III,..
7.1, :. j I (= ) : [Ω, Ω + dω] dw dω = sin θ dθ dφ dw j I [1/s] [1/s m 2 ] = dσ [m2 ]. dσ dω [m2 ] :., σ tot = dσ = dω dσ dω [m2 ] :.
2.4 章では非定常状態の摂動論を用いて 入射平面波 eik x 摂動 ON 入射平面波 + 散乱平面波 X k0 0 ck0 eik x における振幅 ck0 (t) を計算し, Z dw = 黄金律 ' ck0 2 t 微小立体角 2π dω ρ(²k) hk0 V ki 2 h 微分断面積を計算した. dk0 7 章では 入射平面波 + 散乱波 からなる定常状態として考え, より厳密に考察する.
7.2 Schödinge : 2 ϕ(x) = 2mɛ h 2 ϕ(x). 2 = x 2 + 2 y + 2 z ϕ(x) = X(x)Y (y)z(z) e ikxx e ikyy e ikzz = e ik x, k 2 = 2mɛ h 2. ϕ(x) = k a k eik x. : 2 2 = 1 2 1 2L2 (θ, φ) ϕ(x) = R()Θ(θ)Φ(φ) = χ() Yl m (θ, φ). L 2 Yl m (θ, φ) = l(l + 1) Y m l (θ, φ), ( 1 2 ) l(l + 1) χ() 2 2 = 2mɛ h 2 χ().
( 2 l(l + 1) 2 2 ) l(l+1) 2 1 k ( 2 2 l(l+1) ) 2 χ() = k 2 χ() 1. χ() 0 χ() l+1 l. 1 2 k 2χ() k2 χ() l χ() e ik e ik., : Bessel χ() = j l (k) l 1 sin ( k l 2 π) n l (k) l 1 1 cos ( k l 2 π) ( 0) ( ) Yl m : ϕ(x)= (A lm j l (k) + B lm n l (k)) Yl m (θ, φ) l,m ( A e ik lm + B e ik ) lm Yl m (θ, φ). ( 1 k ) l,m
Bessel R() = χ() = j l (k) l 1 sin ( k l 2 π) n l (k) l 1 1 cos ( k l 2 π) ( ) 2 l(l + 1) 2 2 χ() = k 2 χ(). (ρ k) j 0 (ρ) = sin ρ ρ, j sin ρ ρ cos ρ 1(ρ) = ρ 2, n 0 (ρ) = cos ρ ρ, n cos ρ ρ sin ρ 1(ρ) = ρ 2,
7.3,. : z e ikz : ( ) e ik l,m A lm Y l m (θ, φ) f(θ, φ)eik z φ f(θ) eik ϕ(x) e ikz + f(θ) eik
ϕ(x) e ikz + f(θ) eik z j I = h ( (e ikz ) ) 2im z (eikz ) = hk m. (θ, φ) j = h ( ) ( ) f(θ) eik f(θ) eik 2im = hk f(θ) 2 m 2. = ds = 2 dω dw = j ds = hk m f(θ) 2 dω. f(θ) : dσ = dw σ tot = j I = f(θ) 2 dω, f(θ) 2 dω.
7.4 j j = h 2im ( ϕ ϕ ϕ ϕ ) = h m Im ( ϕ ϕ ) = e x x + e y y + e z z 1 = e + e θ ( ) + e 1 φ ( ) ϕ(x) e ikz + f(θ) eik ϕ j e z ik e ikz + e ik eik f(θ) +, hk m e z + hk f(θ) 2 m 2 e + hk ( f(θ) m (e z + e ) Re = j I + j + j int. ) eik e ikz +
0 0 = ds j = ds j I + ds j I + ds j + ds j int 0 + σ tot + Re = 2π Re 1 1 2 dω e (e z + e ) f(θ) eik( z). dτ (τ + 1)f(τ)eik(1 τ) cos θ { τ < 1 0 1 2π Re (1 + 1)f(τ = 1) dτ e ik(1 τ)} = 4π Re ( f(0) 1 ) ik = 4π k Imf(0). σ tot = 4π k Im f(0) =
7.5 Hamiltonian H = h2 2m 2 +V () = h2 2m (L 2, L z ) : ( 1 2 L2 2 2 [H, L 2 ] = 0 = [H, L z ] ) +V (), (l, m). (R() χ()/) ( 2 l(l + 1) 2 2 2m h 2 V () ) χ() = k 2 χ(). }{{} U()
Ã! 2 `(` + 1) 2 χ() U () χ() = k 2 2 自由粒子 U () = 0. χ() 原点で有界な解は j` 型のみ: = j`(k) µ `π sin k. 2 ( = 0 ( a) 短距離力 U (). 6= 0 ( < a) 原点に用いられない波動関数には n` 型が許される: χ( a) = A j`(k) + B n`(k) µ µ `π `π A sin k B cos k 2 2 A : B cos δ` : sin δ` µ `π sin k + δ`. 2 ポ テ ン シャル の 影 響 は, 遠方では自由球面波に 比 べ て の位 相 の ず れ δ` のみ.
1: V () = { 0 ( a) ( < a) 2: 0! = R(a) = A j l (ka) + B n l (ka) tan δ l = B A = j l(ka) n l (ka). V () = { 0 ( a) V 0 ( < a) R(a) = j l (Ka)! = A j l (ka) + B n l (ka) R (a) = K j l (Ka) =! A k j l (ka) + B k n l (ka) ( ) ( ) 1 ( ) A jl (ka) n = l (ka) jl (Ka) B j l (ka) n l (ka) K k j l (Ka) tan δ l = B A.
7.6 {Yl m } : x f(x) = R l () Yl m (θ, φ). l,m (1). e ikz = e ikcos θ = l l R l () Y 0 l (θ) a l j l (k) P l (cos θ) Taylo = i l (2l + 1) j l (k) P l (cos θ) l l i l (2l + 1) sin ( k lπ 2 k ) P l (cos θ). (2). eik f(θ). (3). δ l ) ϕ(x) l i l sin ( k lπ 2 (2l+1) A + δ l l k P l (cos θ).
e ik eik f(θ) = l=0 (1) + (2) = (3) A l = e iδ l. (2l + 1) eiδ l sin δ l P l (cos θ) k f(θ). : : dσ dω = f(θ) 2. σ tot = 4π k l 2 (2l + 1) sin2 δ l l = 4π k σ l. Im f(0). δ l
(cf. 1: ) k 0 sin δ 1,2,... 0, k sin δ 0 ( α : ). k l = 0 : f(θ) k 0 α, σ tot k 0 4πα 2. k 1/a l 1.
7.7 l σ l (k) = 4π(2l + 1) sin2 δ l (k) { 0 (l 1) k 0 4πα 2 (l = 0) k 0. ( sin δ l 1) δ l (k) = nπ k 0 Ramsaue δ l (k) = π 2 + nπ k 4π(2l + 1) R k 2 k 2
2:, l = 0 { 0 ( a) V () = V 0 ( < a) k = K = 2mɛ h 2m(V0 +ɛ) h 0 < a a χ() = C sin K + D cos K A sin k + B cos k C sin K = A sin(k + δ) χ(a) = C sin Ka A sin(ka + δ) χ (a) = CK cos Ka A k cos(ka + δ) : K cot Ka = k cot(ka + δ). k 0, δ \ 0 : K 0 cot K 0 a k cot δ, K 0 2mV0. h
: σ 0 (k) = 4π k 2 sin2 δ = 4π k 2 (cot 2 δ + 1) 4π (K 0 cot K 0 a) 2 + k 2. cot Ka = 0 Ka = 1 2 π, 3 2 π, k = 0. l = 1 k 2.
[1] : (I), (II) ( ). [2], : ( ). [3] : I, II ( ). [4], : I, II ( ).