filename=quantum-dim110705a.tex 1 1. 1, [1],[],[]. 1980 []..1 U(x, y, z; t), p x ˆp x = h i x, p y ˆp y = h i y, p z ˆp z = h i z (.1) Ĥ ( ) Ĥ = h m x + y + + U(x, y, z; t) (.) z (U(x, y, z; t)) (U(x, y, z)) ĤΨ(x, y, z; t) = i h Ψ(x, y, z; t), () t (.) Ĥψ(x, y, z) = Eψ(x, y, z), (E : ) (.4) Ψ(x, y, z; t) = ψ(x, y, z)exp( iet/ h)() (.5) = x + y + z (.6) 1
(x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ) = r + r r + 1 r sin θ = 1 r r (r r ) + 1 = 1 r r r + 1 r sin θ θ (sin θ θ ) + 1 θ (sin θ θ ) + 1 r sin θ θ (sin θ θ ) + 1 r sin θ r sin θ ϕ (.7) r sin θ ϕ (.8) ϕ (.9).9 (x = r cos ϕ, y = r sin ϕ, z) = r + 1 r r + 1 r ϕ + (.10) z = 1 r r (r r ) + 1 r ϕ + (.11) z (.14 x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ (.1) r = ( < x <, < y <, < z < ), (.1) x + y + z, tan ϕ = y x, tan θ = x + y, (.14) z (0 r <, 0 θ π, 0 ϕ π). (.15) r x = x (x + y + z ) 1/ = 1 (x + y + z ) 1/ 1 x = x r = sin θ cos ϕ, y, z r x = sin θ cos ϕ, r y = sin θ sin ϕ, r z ϕ x ( ) y = tan ϕ x x x = cos θ (.16) y ( ) ϕ 1 x = x cos ϕ ϕ x = sin ϕ r sin θ y, z ϕ θ ϕ x = sin ϕ r sin θ, ϕ y = cos ϕ r sin θ, ϕ z = 0, (.17)
θ x cos θ cos ϕ =, r θ y cos θ sin ϕ =, r θ z = sin θ r (.18) x, y, z r, θ, ϕ x, y, z = ( ) r + ( ) θ + ( ) ϕ, x x r x θ x ϕ = ( ) r + ( ) θ + ( ) ϕ, y y r y θ y ϕ = ( ) r + ( ) θ + ( ) ϕ z z r z θ z ϕ r, θ, ϕ.16,(.17),(.18) x, y, z x = (sin θ cos ϕ) r + y z = (sin θ sin ϕ) ( cos θ sin ϕ r + r = (cos θ) ( ) sin θ r r θ ( ) ( ) cos θ cos ϕ sin ϕ r θ r sin θ ϕ, (.19) ) ( ) cos ϕ θ + r sin θ ϕ, (.0) (.1).19 = r x x r ( ) + θ x x θ ( ) x + ϕ x ϕ ( ) x (.) ( U(x, y, z) Ĥ (r, ϕ, θ) p r = mṙ, p ϕ = mr θ, pϕ = mr sin θ ϕ Ĥ = m (ẋ + ẏ + ż ) + U(x, y, z) = m (ṙ + r θ + r sin θ ϕ ) + U(r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) = 1 m (p r + p θ r + p ϕ r sin ) + U(r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) (.) θ p r ˆp r = h i r, p θ ˆp θ = h i θ, p ϕ ˆp ϕ = h i ϕ (.4)
= ( r + 1 r θ + 1 r sin θ ϕ ) (.5).7.8.7.8 [4],[5],[6] bounded π π. xyz (x, y, z) (x + dx, y + dy, z + dz Ψ(x, y, z; t) dx dy dz (.6) xyz + + + Ψ (x, y, z; t)ψ(x, y, z; t) dx dy dz = 1 (.7) (r, ϕ, θ) (r + dr, ϕ + dϕ, θ + dθ Ψ(r, ϕ, θ; t) r sin θ drdθ dϕ (.8) π π 0 0 0 Ψ (r, ϕ, θ; t)ψ(r, ϕ, θ; t) r sin θdr dθdϕ = 1 (.9) 4
r sin θ (0 r <, 0 θ π, 0 ϕ π) Jacobian dxdydz = r sin θdrdθdϕ dxdydz = Jdrdθdϕ, (.0) x x x r θ ϕ sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ D(x, y, z) J D(r, θ, ϕ) = y y y r θ ϕ = sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ cos θ r sin θ 0 z r z θ z ϕ = r sin θ. (.1) dr, r sin θdθ, rdϕ Ψ. ( ().1, ˆl = h i r = ˆl x i + ˆl y j + ˆl z k (.1) ˆl x = h i (y z z y ), ˆly = h i (z x x z ), ˆlz = h i (x y y x ).(.) ˆl ˆl x + ˆl y + ˆl z. (.) hˆl = h i r 5
[ˆl x, ˆl y ] = i hˆl z, [ˆl y, ˆl z ] = i hˆl x, [ˆl z, ˆl x ] = i hˆl y, (ˆl ˆl = i hˆl) (.4) [ˆl, ˆl x ] = [ˆl, ˆl y ] = [ˆl, ˆl z ] = 0, ([ˆl, ˆl] = 0). (.5) z Y lm x, y ( ˆl ± ˆl x ± iˆl y, ˆl ± = ˆl. (.6) [ˆl z, ˆl ± ] = ± hˆl ±. (), (.7) [ˆl +, ˆl ] = hˆl z, (.8) [ˆl, ˆl ± ] = 0, (.9) ˆl ˆl ˆl = ˆl z + 1 (ˆl +ˆl + ˆl ˆl+ ), (.10) = ˆl ˆl+ + ˆl z + hˆl z, (.11) = ˆl +ˆl + ˆl z hˆl z. (.1) :. ( ˆl x = i h sin ϕ ) cos θ cos ϕ + (.1) θ sin θ ϕ ( ˆl y = i h cos ϕ ) cos θ sin ϕ + (.14) θ sin θ ϕ ˆl z = h i ϕ, (.15) ( ˆl ± = he ±iϕ ± ) θ + i 1 () (.16) tan θ ϕ { ( ˆl = h 1 sin θ ) + 1 } sin θ θ θ sin θ (.17) ϕ 6
( ˆl x = h ( y i z z ), ˆly = h ( z y i x x ), ˆlz = h z i ( x y y ) (.18) x (.19),(.0),(.1), (.1),(.14),(.15) (.1) ( ) cos θ ˆl x = h [ cos ϕ sin θ θ + sin ϕ θ sin ϕ cos ϕ(1 + cos θ) sin θ ϕ ( ) ( ) ( ) cos θ + cos ϕ sin ϕ cos θ sin θ ϕθ + cos ϕ ], (.19) sin θ ϕ ( ) cos θ ˆl y = h [ sin ϕ sin θ θ + cos ϕ θ + sin ϕ cos ϕ(1 + cos θ) sin θ ϕ (.0) ( ) ( ) ( ) cos θ cos ϕ sin ϕ cos θ sin θ ϕθ + sin ϕ ], sin θ ϕ ˆl z = h ϕ (.1) (.19), (.0),(.1).17. ˆl Y (θ, ϕ) = λy (θ, ϕ), (.) Y (θ, ϕ) = Θ(θ)Φ(ϕ), (.) ϕ Φ(ϕ) = m Φ(ϕ), (.4) Φ m (ϕ) = 1 exp(imϕ), (m = 0, ±1, ±,.). π (.5) m z Φ m (ϕ) { 1 h (sin θ θ ) sin θ θ Θ(θ) Φ(ϕ) + Θ(θ) } sin θ ( m )Φ(ϕ) = λθ(θ)φ(ϕ) (.6) 1 d (sin θ ddθ ) ( ) sin θ dθ Θ(θ) + λ m sin Θ(θ) = 0. (.7) θ ξ = cos θ dξ = sin θdθ d dθ = dξ d dθ dξ = sin θ d dξ. (.8) 7
Θ(θ) P (ξ)(= P ) [ d (1 ξ ) d ] ( ) P + λ m P = 0 (.9) dξ dξ 1 ξ ( ) (1 ξ ) d P dp ξ dξ dξ + λ m P = 0. (.0) 1 ξ (Legendre) λ = l(l + 1), l = 0, 1,, P m l (ξ) ˆl, ˆl z Y lm (θ, ϕ) ( 1) m+ m l + 1 (l m )! 4π (l + m )! P m l (cos θ) e imϕ, (.1) Y lm(θ, ϕ) = ( ) m Y l m (θ, ϕ). (.) Y lml (θ, ϕ) spherical harmonics π 0 sin θdθ π : () 0 dϕ Ylm(θ, ϕ) Y l m (θ, ϕ) = δ ll δ mm. (.) Y 00 (θ, ϕ) = 1 4π, (.4) Y 1,±1 (θ, ϕ) = 1 π sin θe±iϕ, (.5) Y 1,0 (θ, ϕ) = 1 cos θ, (.6) π Y,± (θ, ϕ) = 1 5 4 π sin θe ±iϕ = 1 5 8 π (1 cos θ)e±iϕ, (.7) Y,±1 (θ, ϕ) = 1 5 π cos θ sin θe±iϕ = 1 5 4 π sin θe±iϕ, (.8) Y,0 (θ, ϕ) = 1 4 5 π ( cos θ 1) = 1 8 5 (1 + cos θ), (.9) π (.40) l + 1 Y lm (0, 0) = 4π δ m0, (.41) l + 1 Y lm (0, ϕ) = 4π δ m0. (.4) 8
.4 m l = 0 l 0 Y lm (θ, ϕ), Y l m (θ, ϕ) 1. l = 1(p ) Y px 1 (Y 1, 1 Y 1,+1 ) = 4π Y py i (Y 1, 1 + Y 1,+1 ) = Y pz Y 1,0 = sin θ cos ϕ = 4π x r, (.4) y 4π r, (.44) sin θ sin ϕ = 4π 4π cos θ = z 4π r. (.45) p. l = (d Y dzx 1 15 (Y, 1 Y,1 ) = 16π Y dyz i 15 (Y, 1 + Y,1 ) = 15 zx sin(θ) cos ϕ = 4π r, (.46) 15 yz sin(θ) sin ϕ = 16π 4π r, (.47) 15 x y (.48), 16π r Y dx y 1 15 (Y, + Y, ) = 16π sin θ cos(ϕ) = 9
Y dxy Y dz Y,0 = i 15 (Y, Y, ) = 5 16π ( cos θ 1) = 15 xy 16π sin θ sin(ϕ) = 16π r, (.49) ( ) z 4π r 1. (.50) 1. (. ˆl ˆl z. n.5 ˆl Ylm = l(l + 1) h Y lm, (l = 0, 1,, ) (.51) ˆl z Y lm = m hy lm, ( l m l), (.5) ˆl ± Y lm = l(l + 1) m(m ± 1) hy lm±1, () (.5) = (l m)(l ± m + 1) hy lm±1, () (.54) θ, ϕ (.51),(.5), (.5) : (A) (.51).0 (B) (.5m ˆl, ˆl z Y lm Y lm Y l m = δ ll δ mm (.55) ˆl Y lm ˆl x + ˆl y + ˆl z Y lm = Y lm ˆl xˆl x Y lm + Y lm ˆl y ˆl y Y lm + m h = (ˆl x Y lm ) (ˆl x Y lm ) + (ˆl y Y lm ) (ˆl y Y lm ) + m h 0 (.56) 10
l 0 l(l + 1) 0. (.57).9ˆl ± Y lm ˆl l ˆl (ˆl± Y lm ) = ˆl ±ˆl Ylm = l(l + 1) h (ˆl ± Y lm ) (.58).7 ˆl z ˆl+ Y lm = (ˆl +ˆlz + hˆl + )Y lm = m hˆl + Y lm + hˆl + Y lm = h(m + 1)ˆl + Y lm (.59) ˆl + Y lm m ˆl z ˆl z ˆl Y lm = h(m 1)ˆl Y lm (.60) ˆl Y lm m ˆl z ˆl + raising operator ˆl lowering operator ˆl ± Y lm = C ± (l, m)y lm±1 (.61) ˆl ± Y lm ( (ˆl ± Y lm ) (ˆl ± Y lm ) 0. (.6) (.6),(.11), (.1), (.5), (.5) (ˆl ± Y lm ) (ˆl ± Y lm ) = Y lm (ˆl ± ) ˆl± Y lm = Y lm (ˆl )ˆl ± Y lm = Y lm (ˆl ˆl z ± hˆl z ) Y lm 0. (.6) l(l + 1) m + m, (.64) l(l + 1) m m (.65) l(l + 1) 0 l 0.64.65 l m l (.66) 11
m m min ) ˆl Y lmmin = 0 (.67) (.1) Y lmmin l(l + 1) h = (m min ) h m min h (.68) m m max ) ˆl + Y lmmax = 0 (.69) (.11) Y lmmax l(l + 1) h = (m max ) h + m max h (.70) m min = l, m max = +l m ˆl + 1 (a) l m (l + 1) (b) m m = l, (l 1),, 1, 0, 1,,, l 1, l (.71) (C).5.61 C ± (lm).61 C ± (lm) Y l,m±1 Y l,m±1 = Y lm ˆl ˆl± Y lm = Y lm (ˆl ˆl z hˆl z )Y lm = [l(l + 1) m(m ± 1)] h = [(l m)(l ± m + 1)] h (.7) C + (lm) = h (l m)(l + m + 1), (.7) C (lm) = h (l + m)(l m + 1) (.74) 1
(D) m l ˆl l h l(l + 1) h ( x, y l x, l y.6 ( l x ) Y lm (ˆl x ) Y lm ( Y lm ˆl x Y lm ), (.75) ( l y ) Y lm (ˆl y ) Y lm ( Y lm ˆl y Y lm ). (.76) ( l x ) = 1 4 Y lm (ˆl + + ˆl + ˆl +ˆl + ˆl ˆl+ ) Y lm 1 Y lm (ˆl + + ˆl ) Y lm = h [(l + m)(l m + 1) + (l m)(l + m + 1)] 4 = h (l + l m ). (.77) ( l y ) = h (l + l m ) (.78) ( l x ) + ( l y ) = h (l + l m ) (.79) m = l l l = 0 4 U(r), (r = x + y + z ), [ h m + U(r)]ψ(x, y, z) = Eψ(x, y, z), (E : ) (4.1) (x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ) = r + r r + 1 r sin θ = 1 r r (r r ) + 1 r sin θ 1 θ (sin θ θ ) + 1 θ (sin θ θ ) + 1 r sin θ r sin θ ϕ (4.) ϕ (4.)
(.17) = r + r r ˆl r h = 1 r r (r r ) r ˆl. (4.4) h ψ R(r) Y lm (θ, ϕ) (4.1) ψ(x, y, z) = R(r)Y lm (θ, ϕ). (4.5) ˆl Ylm (θ, ϕ) = l(l + 1) h Y lm (θ, ϕ) (4.6) R(r) [ h d R m dr + dr r dr ] l(l + 1) R + U(r)R = ER (4.7) r r [ h d R m dr + ] ] dr l(l + 1) h + [U(r) + R = ER (4.8) r dr mr U(r) l(l + 1) h (4.9) mr R(r) U(r) l (4.8) χ(r) R(r) = χ(r), χ(r) rr(r), (4.10) r 1 d χ(r) = d R(r) + dr(r). (4.11) r dr dr r dr 4.104.114.8 χ(r) [ h d ] ] χ(r) l(l + 1) h + [U(r) + χ(r) = Eχ(r). (4.1) m dr mr 1 14
[1] 1974 [] 1994 [] 1995 [4] [5] 5 [6] III 16 15