m = 1 2 r ev = e 2m r p = e 2m L (11.25)
62 µ = e (L + 2s) (11.26) 2m m B U I = m B (11.27) B j U I U I = j A d 3 r (11.28) Ampere B = µ 0 j j B A U I U I = 1 ( B) A d 3 r (11.29) µ 0 (A B) = ( A) B ( B) A U I U I = 1 ( A) B d 3 r + 1 µ 0 µ 0 (A B) d 3 r (11.30), U I U I = 1 ( A) B d 3 r (11.31) µ 0
63 54 A m A = µ 0 m r A 4π r 3 A = µ [ ] 0 m 3r(m r) 4πδ(r)m (11.32) 4π r3 r 5 U I δ U I = m B m 1 m 2 m 1 m 2 r m 2 B B = µ [ 0 m2 4π r 3r(m ] 2 r) 3 r 5 (11.33) U I = m 1 B U I = µ [ 0 (m1 m 2 ) 3(m ] 1 r)(m 2 r) 4π r 3 r 5 (11.34) p 1 p 2 r U I = 1 [ (p1 p 2 ) 3(p ] 1 r)(p 2 r) 4πε 0 r 3 r 5 (11.35)
64 Faraday Faraday E = B t (Faraday ) (1.1c) S Stokes E ds = E dr = V (12.1) S C V C B S t ds = d B ds dt S Faraday V = d dt S B ds (12.2a) Φ Φ = B ds (12.3) S
65 56 Φ Faraday V = dφ dt (12.2b) 12.2 Faraday B = A E = ( A) (12.4) t ( E + A t ) = 0 (12.5) φ E = A t φ (12.6) Faraday 12.3 m v m r 0 v r 0 = vt r A(r) = µ 0 m (r vt) 4π r vt (12.7)
66 E = A t E(r) = µ [ 0 m v 4π r 3 3m r(r v) r 5 ] (12.8) r vt r 12.4 J Φ Φ = LJ (12.9) L Biot- Savart B = µ 0J dr (r r ) 4π r r 3 J L C U = 1 2 V j A d 3 r = 1 2 J C A dr = 1 2 J S A ds = 1 2 J S B ds (12.10) U = 1 2 JΦ = 1 2 LJ 2 (12.11) (1) a J x y
67 58 r = 0 B = µ 0J r dr (12.12) 4π r 3 r = ae r, dr = adθe θ, r dr = a 2 dθe z B = µ 0J 4π 2πa 2 Φ Φ = S a 3 e z = µ 0J 2a e z (12.13) B ds µ 0J 2a πa2 = µ 0πa 2 J (12.14) L L µ 0πa 2 (12.15) (2) a z J z b ( a < b) z J z l a < r < b Stokes Ampere B θ = µ 0J a < r < b (12.16) 2πr U = 1 2µ 0 B 2 d 3 r = 1 b 2π l B θ 2 rdr dθ dz = µ 0J 2 ( ) l b 2µ 0 a 0 0 4π ln a (12.17)
68 U = 1 2 LJ 2 L L = µ ( ) 0l b 2π ln a (12.18)
69
70 Maxwell Maxwell E = ρ ε 0 (Gauss ) (1.1a) B = 0 E = B t ( ) (1.1b) (Faraday ) (1.1c) B = µ 0 j + 1 E (Ampere Maxwell ) (1.1d) c 2 t Ampere Maxwell Ampere B = µ 0 j B = µ 0 j = 0 (13.1) ρ t + j = 0 (13.2) Ampere Maxwell (1.1d) B = µ 0 j + 1 c 2 E t = µ 0 j + 1 c 2 ɛ 0 ρ t = 0 (13.3) c = 1 ɛ0 µ 0
71 j d Ampere-Maxwell E j d = ε 0 (13.4) t 13.2 Newton ( ) d 1 dt 2 mṙ2 = F ṙ (13.5) W 0 W 0 = F ṙ (13.6) W 0 = F ṙ = eṙ (E + ṙ B) = eṙ E (13.7) N ρ W 0 W 0 = ρṙ Ed 3 r = j Ed 3 r (13.8) Ampere-Maxwell W 0 = 1 ( B E (E B) 1 ) E µ 0 c 2 t E d 3 r (13.9) Faraday Poynting S = 1 µ 0 E B (13.10) W 0 [ ( ) 1 W 0 = B 2 + ( ) ] ε0 t 2µ 0 t 2 E 2 + S d 3 r (13.11)
72 Sd 3 r = S n ds (13.12) V S S : RC RC C R V a d E C E = V d, C = ε 0πa 2 d (13.13) J V = RJ + Q C = RdQ dt + Q C (13.14) t = 0 Q = 0 Q = CV ( ) 1 e t RC (13.15) J J = dq dt = V t R e RC (13.16) z E = σ ε 0 e z = Q ε 0 πa 2 e z = V C ( ) t 1 e ε 0 πa 2 RC ez (13.17) j d = ε 0 E t = V Rπa 2 e t RC ez (13.18) Ampere- Maxwell r B dr = µ 0 i d πr 2 (13.19) C
73 64 B = µ 0i d r 2 e θ = µ 0r t 2πa 2 R e RC eθ (13.20) r = a Poynting S S = 1 E B = 1 V C ( ) t 1 e µ 0 µ 0 ε 0 πa 2 RC ez µ 0r t 2πa 2 R e RC eθ (13.21) S = V 2 ( ) t 2πaRd e RC 1 e t RC er (13.22) r 0 S n dt = CV 2 4πad (13.23) Poynting E tot = S n ds = CV 2 4πad 2πad = 1 2 CV 2 (13.24) : LC LC C L V a d J V = L dj dt + Q C = Q Ld2 dt + Q 2 C (13.25) t = 0 Q = 0, J = 0 Q = CV (1 cos ωt), ω = 1 LC (13.26) J J = dq dt = V Cω sin ωt (13.27)
74 E = V C ε 0 πa 2 (1 cos ωt) e z (13.28) j d = ε 0 E t = ωv C πa 2 sin ωt e z (13.29) B = µ 0rωV C 2πa 2 sin ωt e θ (13.30) r = a Poynting S S = 1 µ 0 E B = 1 µ 0 V C ε 0 πa (1 cos ωt) e 2 z µ 0ωV C sin ωt e θ (13.31) 2πa S = ωcv 2 2πad (1 cos ωt) sin ωt e r (13.32)
75 Maxwell ρ = 0 j = 0 Ampere-Maxwell B = 1 E (13.33) c 2 t ( A) = 1 ( ) A c 2 t t + φ (13.34) Gauss 2 φ = 0 φ = 0 (13.35) A = 0 (13.34) ( ) A = 0 (13.36) c 2 t 2 A = k,λ ɛ(k, λ) 2ωk V ( c k,λ e ikx + c k,λ e ikx) (13.37) kx = ω k t k r (13.38) ω k = c k (13.39) ɛ(k, λ) A = 0 k ɛ(k, λ) = 0 (13.40)
76 (13.37) A k (13.37) c k,λ, c k,λ c k,λ 1 (13.37) 1 k, λ A 0 = ɛ(k, λ) 2ωk V e ikx (13.41) k 13.4 LC A e H I = j Ad 3 r (13.42) W dh I dt = [ j t A + j A t ] d 3 r (13.43) φ E = A t (13.44) W j W = t Ad3 r + j Ed 3 r (13.45)
77 68 (13.8) W 0 W 1 W 1 = j t Ad3 r (13.46) j t Zeeman Hamiltonian H = e 2m σ B 0 (13.47) m B 0 r B 0 z B 0 = B 0 e z j j = e m ψ ˆpψ ˆp = i j t = e [ ψ m t ˆpψ + ψ ˆp ψ ] = e t 2m B 0(r) (13.48) 2 e W 1 = 2m ( B 0(r)) Ad 3 r (13.49) 2 A B 0 A B 0 B 0 B 0 A A B 0 B 0
78 p 1 1s 1 2p s 1 2 A(x) = k 2 λ=1 1 2V ωk ɛ(k, λ) [ c k,λ e ikx + c k,λ eikx] (13.50) ω k = k ɛ(k, λ) ɛ(k, λ) k = 0, ɛ(k, λ) ɛ(k, λ ) = δ λ,λ (13.51) c k,λ c k,λ [c k,λ, c k,λ ] = δ k,k δ λ,λ (13.52) c k,λ c k,λ Fock c k,λ 0 = 0 (13.53) 0 c k,λ c k,λ 0 = k, λ (13.54) k λ ɛ µ (k, λ) ɛ µ (k, λ) ɛ µ (k, λ)
79 70 A µ Lagrange µ F µν = µ µ A ν ν µ A µ = 0 (13.55) A µ (x) = k 2 λ=1 1 2V ωk ɛ µ (k, λ) [ c k,λ e ikx + c k,λ eikx] (13.55) k 2 ɛ µ (k ν ɛ ν )k µ = 0 (13.56) 3 {k 2 g µν k µ k ν }ɛ ν = 0 (13.57) ν=0 ɛ µ (k, λ) det{k 2 g µν k µ k ν } = 0 (13.58) k 2 = 0 det{ k µ k ν } = 0 k 2 = 0 (13.56) k µ ɛ µ = 0 (13.59) Lorentz Lorentz Coulomb k ɛ = 0 ɛ 0 = 0 ɛ µ (k, λ) (13.51) Dirac Dirac (det{α k+mβ E} = 0) (E = ± k 2 + m 2 )
80 Dirac Dirac A µ ɛ µ (k, λ) Dirac ( ) (Lorentz ) ɛ(k, λ) ɛ(k, λ) ɛ(k, λ ) = δ λ,λ E(r, t) B(r, t) A(r, t) k, λ A 0 = ɛ(k, λ) 2ωk V e ikx (13.60)
81 72 ɛ(k, λ) ɛ(k, λ) k = 0, ɛ(k, λ) ɛ(k, λ ) = δ λ,λ z ɛ(k, 1) = e x, ɛ(k, 2) = e y (13.61) x x 1 0 x (1) (13.61) ɛ(k, 1) = 1 2 (e x + e y ), ɛ(k, 2) = 1 2 (e x e y ) (13.62) Compton
82 (2)
83 Lorentz S(t, x, y, z) S (t, x, y, z ) Lorentz x = x, y = y, z = γ(z + vt), t = γ (t + v ) c z (13.51) 2 v S z c γ γ = 1 1 ( v c )2 (13.52) p x = p x, p y = p y, p z = γ (p z + v ) c E, E = γ (E + vp 2 z ) (13.53) E 2 p 2 c 2 = E 2 p 2 c 2 (13.54) E = p 2 c 2 + (mc 2 ) 2 (13.55)
84 Maxwell Lorentz Maxwell Lorentz Maxwell ( ) A = 0 (13.56) c 2 t c 2 2 t 2 = 2 1 c 2 2 t 2 (13.57) Lorentz
85
86 Maxwell Maxwell Hamiltonian Hamiltonian H = 1 2m (p ea)2 Ze2 r (14.1) Hamiltonian Dirac H = 1 2m (σ (p ea))2 Ze2 r (14.2) (σ (p ea)) 2 = (p ea) 2 + ieσ p A (14.3)
87 78 Hamiltonian H = 1 2m (p ea)2 e 2m σ B Ze2 r (14.4) B Zeeman Hamiltonian Hamiltonian e H = 1 2m p2 Ze2 e (L + σ) B (14.5) r 2m L 14.2 Zeeman Lamour Hamiltonian H H = e (L + σ) B (14.6) 2m Hamilton ṙ = H p, ṗ = H r (14.7) m r = eṙ B (14.8) z B x ( ) eb 2 ẍ + x = C m C x x 0 = A sin ωt + B cos ωt
88 14.2. Zeeman Lamour 79 y y 0 = A cos ωt B sin ωt ω = eb m x 0, y 0, A, B T T = 2π ω = 2πm eb ω Lamour Frequency
89
90 81 A A.1 e e A.1.1 e Lagrangian Q A e
91 82 A W ± W ± ± e g W ± O(g 2 e 2 ) A.1.3 Yes
92 A.2. : 83 A.2 : A.2.1 A.2.2 A.2.3
93 84 A
94 85 B Lagrange Lagrange Lagrange B.1 Lagrange (x, y, x) (r, θ, ϕ) Newton m r = U(r) Lagrange q i = (q 1, q 2, q 3 ) Lagrangian L(q i, q i ) Lagrange Lagrangian d L = L (i = 1, 3) dt q i q i L = 3 i=1 1 2 mq i 2 U(q 1, q 2, q 3 ) Newton B.1.1 Newton
95 86 B Lagrange (x, y, z) (q 1, q 2, q 3 ) x = x(q 1, q 2, q 3 ), y = y(q 1, q 2, q 3 ), z = z(q 1, q 2, q 3 ) ẋ = x q 1 q 1 + x q 2 q 2 + x q 3 q 3 ẏ = y q 1 q 1 + y q 2 q 2 + y q 3 q 3 ż = z q 1 q 1 + z q 2 q 2 + z q 3 q 3 x = ẋ, q i q i y = ẏ, q i q i z = ż, (i = 1, 2, 3) q i q i V = V (x, y, z) I I V ( V x = + V y + V ) z q i x q i y q i z q i Newton mẍ = V x, mÿ = V y, m z = V I ( z I = m ẍ x + ÿ y + z z ) q i q i q i { ( d I = m ẋ x ) ẋ ẋ + d ( ẏ y ) ẏ ẏ + d ( ż z ) ż ż } dt q i q i dt q i q i dt q i q i x = ẋ q i q i { ( d I = m ẋ ẋ ) ẋ ẋ + d ( ẏ ẏ ) ẏ ẏ + d ( ż ż ) ż ż } dt q i q i dt q i q i dt q i q i T = 1 2 m(ẋ2 + ẏ 2 + ż 2 ) I I = d dt ( ) T q i T q i = V q i V V = 0 Lagrangian L = T V q i ( ) d L = L (i = 1, 2, 3) dt q i q i
96 B.1. Lagrange 87 Lagrange Newton Lagrange B.1.2 Lagrange Lagrange S = tb t a L(q i, q i )dt Euler δs = 0 δs δs = tb t a ( 3 L δq i + L ) 3 δq i dt = i=1 q i q i i=1 tb t a ( L d ) L δq i dt = 0 q i dt δq i (t a ) = δq i (t b ) = 0 δq i L d L = 0 (i = 1, 3) q i dt q i Euler Lagrange Lagrange q i
97 88 B Lagrange B.2 Maxwell Lagrangian Lagrangian Lagrangian E B.2.1 Lagrangian L = 1 2 mṙ2 Lagrangian Lagrangian L = 1 2 mṙ2 + e (ṙ A φ) Lagrangian A = A + χ, L = 1 2 mṙ2 + e (ṙ A φ ) e dχ dt φ = φ χ t Lagrangian e dχ Lagrange dt Lagrangian B.2.2 Lagrangian Lagrangian Lagrange m r = ee + eṙ B Coulomb Lagrgange x
98 B d L dt ẋ = L x L ẋ = mẋ + ea x [ d L dt ẋ = mẍ + e Ax x ẋ + A x y ẏ + A x z ż + A ] x t [ L x = e Ax x ẋ + A y x ẏ + A z x ż A ] 0 x Lagrangian e Maxwell Lagrangian Lagrangian Coulomb
99 90 B Lagrange B.3 Lagrange Schrödinger ψ(t, r) Lagrangian B.3.1 Lagrangian Schrödinger Lagrangian L = iψ ψ t 1 ψ ψ ψ V ψ 2m x k x k h = 1 c = 1 V k k = 1, 2, 3 1 ψ ψ = 2m x k x k 3 k=1 1 ψ ψ = 1 2m x k x k 2m ψ ψ Lagrange µ L ( µ ψ) L t ψ + L x k ( ψ x k ) = L ψ Bjorken-Drell µ µ = 0, 1, 2, 3 µ ( x 0, x 1, x 2, x 3 ) = ( t, x, y, ) ( ) = z t, x µ = (t, r), x µ = (t, r), p µ = (E, p), p µ = (E, p)
100 B.3. Lagrange 91 A µ B µ A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 x µ x µ = t 2 r 2, p µ p µ = E 2 p 2, x µ p µ = te r p Bjorken-Drell B.3.2 Lagrange Lagrange S = L(ψ, µ ψ)d 4 x Lagrange δs = [ L ψ δψ + L ] [ ( µ ψ) δ( µψ) d 4 L x = ψ µ ( )] L δψd 4 x = 0 ( µ ψ) ( ) L L ψ = µ ( µ ψ) ψ Lagrange ψ Lagrange B.3.3 Schrödinger Schrödinger ) h2 ( i h + t 2m 2 V (r) ψ(t, r) = 0 Schrödinger h h = 1 c = 1
101 92 B Lagrange B.3.4 Hamiltonian Lagrangian Hamiltonian Hamiltonian Lagrangian T µν T µν L ( µ ψ) ν ψ + L ( µ ψ ) ν ψ Lg µν µ T µν = 0 T µν Hamiltonian H T 00 = L ( 0 ψ) 0ψ + L ( 0 ψ ) 0ψ L Schrödinger Lagrangian Hamiltonian H Hd 3 r = [ ] 1 2m ψ ψ + ψ V ψ d 3 r H = [ 1 ] 2m ψ 2 ψ + ψ V ψ d 3 r Schrödinger Hamiltonian Hamiltonian Hamiltonian Hamiltonian H 0 = h2 2m 2 + V (r)
102 B.3. Lagrange 93 Hamiltonian Hamiltonian Hamiltonian Hamiltonian H 0 ψ Schrödinger Hamiltonian
103 94 B Lagrange B.4 QED+ Lagrangian m Dirac Lagrangian L = ψ(i µ γ µ m)ψ = ψ i [γ 0 (i µ γ µ m)] ij ψ j ψ 1 (r, t) ψ ψ(r, t) = 2 (r, t) ψ 3 (r, t) ψ 4 (r, t) ψ (r, t) = ( ψ 1(r, t) ψ 2(r, t) ψ 3(r, t) ψ 4(r, t) ) ψ ψ ψ γ 0 m Dirac (i t ) + i α mβ ψ(r, t) = 0 B.4.1 Lagrangian Dirac B.4.2 Lagrangian Lagrangian
104 B.4. QED+ Lagrangian 95 Lagrangian [1, 2] m ψ A µ G Lagrangian L = i ψγ µ µ ψ e ψγ µ A µ ψ m(1 + gg) ψψ 1 4 F µνf vµν µg µ G G Lagrangian log
105
106 97 C Kepler Kepler Kepler GPS C.1 1 V (x) x x 0 V (x) x = x 0 V (x) = V (x 0 ) V (x 0 )(x x 0 ) 2 +
107 98 C Kepler V (x) F (x) = x = V (x 0 )(x x 0 ) x x 0 x k V (x 0 ) F F = kx Newton mẍ = kx k ω m ẍ + ω 2 x = 0 x = A sin ωt + B cos ωt A B t = 0 x = 0 ẋ = v 0 x = v 0 ω sin ωt sin cos T T = 2π ω
108 C.2. Kepler 99 C.2 Kepler Kepler Newton m r = G Mmr r 3 ṙ mṙ r = G Mmṙ r r 3 m dṙ 2 2 dt = d GMm dt r E E = 1 2 mṙ2 GMm r L L = r mṙ L dl dt = ṙ mṙ + r m r = r GMmr r 3 = 0 a a = 0 Newton L L z x y r r L = r r mṙ = mr r ṙ = 0 L x y L z l = L 2 l = L = L z = mr 2 ϕ
109 100 C Kepler 2 x = r cos ϕ, y = r sin ϕ ẋ = ṙ cos ϕ r ϕ sin ϕ, ẏ = ṙ sin ϕ + r ϕ cos ϕ E E = 1 2 m ( ṙ 2 + r 2 ϕ 2) GMm r l = mr 2 ϕ E = 1 ( ) 2 m ṙ 2 + l2 m 2 r 2 GMm r r E C.2.1 Kepler ( ) Kepler E l (r, ϕ) l = mr 2 ϕ E = 1 ( ) 2 m ṙ 2 + l2 GMm m 2 r 2 r dr ṙ = ϕ = dϕ = ṙ ϕ ( dr dϕ = r2 2m E α ) l r l2 2mr 2 α = GMm r = 1 s dϕ = lds 2m ( ) = E αs l2 s 2 2m ( 2mE l 2 ds 2mα l 2 s s 2 )
110 C.2. Kepler 101 s dϕ = ds (s s 1 )(s 2 s) s 1 s 2 s 1 = mα (mα ) 2 l 2mE + 2 l 2 l 2 s 2 = mα (mα ) 2 l + 2mE + 2 l 2 l 2 s = s 1 + (s 2 s 1 ) sin 2 θ ϕ 0 ϕ = 2θ θ s s r A r = 1 + ε cos ϕ ϕ 0 ϕ 0 = π A ε A = l2 mα, ε = 1 + 2El2 mα 2 E ε < 1 Kepler Kepler S = 1 2 r2 ϕ ds dt = 1 2 r2 ϕ l = mr 2 ϕ ds dt = l 2m Kepler
111 102 C Kepler Kepler a b a = A 1 ε, b = A 2 1 ε 2 S S = πab T S = l 2m T S = πab = πa2 (1 ε 2 ) 3 2 = π Aa 3 2 = l 2m T T 2 a 3 Kepler ϕ = l T mr 2 l T 2π 2π dt = r 2 dϕ = A 2 1 m (1 + ε cos ϕ) 2 dϕ ε l m T = 2πA2 (1 ε 2 ) 3 2
112 C C.3 [1, 2] V (r) = GmM + 1 ( ) GmM 2 r 2mc 2 r Newton m r = GmM + l2 r 2 mr + G2 M 2 m 3 c 2 r 3 L L 2 l 2 + G2 M 2 m 2 ω R ω l mr 2, R ab = c 2 l 2 GMm 2 (1 ε 2 ) 3 4 a, b ε ω Ω 2 ω 2 + G2 M 2 ( = ω G2 M 2 ) ω 2 (1 + η) c 2 R 4 c 2 R 4 ω 2 η η = G2 M 2 c 2 R 4 ω 2 r = A ε A = L2 GMm 2, ε = A 1 + ε cos ( L l ϕ) 1 + 2L2 E m(gmm) 2
113 104 C Kepler ϕ = l T mr 2 l T 2π 2π dt = r 2 dϕ = A 2 1 ( ( m ε cos L l ϕ)) 2 dϕ ωt = 2π(1 + 2η) (1 εη) 2π{1 + (2 ε)η} ( ) ε ( ) T (2 ε)η T th ( ) ω (*) (2 ε)η ω th ω C.3.1 ω θ 42 per 100 year 0.24 δθ δθ ω ω η = G2 M c 2 R ω2
114 C R = m, M = kg δθ th ( ) ω δθ th ω th C.3.2 GPS GPS GPS GPS GPS η = G2 M c 2 R ω2 R = m, M = kg, ω = GPS ( ) T T th GPS GPS [3] ( T ) T
115 106 C Kepler ) T GR ( ) ( T GPS GPS θ = 2π(2 ε)η GPS l GP S (one year) = l m l GP S (one year) = θ R m GPS Newton GPS C.3.3 R M ω R = m, M = kg, ω = T T = (2 ε)η T Orbital Motion = s/year ( )
116 C T Obs Orbital Motion ± s/year (**) Newcomb [4] Newtonian
117
118 109 D, Z = n e βe n, β = 1 kt E n n < E >= n E n e βe n Z = β log Z E n D.1 ( ) D.1.1 W (E t ) E t
119 110 D E p(e) p(e) = W (E t E) W (E t ) = exp [ln(w (E t E) ln W (E t )] exp p(e) W (E) E t E t E E E t D.1.2 E W (E) W (E t E) W (E t ) W (E) E W (E) W E E t p(e) exp exp [ [ ln ( W (E t ) ( ) ln W (E) E ( ) W (E) E E=E t E ] E=E t E ) ln W (E t ) p(e) exp[ ae] ] a = 1 kt = ln W (E) E
120 D D.1.3, T ds = de + pdv ds = 1 T de + p T S E = 1 T S S dv = de + E V dv ln W (E) E = 1 kt S = k ln W 1 a kt Log, S S(E t ) = k ln W (E t ) E E t S S S
121 112 D D.2 D.2.1 Planck
122 D D.2.2 Dirac D He 4 He
123 114 D D.3 e βe Z = N dxdpe βh H Hamiltonian N Hamiltonian Hamilton Newton H D.3.1 Z = n e βen = e β hω(n+ 1 2 ) = n=0 1 2 sinh( β hω 2 ) h 0 D.3.2
124 115 E E.1 E.1.1 f(x, y) f(x, y) x f(x, y) y lim x 0 lim y 0 f(x + x, y) f(x, y) x f(x, y + y) f(x, y) y (y x ) (x y ) E.1.2 f (x(t), y(t), t) df (x(t), y(t), t) dt lim t 0 f (x(t + t), y(t + t), t + t) f (x(t), y(t), t) t
125 116 E df (x(t), y(t), t) dt = lim t 0 { f (x(t + t), y(t + t), t + t) f (x(t), y(t + t), t + t) x f (x(t), y(t + t), t + t) f (x(t), y(t), t + t) y + y t } f (x(t), y(t), t + t) f (x(t), y(t), t) + t x x(t + t) x(t), x t y y(t + t) y(t) df (x(t), y(t), t) dt = f dx x dt + f dy y dt + f t
126 E E.2 E.2.1 ( ) (x, y, z) E.2.2 r x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dxdydz = 0 π r 2 dr 0 2π sin θdθ ( 3 ) 0 dφ : dω = sin θdθdφ E.2.3 x = r cos φ y = r sin φ z = z dxdydz = 0 2π rdr dθ dz 0
127 118 E E.3 (1 + x) α = 1 + αx α(α 1)x2 + e x = 1 + x x2 + sin x = x + cos x = x2 + ln(1 + x) = x + E π e ax dx = 1 a x n e ax dx = ( 1) n n a n 1 a = n! π e ax2 dx = a x 2n+1 e ax2 dx = 1 2 n! a n+1 x 2n e ax2 dx = ( 1) n n 1 a n 2 f(x)dx = 1 a 2 π 2 (x 2 + a 2 ) 3 2 dx = 2 (x 2 + a 2 ) 3 2 a 2 sin 2 θdθ = π 0 dt a + bt = 2 b π 2 a n+1 π a = (2n 1)!! π 2 n+1 a n+ 1 2 f(a tan θ) cos θdθ (x = a tan θ) cos 2 θdθ = π 2 ( ) a + b a b
128 E E.4.1 δ δ(x)dx = 1 f(x)δ(x a)dx = f(a) δ(ax) = 1 a δ(x) δ(x 2 a 2 ) = 1 [δ(x a) + δ(x + a)] 2 a δ(r) δ(x)δ(y)δ(z) E.4.2 a (b c) = (a b) c ( A) = ( ) A = 0 a (b c) = (a c)b (a b)c ( A) = ( A) ( )A (A B) = B ( A) A ( B) (σ A)(σ B) = A B + iσ A B σ : Pauli matrices a b = a b cos θ a b = a b sin θ
129 120 E E = e x x + e y y + e z z = e r r + e 1 θ r θ + e 1 ϕ r sin θ ϕ 2 = 1 r 2 r r2 r + 1 [ 1 r 2 sin θ θ sin θ θ ] sin 2 θ ϕ 2 1 r r = r r r r r r = 4πδ(r r ) E.4.4 a b = a x b x + a y b y + a z b z E div E = E x x + E y y + E z z E.4.5 a b = (a y b z a z b y )e x + (a z b x a x b z )e y + (a x b y a y b x )e z B rot B = ( Bz y B ) ( y Bx e x + z z B ) ( z By e y + x x B ) x e z y
130 E E.5 E.5.1 (x, y, z) Gradient: A 0 = A 0 x e x + A 0 y e y + A 0 z e z Laplacian: 2 A 0 A 0 = 2 A 0 x + 2 A 0 2 y + 2 A 0 2 z 2 A = A x e x + A y e y + A z e z Divergence: diva A = A x x + A y y + A z z Rotation: rota A = ( Az y A ) ( y Ax e x + z z A ) ( z Ay e y + x x A ) x e z y E.5.2 (r, ϕ, z) Gradient: A 0 = A 0 r e r + 1 A 0 r ϕ e ϕ + A 0 z e z Laplacian: 2 A 0 = 1 r ( r A ) A 0 r r r 2 ϕ + 2 A 0 2 z 2
131 122 E A = A r e r + A ϕ e ϕ + A z e z A r = A x cos ϕ + A y sin ϕ, A ϕ = A x sin ϕ + A y cos ϕ Divergence: A = 1 r r (ra r) + 1 A ϕ r ϕ + A z z Rotation: A = ( 1 A z r ϕ A ) ( ϕ Ar e r + z z A ) ( z 1 e ϕ + r r r (ra ϕ) 1 r ) A r e z ϕ E.5.3 (r, θ, ϕ) Gradient: A 0 = A 0 r e r + 1 A 0 r θ e θ + 1 A 0 r sin θ ϕ e ϕ Laplacian: 2 A 0 = 1 r 2 r ( r 2 A 0 r ) + 1 r 2 sin θ ( sin θ A ) 0 + θ θ 1 2 A 0 r 2 sin 2 θ ϕ 2 A = A r e r + A θ e θ + A ϕ e ϕ A r = A x sin θ cos ϕ + A y sin θ sin ϕ + A z cos θ A θ = A x cos θ cos ϕ + A y cos θ sin ϕ A z sin θ A ϕ = A x sin ϕ + A y cos ϕ
132 E Divergence: A = 1 r 2 r (r2 A r ) + 1 r sin θ θ (sin θa θ) + 1 A ϕ r sin θ ϕ Rotation: A = 1 r sin θ ( θ (sin θa ϕ) A ) θ e r + 1 ϕ r ( r (ra θ) A ) r e ϕ θ + 1 r ( 1 A r sin θ ϕ ) r (ra ϕ) e θ E.5.4 e r = sin θ cos ϕ e x + sin θ sin ϕ e y + cos θ e z e θ = cos θ cos ϕ e x + cos θ sin ϕ e y sin θ e z e ϕ = sin ϕ e x + cos ϕ e y e x = sin θ cos ϕ e r + cos θ cos ϕ e θ sin ϕ e ϕ e y = sin θ sin ϕ e r + cos θ sin ϕ e θ + cos ϕ e ϕ e z = cos θ e r sin θ e θ
133 124 E E.6 z = x + iy = r(cos θ + i sin θ) = re iθ E.6.1 Euler e iθ = cos θ + i sin θ cos θ = 1 2 (eiθ + e iθ ) sin θ = 1 2i (eiθ e iθ ) E.6.2 f(z) f(z) = n= c n z n n f(z) = z R θ = 0 C f(z) C C f(z)dz = c n n= C z n dz z = Re iθ dz = ire iθ dθ C f(z)dz = 2π c n n= 0 { 0 n 0 ir n+1 e i(n+1)θ dθ = 2πi c 1 n = 1 f(z) n = 1 z = 0 c 1
134 E e ipx dx : (p > 0, a > 0) x 2 + a2 C e ipz dz z 2 + a R ( C) 2 C e ipz z 2 + a 2 dz = R R e ipx π ipr cos θ pr sin θ x 2 + a dx + e ire iθ dθ 2 0 R 2 e 2iθ + a 2 R θ 0 < θ < π sin θ e ipr cos θ pr sin θ R C e ipz dz = 2πie pa z 2 + a2 2ai = πe pa a e ipx πe pa dx = x 2 + a2 a
135 126 E E.7 sin(x ± y) = sin x cos y ± cos x sin y cos(x ± y) = cos x cos y sin x cos y tan(x ± y) = tan x ± tan y 1 tan x tan y a sin θ + b cos θ = a 2 + b 2 sin(θ + α) = a 2 + b 2 (sin θ cos α + cos θ sin α) ( ) a cos α = a2 + b, sin α = b 2 a2 + b 2 sin A + sin B = 2 sin A+B cos A B 2 2 sin A sin B = 2 cos A+B sin A B 2 2 cos A + cos B = 2 cos A+B cos A B 2 2 cos A cos B = 2 sin A+B sin A B 2 2 sin 2θ = 2 sin θ cos θ sin 2 θ = 1 (1 cos 2θ) 2 cos 2 θ = 1 (1 + cos 2θ) 2
136 E E.8 : e x e y = e (x+y), (e x ) y = e xy, e = log xy = log x + log y, log x y = y log x, ln x log e x : de x dx = ex d ln x dx = 1 x E.9 Taylor e x = 1 + x + 1 2! x ! x n! xn +... log(1 + x) = x 1 2 x x3 1 4 x x = 1 x + x2 x 3 + x sin x = x 1 3! x ! x5 + cos x = x ! x4 + e ix = 1 + ix 1 2 x2 i 1 3! x ! x4 + i 1 5! x5 + = cos x + i sin x
137 128 E E.10 C A dr C (A x dx + A y dy + A z dz) C : a A θ = A θ (r) A dr = Aθ (r)2πa E.11 S A ds S A n ds ds : A n e r R A n = A r (r) A ds = Ar (r)4πr 2
138 129 [1] Symmetry and Its Breaking in Quantum Field Theory T. Fujita, Nova Science Publishers, 2011 (2nd edition) [2] Fundamental Problems in Quantum Field Theory T. Fujita and N. Kanda, Bentham Publishers, 2013 [3] B.W. Parkinson and J.J. Spilker, Global Positioning System, Progress in Astronautics and Aeronautics (1996) [4] Simon Newcomb, Tables of the Four Inner Planets, 2nd ed. (Washington: Bureau of Equipment, Navy Dept., 1898).
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39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
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p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
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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 :
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[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
) 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
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
量子力学Ａ
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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
Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
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. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
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ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
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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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)
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Chap11.dvi
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量子力学 問題
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24 I ( ) 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
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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),