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 + α 2 ) (ω 1 < ω 2 ) ω 2 1 ω2 2 A 1/B 1 A 2 /B 2 (5) l = l m m ω 1 ω 2 g ω 1 = 1 1 m g, ω 2 = 1 + 1 m l 2 m l 2 m
(6) t = 0 θ = 0 θ = 0 ϕ = ϕ 0 ϕ = 0 A 1 A 2 B 1 B 2 θ(t) ϕ(t) t t t = 0 4π ω 2 ω 1 θ(t) ϕ(t) ( cosα + cosβ = 2cos α+β 2 cos α β 2 α+β α β, cosα cosβ = 2sin sin ) 2 2 g θ l m ϕ l' m'
¼µ ½µ 5678# ¼ Õ'(Ñ Á¹¾ µ ¾µ./9:;<=&>?@# 0$1#)2.34# ¼ ¼ µ ¼!"#$%& ¼ NMO>PQ@R# ¼µEF%GHIJKLM # Õ)*+$),#+$-./) Ú Ú ¼ ¼ ¼µ& Ú OS.T3ÜU,5VWXY3Z$G [G µ -ABCD ¼ ¼ rs&>_`der# T5V9:R#\R]^ µ µ_àbc3g\der# µ Ú ¼ k"#\g 567lGmnopqVG µ µk )+&> fg] hi hij./ ¼ ¼ µµ ½¼¼ µ µ Ú ¼ ¼ ¼ ½
I-3 (100 ) 1/2 s ( ) ( ) ( ) 0 1 0 i 1 0 σ 1 =, σ 2 =, σ 3 =, 1 0 i 0 0 1 s = h σ 2 s [s 1, s 2 ] = i hs 3, [s 2, s 3 ] = i hs 1, [s 3, s 1 ] = i hs 2, 1/2 µ = γs z B 0 x-y ω B 1 (t) 0 B 1 cos(ωt) B 0 = 0, B 1 (t) = B 1 sin(ωt), (A) 0 B 0 B0 B1(t) = s H(t) = µ (B 0 + B 1 (t)), t = 0 z t z P (t)
1/2 ( ) ψ + (t) ψ(t) =, ψ (t) i h ψ(t) = H(t)ψ(t), t ω 0 = γb 0, ω 1 = γb 1, (1) ( (B ) 1 = 0) ψ 0+ (t) ψ 0 (t) = ψ 0 (t) (2) ( B 1 0 ) (1) ψ 0 a + (t)ψ 0+ (t) ψ(t) = a ± (t) a (t)ψ 0 (t) (3) t = 0 z a ± (t) t z P (t) (4) P (t) (A) B 1 ω
18 II ( ) (1) II-1 II-2 II-3 (2) (3) II-1 ( ) (100 ) ( p T n p = k BTn 1 bn an2 (A) a, b k B (2) T p v v =1=n v g v l v p = p cx (T ) μ f v g v l p cx df = pdv (Maxwell (3) (A) K T =(@n=@p) T =n nk B K T = (1 bn)2 T T s (n) (B) n T s (n) T = T s (n) (4) T T c v = v g v l T s (n) n
T = T s (n) n c T c a,b n n c T T c K T (5) 2 dp = L (C) dt T v L = T s s = s g s` (C) P B P cx A v l v v g
II-2 ( ) (100 ) f(t) t M α f(t) Me αt f(t) χ(ζ) χ(ζ) = 0 f(t)e iζt dt (A) (1) χ(ζ) n χ (n) (ζ) ζ χ (n) (ζ) M n! (α + ζ 2 ) n+1 (B) ζ 2 ζ N N a i a i i=1 i=1 (2) ζ χ(ω) = 1 iπ P χ(ζ) ζ ω dζ (C) ω P F (x) x 0 x 1 < x 0 x 2 > x 0 x2 [ x0 ε x2 ] P F (x)dx = lim F (x)dx + F (x)dx x ε +0 1 x 1 x 0 +ε (3) χ(ζ) χ 1 (ζ) χ 2 (ζ) χ(ζ) = χ 1 (ζ) + iχ 2 (ζ) χ 1 (ζ) χ 2 (ζ) ζ (C) χ 1 (ω) = 2 π P χ 2 (ω) = 2 π P 0 0 ζχ 2 (ζ) ζ 2 ω 2 dζ ωχ 1 (ζ) ζ 2 ω 2 dζ (D) (E)
(4) f(t) df(t) dt + af(t) = Ae bt (F) f(t) χ(ζ) a b A a > b f(t) f(0) = 0
II-3 ( ) (100 ) (1) v v = rota, divv = 0 (A) A diva = 0 ω = rotv 2 A = ω A(r) = 1 4π d 3 r ω(r ) r r (2) e z z ω = ωe z S Γ ωs S 0 xy Γ ( ω = Γδ(x)δ(y) ) (3) (x j, y j ) Γ j j (1 j N) (2) z j x j + iy j, z j = x j iy j (B) dz j dt = 1 2πi N k=1,k j Γ j z j z k (C)
(4) (3) (C) Γ j dx j dt = H y j, Γ j dy j dt = H x j ( H (5) (D) N j=1 Γ j (x j dy j dt dx j dt y j) (E)
18 III (3 ) (1) III-1 III-9 9 3 (2) (3) (4) III-1 III-2 III-3 III-4 III-5 III-6 III-7 III-8 III-9
III-1 (100 ) U(r) m (1) (2) r θ (3) U(r) =g/r g <0 u =1/r u θ u 2 (4) r θ r = a 1+b cos θ a, b (5) b<1 a/(1 b 2 ) (6) U(r) =gr n g >0 g<0 n g n (7) (6) r 0 U(r) U (eff) (r) r r 0 ω r (8) (7) ω r g ω r θ ω θ (9) (8) n = 1 (n =2) n n (A)
III-2 (100 ) m e A H = 1 ( i h + ea)2 2m x y z L x L y L z (1) A = (0, Bx, 0) z B (2) z (3) x y y x (4) m K u N (x) = C N exp ( ) ( x2 x H 2l 2 N l ε N = (N + 1/2) hω c x E x N 0 C N N ω c = K/m l = 4 h 2 /mk H N (x/l) (5) E x ( ) L x l L y l (6) χ = Bxy χ )
III-3 ( : ) (100 ) N V f(ɛ) f(ɛ) = 1 e β(ɛ µ) 1 ɛ, µ β k B T β = 1/(k B T ) (1) µ (a) µ (b) µ = 0 (2) p m ɛ = p 2 /2m N d T N /V F d/2 (α) 0 dx xd/2 1 e x+α 1 T N d S d (S 3 = 4π, S 2 = 2π ) (3) (2) d α (4) (a) T c (b) T T c n c n N/V T, T c ζ(z) n z, Γ(x) n=1 0 t x 1 e t dt, ( 1 Γ = 2) π
(5) ɛ = cp (2) (3)
III-4 (100 ) (1) u(x, y) 2 ( ) 2 x + 2 u(x, y) = 0, 2 y 2 (r, θ) (0 r <, π θ < π) x = r cos θ, y = r sin θ, (2) (1) u(r, θ) r = 0 (3) (2) r = r 0 u(r 0, θ) = f(θ), f(θ) (4) (3) f(θ) [ π, π] f(θ) = θ, ( π θ < π),
3,456789:;7<=>?!@ &Ü'!()*+,-./0 Ü(12 ܵ,+!"#$% ÁÁÁ¹ µ HI,+,<=7MNOP A- ܵBCDÔ ÜµEFÚ ÜµGDHIJ.KL XYEQK0QKHI' Z[:;70\->?J]:;7& ',& '0\^_!<= ½µ89:;7& '0\`>&abcdeT'JXfTLg 3H Ô hij> ÚQKQHIORSTUVWH» HI& >?!`J Ú ½¼¼µ ½ Ü Úµ ¼ Ú Ú Ü Ô Ü ¼ Ô Ü Ô µ ¼ ÓÒ Ø ÒØ µ µ µ Ú µ Ô Ã ÓÒ Ø ÒØ µ,opturpvmwx+rp&yz'!{ GDM}~G, <=Hklmn3,4opqErpsE,+!š <=œˆ! > Ž db,ˆ!{ }~GxN,@Oƒ +>?!@# ˆ ŠMIX>?@&Z Œ ' @!./0MÜ,! -I,3 &Ü Ü µh Ü ¼
¼@u Gn I3, Ô, n ¼µ3, A-~G ¾µGÚ ¼M~G<=HIOtP Ü ÜµM?@t<=!\ > ¼(H ¾ Ü,! + Ú ¾ µgnm~g! - @3,MH4 Ü Ô µgn¾ Ú ¾,O!Ü,3,A-Ü!{ ~G,+ µ Ü Ü ¼!Qj J3!ˆ IuJ- >!Ú ½X3H ¼H µž dbop&z'hgú ¼X3H~G \nˆ Ú ¼ ¼ µ,-rp&yz'hgn&g'ú ½ µ,+gú ½QKQ, µ Ü Ü µ Ü Ü µ Ü Ü Ú Ú Ú Ú ~G ½>O\ 4ˆ Ú ½
III-6 ( ) (100 ) (1) (a) (b) m kg N A 6.0 10 23 /mol (c) 1 e 1.6 10 19 C h 6.6 10 34 J s (2) (a) k 1 k 2 P E k 1 k 2 m (b) k 1 k 2 0 180 P P k 1 k 1 k 2 k 2 k 1 k 2 (c) k 2 P E k 1 P E (d) E ph P ph P = P ph E = E ph v E ph = vp ph E > 0 k 1
III-7 ( : ) (100 ) (1)! ρ f R (a)!; ρ; f; R [L] [M] [T] ( ) (b)!; f; ρ; R! / f A ρ B R C A; B; C (2) 2 1 40 1 (a) X μ P (X; μ) P (X; μ) = μx X! e μ ff 2 (b) 1 2 e 2 = 0.14 (c) 2 χ 2 2 χ 2 0 11 1 12 2 11 3 4 4 2 5 0 1
2 χ 2 : ffi ( )
III-8 ( ) (100 ) ( e m) b v (dw/dν) dw dν = 4πe2 3ɛ 0 c 3 ˆ r(ω) 2 ν ω = 2πν ɛ 0 c r r 2 ˆ r(ω) r τ τ = b/v ˆ r(ω) { 1 ˆ r(ω) = rdt ω τ 1 2π (B) 0 ω τ 1 (1) dw/dν ω τ 1 ω τ 1 (B) (2) v ( ) b b+db n e n p ω τ 1 b b min b max d 3 W/dνdV dt T d 3 W dνdv dt = e 6 2π 6π 2 ɛ 3 0mc 3 3mkT n en p e hν/kt g ff (C) k h g ff d 3 W/dνdV dt α ν (A) B ν = 2hν3 c 2 1 exp (hν/kt ) 1 (D) 4πα ν B ν
(3) hν kt exp( hν/kt ) hν/kt 1 L τ ν τ ν = α ν L = 1.6 10 11 ( ne ) ( np 1 m 3 ) ( L 1 m 3 1 m ) ( ) 3/2 T ( ν ) 2 (E) 1 K 1 Hz I ν ( I ν = 2kν2 T (1 e τ ν ν ) ( ) 2 T ) = 3.1 10 40 (1 e τ ν ) c 2 1 Hz 1 K (J s 1 m 2 Hz 1 Sr 1 ) (F) (4) τ ν = 1 ν ν c ν ν c ν (5) 3 10 16 m 10 4 K J s -1 m -2 Hz -1 Sr -1 10-21 10-22 10-23 10-24 10 6 10 7 10 8 Hz
III-9 ( ) (100 ) T 5.0 10 3 K λ = 5 10 7 m T = 5.0 10 3 K (continuum) c = 3.0 10 8 m/s h = 6.6 10 34 J s k = 1.4 10 23 J/K e = 1.6 10 19 C χ = χ 0 ev T = T 0 K log 10 (e χ/kt ) = 5.0 10 3 χ 0 /T 0 (1) 13.6 ev n = 2 n = 3 n = 1 10.2 ev 12.1 ev (2) λ = 5 10 7 m n = 3 (3) T = 5.0 10 3 K n = 3 10 (4) H + N (v, v + dv) dn + dn + N = 2g+ 4πm 3 ev 2 dv exp( χ + m ev 2 /2 ) (A) g N e h 3 kt g g + N e χ m e v N + N = 2g+ (2πm e kt ) 3/2 e χ/kt (B) g N e h 3 χ = 13.6 ev T = 5.0 10 3 K g + /g = 1/2 10 6
(5) 2 H 1 0.7 ev (B) H g 1 (6) n = 3 T = 5.0 10 3 K