6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civit
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1 6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civita ɛ 123 =1 0 r p = 2 2 = (6.4) Planck h L p = h ( h i =, h, h ) i r 1 i r 2 i r 3 (6.5) f(r) [ r i,p j ] f = r i h i f h h f h (r r j i r i f)=r i δ j i r ij j i r i 105 h i f (6.6) r j
2 106 6 [ r i,p j ]=δ ij i h (6.7) (6.7) (6.3) (6.7) [ L i,l j ] = 1 h 2 = 1 h 2 = i h h 2 klmn klmn ( Levi Civita ɛ ikl ɛ jmn [ r k p l,r m p n ] ɛ ikl ɛ jmn (r m [ r k,p n ] p l + r k [ p l,r m ] p n ) ) ɛ ikl ɛ jln r k p n klm ɛ ikl ɛ jmk r m p l kln (6.8) ɛ ijk ɛ lmk = δ il δ jm δ im δ jl (6.9) k [ L i,l j ]= ī ) ( δ h ij r p + r i p j + δ ij r p r j p i = ī ) (r h i p j r j p i (6.10) : [ L i,l j ]= k iɛ ijk L k (6.11) [ L x,l y ]=il z, [ L y,l z ]=il x, [ L z,l x ]=il y (6.12) L 2 L (6.11) [ L 2,L i ] = [ L j L j,l i ]= ) (L j [ L j,l i ]+[L j,l i ] L j j j = iɛ jik L j L k + (6.13) iɛ jik L k L j jk jk j k ɛ jik [ L 2,L i ]=0 (i =1, 2, 3 ) (6.14) L 2 L L 2 L z
3 x, y, z r, θ, φ x = r sin θ cos φ y = r sin θ sin φ (6.15) z = r cos θ x = r x y z = r y r + θ x = sin θ cos φ r r + θ y = sin θ sin φ r = r z r + θ z = cos θ r sin θ r θ + φ x φ cos θ cos φ + r θ + φ y φ cos θ sin φ + r θ + φ z θ φ L x, L y, L z ( L x = i L y = i sin φ cos θ cos φ + ( θ sin θ cos φ cos θ sin φ θ sin θ ) φ φ θ sin φ r sin θ θ + cos φ r sin θ φ φ ) L z = i φ (6.16) (6.17) θ φ r L 2 [ ( 1 L 2 = sin θ ) + 1 ] 2 sin θ θ θ sin 2 θ φ 2 (6.18) L 2 Y (θ, φ) λ L 2 Y (θ, φ) =λy(θ, φ) (6.19) Y (θ, φ) =Θ(θ) Φ(φ) (6.20)
4 108 6 sin2 θ Θ Φ 1 sin θ ( d dθ sin θ dθ dθ ) λ sin 2 θ = 1 Φ d 2 Φ dφ 2 = µ2 (6.21) d 2 Φ dφ 2 + µ2 Φ = 0 (6.22) cos µφ sin µφ Φ(φ+2π) = Φ(φ) µ Θ z = cos θ [ ] (1 z 2 ) d2 Θ dθ 2z dz2 dz + λ µ2 1 z 2 Θ = 0 (6.23) z = ±1 θ =0,π Legendre P µ l (cos θ) (µ, l 0, µ l ) (6.24) λ = l(l +1) Y (θ, φ) 0 l P l (cos θ), P µ l (cos θ) cos µφ, P µ l (cos θ) sin µφ ( µ =1, 2,,l) (6.25) (2l +1) cos µφ sin µφ cos µφ ± i sin µφ =e ±iµφ µ =1, 2,,l (6.26) µ m (2l +1) (6.25) Y lm (θ, φ) =( 1) m+ m 2 [ 2l +1 2 ] (l m )! 1/2 P m 1 l (cos θ) (l + m )! (2π) 1/2 eimφ (6.27) L 2 L z L 2 Y lm (θ, φ) = l(l +1)Y lm (θ,φ) L z Y lm (θ, φ) = my lm (θ, φ) (6.28) l =0, 1, 2, m = l, (l 1),,l 1, l 2π 0 dφ π 0 sin θ dθy lm (θ,φ) Y l m (θ, φ) =δ ll δ mm (6.29) (6.27) Y lm (θ, φ) =( 1) m Y l m (θ,φ) (6.30)
5 n θ R(n,θ) ψ ψ = R(n,θ) ψ (6.31) R(n,θ) unitary [R(n,θ)] R(n,θ) = 1 (6.32) θ 0 R(n,θ) 1 Hermite S(n,θ) R(n,θ) = exp [ is(n,θ) ] (6.33) J δθ (6.33) ψ ψ = R(n,δθ) ψ =[1 is(n,δθ)]ψ (6.34) k n = e k ψ ψ = R(e k,δθ) ψ =[1 is(e k,δθ)]ψ (6.35) δθ Hermite S(e k,δθ) δθ J k S(e k,δθ)=δθ J k (6.36) J k k n J x, J y, J z J S(n,δθ)=(n J) δθ (6.37) = hj Hermite J k (6.34) ψ ψ = R(n,δθ) ψ =[1 i (n J) δθ ] ψ (6.38)
6 110 6 R(n,θ) ψ = exp [ i (n J) θ)] ψ (6.39) J J x y R x = exp ( iδθ x J x ) = 1 iδθ x J x (6.40) R y = exp ( iδθ y J y ) = 1 iδθ y J y R x R y R y R x = (1 iδθ x J x )(1 iδθ y J y ) (1 iδθ y J y )(1 iδθ x J x ) = δθ x δθ y (J x J y J y J x ) (6.41) 6.1 z δθ x δθ y z z y y x δθ y 1 x δθ y 1 x x x δθ x δθ y 6.1: R y R x R x R y R x R y R y R x = exp ( iδθ x δθ y J z ) 1= iδθ x δθ y J z (6.42) [ J x,j y ]=ij z (6.43) [ J i,j j ]= k iɛ ijk J k (6.44)
7 [ J 2,J x ]=[J 2,J y ]=[J 2,J z ] = 0 (6.45) unitary { exp ( i n J) } θ =0 θ θ J x, J y. J z unitary SU(2) { J x,j y J z } SU(2) Lie [ J i,j j ]= k iɛ ijk J k ɛ ijk O(3) O(3) SU(2) Fermi 1/2
8 J 2 J 2 J z (6.44) J ± J ± = J x ± ij y (6.46) J x J y J 2 (6.44) [ J z,j ± ]=±J ±, [ J,J + ]= 2J z (6.47) J x J y Hermite Hermite (J ± ) = J (6.48) J J + = J 2 J z (J z +1), J + J = J 2 J z (J z 1), (6.49) J 2 λ j J z m ψ jm : J 2 ψ jm = λ j ψ jm J z ψ jm = mψ jm (6.50) λ j m J 2 Jz 2 = Jx 2 Jy 2 ψ j (6.50) ( λ j m 2 ) ψ jm =(J 2 x + J 2 y ) ψ jm (6.51) ψ jm λ j m 2 = jm ( Jx 2 + J y 2 ) jm 0 (6.52) Hermite jm J 2 x jm = j m jm J x j m j m J x jm = j m j m J x jm 2 0 (6.53)
9 J z m λ j m λ j (6.54) J z m 1J z (6.47) ψ jm J z J ± ψ jm =(J ± J z ± J ± ) ψ jm =(m ± 1) J ± ψ jm (6.55) J ± ψ jm ψ jm±1 J z m 1 J 2 J ± J 2 λ j J 2 λ j J z m m m 1 m 2 m 1 m 2 J + ψ jm1 =0, J ψ jm2 = 0 (6.56) J J + (6.49) [ λ j m 1 (m 1 +1)]ψ jm1 =0, [ λ j m 2 (m 2 1) ]ψ jm2 = 0 (6.57) λ j (m 1 + m 2 )(m 1 m 2 + 1) = 0 (6.58) m 1 m 2 0 m 2 = m 1 (6.59) (1) m 1 (2) m m 2 = m 1 m 0 m 1 m 2 =2j j =0, 1 2, 1, 3 2, 2, (6.60) (6.57) J 2 m (2j +1) λ j = j(j + 1) (6.61) m = j, (j 1),,j 1, j (6.62)
10 114 6 j m (2J +1) (2j +1) { jm } m = j, (j 1),,j 1, j jm R(n,θ) jm (2j +1) { jm } j (2j +1) Casimir J 2 SU(2) Casimir Cartan SU(2) 1 J 2 j(j +1) (2j +1) (2j +1) SU(2) Cartan SU(2) Cartan J z m m = j Cartan
11 j (2j +1) (2j +1) j j j j+1. jj 1 jj J 2 = J + J + Jz 2 J z ψ jm ψjm j(j +1) = jm J + jm 1 jm 1 J jm + m 2 m (6.63) Hermite (6.48) jm 1 J jm = jm J + jm 1 (6.64) jm J + jm 1 jm 1 J jm = jm J + jm 1 2 = j(j +1) m 2 + m (6.65) jm J + jm 1 =e iδ j(j +1) m 2 + m (6.66) e iδ =1 - J + J Hermite (6.64) J + J jm±1 J ± jm = (j m)(j±m + 1) (6.67) J ± ψ jm = (j m)(j±m +1)ψ jm±1 (6.68) (6.55 J ± ψ jm ψ jm±1
12 116 6 J x J y J x = 1 2 (J + + J ) J y = 1 2i (J + J ) (6.69) J x, J y j =1/2 j =1/2 m = 1 2 m =+1 2 Pauli 2 J = 1 2 σ (6.70) Pauli σ z ( ) ( ) ( ) i 1 0 σ x = σ y = σ z = (6.71) 1 0 i Pauli σ 2 x = σ 2 y = σ 2 z = ( ) (6.72) y θ ( R(e y,θ) = exp ( iθj y ) = exp i θ ) 2 σ y θ =2nπ n = cos θ 2 i sin θ 2 σ y (6.73) R(e y, 2nπ) = cos nπ =( 1) n (6.74) 2π 1 2π 1 j =1 j =1 m = 1 m =0 m =+1 3 J x = J y = i 0 i 0 i 0 i 0 J z = (6.75) J z
13 6.5 Clebsch-Gordan 6.5 Clebsch-Gordan 117 J = J 1 + J 2 (6.76) [ J 1i,J 1j ]= k iɛ ijk J 1k [ J 2i,J 2j ]= k iɛ ijk J 2k [ J 1i,J 2j ] = 0 (6.77) J 1 J 2 ψ j1 m 1 ψ j2 m 2 ψ jm J 2 1 ψ j 1 m 1 = j 1 (j 1 +1)ψ j1 m 1 J 1z ψ j1 m 1 = m 1 ψ j1 m 1 J2 2 ψ j2 m 2 = j 2 (j 2 +1)ψ j2 m 2 J 2z ψ j2 m 2 = m 2 ψ j2 m 2 (6.78) J 2 ψ jm = j(j +1)ψ jm J z ψ jm = mψ jm J 2 [ J 2,J z ]=[J 2, J 2 1 ]=[J 2, J 2 2 ] = 0 (6.79) j 1 j 2 j m J 1z J 2z J 2 [ J 2,J 1z ] 0 [J 2,J 2z ] 0 (6.80) m 1 m 2 J m 1 m 2 Clebsch-Gordan ψ j1m 1 ψ j2m 2 ψ j1m 1 ψ j2m 2 j J 2 ψ jm = j 1 m 1 j 2 m 2 jm ψ j1m 1 ψ j2m 2 (6.81) m 1m 2 ψ jm unitary j 1 m 1 j 2 m 2 jm Clebsch-Gordan j j 1 j 2 j 1 + j 2 j = j 1 j 2, j 1 j 2 +1,,j 1 + j 2 1, j 1 + j 2 (6.82)
14 118 6 (6.81) unitary j 1 m 1 j 2 m 2 jm j 1 m 1 j 2 m 2 j m = δ jj δ mm m 1m 2 j 1 m 1 j 2 m 2 jm j 1 m 1j 2 m 2 jm = δ m1 m δ 1 m 2 m 2 jm (6.83) (6.81) ψ j1 m 1 ψ j2 m 2 = jm j 1 m 1 j 2 m 2 jm ψ jm (6.84) m m = m 1 + m 2 0 Clebsch-Gordan ψ j1m 1 ψ j2m 2 = jm j 1 m 1 j 2 m 2 jm ψ jm SU(2) { ψ j1m 1 } m 1 = j 1, j 1 +1,,j 1 1, j 1 { ψ j2 m 2 } m 2 = j 2, j 2 +1,,j 2 1, j 2 (2j 1 +1) (2j 2 +1) (2j +1) { ψ jm } m = j, j +1,,j 1, j j 1 =1 3 j 2 =5/2 6 j j =3/2, 5/2, 7/2 3 6 = m = m 1 + m 2 z j 1 m 1 j 2 m 2 jm =0 m 1 + m 2 m (6.85) (6.76) (6.76) J = J J 2 (6.86)
15 6.5 Clebsch-Gordan (6.86) z J z ψ j1 m 1 ψ j2 m 2 = (J 1z ψ j1 m 1 )(1ψ j2 m 2 )+(1ψ j1 m 1 )(J 2z ψ j2 m 2 ) = m 1 ψ j1 m 1 ψ j2 m 2 + m 2 ψ j1 m 1 ψ j2 m 2 =(m 1 + m 2 ) ψ j1 m 1 ψ j2 m 2 (6.87) z m 1 + m 2 Clebsch-Gordan Clebsch-Gordan j = j 1 +j 2 m = j m 1 = j 1, m 2 = j 2 ψ jj = j 1 j 1 j 2 j 2 jj ψ j1 j 1 ψ j2 j 2 ( j = j 1 + j 2 ) (6.88) Clebsch-Gordan unitary Clebsch-Gordan 1 - j 1 j 1 j 2 j 2 jj =1 (j = j 1 + j 2 ) (6.89) ψ jj = ψ j1 j 1 ψ j2 j 2 ( j = j 1 + j 2 ) (6.90) m = j 1 J = J J 2 (6.91) (6.90) (6.90) (6.68) 2jψjj 1 = 2j 1 ψ j1 j 1 1ψ j2 j 2 + 2j 2 ψ j1 j 1 ψ j2 j 2 1 (6.92) ψ jj 1 = j1 j ψ j2 j 1 j 1 1ψ j2 j 2 + j ψ j 1 j 1 ψ j2 j 2 1 (6.93) Clebsch-Gordan j 1 j 1 1 j 2 j 2 jj 1 = j 1 j 1 j 2 j 2 1 jj 1 = j1 j j2 j ( j = j 1 + j 2 ) (6.94)
16 120 6 j = j 1 + j 2 Clebsch-Gordan j 1=j 1 + j 2 1 z m = j 1=j 1 + j 2 1 (6.93) ψ j 1j 1 = j2 j ψ j1 j 1j 1 1ψ j2j 2 j ψ j 1j 1 ψ j2j 2 1 (6.95) - j m Clebsch-Gordan Clebsch-Gordan aαbβ cγ = δ α+β,γ [ ] (c + a b)!(c a + b)!(a + b c)!(c + γ)!(c γ)!(2c +1) 1/2 (c + a + b + 1)!(a α)!(a + α)!(b β)!(b + β)! ( 1) k+b+β (c + b + α k)!(a α + k)! (c a + b k)!(c + γ k)!k!(k + a b + γ)! k (6.96) k 0 Clebsch-Gordan j 1 m 1 j 2 m 2 jm = ( 1) j1+j2 j j 1 m 1 j 2 m 2 j m = ( 1) j 1+j 2 j j 2 m 2 j 1 m 1 j m = ( 1) j 1 m 1 2j +1 2j 2 +1 j 1m 1 j m j 2 m 2 (6.97)
17 6.6 Wigner-Eckart Wigner-Eckart A A A x, A y, A z j z m A ±1 = 1 (A x ± ia y ), A 0 = A z (6.98) 2 A µ µ z J J µ J ±1 = 1 (J x ± ij y ), J 0 = J z (6.99) 2 Racah { T LM } M = L, L +1,,L 1, L (6.100) (L M)(L ± M +1) [ J ±1,T LM ] = T LM±1 2 (6.101) [ J 0,T LM ] = MT LM L (6.99) J µ Q 2µ = r 2 Y 2µ (θ, φ) µ = 2, 1, 0, 1, 2 (6.102) x, y, z 15 Q 2 ±2 = (x ± iy)2 32π 15 Q 2 ±1 = (x ± iy) z 8π 5 Q 20 = 16π [2z2 x 2 y 2 ] (6.103)
18 122 6 Wigner-Eckart J 2 J z j 1 m 1 T LM j 2 m 2 = 1 2j1 +1 j 2m 2 LM j 1 m 1 j 1 T L j 2 (6.104) reduce reduced matrix element Wigner-Eckart J z Clebsch-Gordan (2j 1 +1) 2j 2 +1 L (2j 1 +1) (2j 2 +1) (2L +1) Clebsch-Gordan
SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
量子力学3-2013
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1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
0406_total.pdf
59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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SO(2)
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C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq
7 2003 6 26 ( ) 5 5.1 F K 0 (q 1,,q N,p 1,,p N ) (Q 1,,Q N,P 1,,P N ) Q i Q i (q, p). (1) P i P i (q, p), (2) (p i dq i P i dq i )df. (3) [ ] Q αq + βp, P γq + δp α, β, γ, δ [ ] PdQ pdq (γq + δp)(αdq +
20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
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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
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