多体問題

Transcription

1 Many Body Problem 997 4, 00 4, RPA RPA RPA RPA Sum Rule A 66 A A A A A A. L. Fetter and J.D. Walecka, Quantum theory of many body systems McGraw Hill, 97 P. Ring and P. Schuck, The nuclear many body problem Springer Verlag, 980 J. W. Negele and H. Orland, Quantum many particle systems Addison Wesley, 988

2 i α i N α α α N α α α N N N i i α i i, α i : α i α j = δ ij. N, N =, α α, α, α α α, α, α α α α α = C α α + C α α., :,,,. α α = C α α α α C α α α i α α α α = C α α α α α α α α = C α α α α α α α α = C = C = / α α = α α α α N α α α N = det α i j = N! N! α α α N α α α N. α N α N α N N

3 i j, i j,, i j, α, α,, α N, 0, α, α,, α N,. α a α { a α, a α } a αa α + a α a α = δ αα. { a α, a α } a α a α + a α a α = 0.3 { a α, a α } a αa α + a α a α = 0.4 N α a αa α n α n α : N α a β n α = N α n α = n α n α a β N α + δ αβ a β n α = n α + δ αβ a β n α.5 N α a β n α = n α δ αβ a β n α.6 a α N α, a α, a α, a α,, N α α Nα = a αa α a αa α = a α a α a α aα = a αa α = N α, n α = n α n α = 0 n α =.6 n α = 0 N α a α n α = 0 = a α n α = 0 N α, a α = 0 N α a α = a αa α = 0, a α n α = 0 = 0 N α vacuum, α a α 0 = 0 N α α N a a α a α N 0.7

4 3 a α.4,, a α a α 3 a α a α a α 3 = a α a α 3 a α = + a α 3 a α a α = a α 3 a α a α a αa α = 0, α i i =,,, N a α α N α α N a = 0 a αn a α a α a α a α a α N 0 = 0 a αn a α a α a α a α a α N 0 a α a α,, a α N a α a α a α a α N 0 = N a α a α N a α 0 = 0 a α α N α α N a = 0 a αn a α3 a α a α a α 3 a α N 0 α α N a = 0 a αn a α3 a α 3 a α N 0. = 0 0 =.8 f i f f i, N F = α β F = N f i.9 i= f αβ a α a β, f αβ α f β.0.0 φ φ = α c α α f φ f φ = αβ α f β c αc β, c, c,,.9.0 αβ = α β β α, α β = α β β α α β F αβ = α β β α f + f α β β α

5 4 f, f, α β f α β = α f α β β α β F αβ = α f α β β + β f β α α β f α α β α f β β α = f α αδ ββ + f β βδ αα f β αδ βα f α βδ αβ., αβ a = a α a β 0, α β a = a α a β 0 a α β F αβ a = µν f µν 0 a β a α a µa ν a αa β 0 a ν a αa β = δ να a αa ν a β = δ ναa β δ νβa α + a αa β a ν a ν a αa β δ 0 = να a β δ νβa α 0 0 a β a α a µ = 0 δ µα a β δ µβ a α a α β F αβ a = f α α 0 a β a β 0 + f β β 0 a α a α 0 f β α 0 a α a β 0 f α β 0 a β a α 0 = f α αδ ββ + f β βδ αα f β αδ βα f α βδ αβ..,. α β F αβ = a α β F αβ a N N, N., F = f i. i= α α N a = a α a α N 0, F = αβ f αβ a αa β,, i j vi, j, vi, j = vj, i V = vi, j = i<j vi, j i j

6 5 V = αβ v α β a αa β a β a α, αβ v α β α β v, α β.3 αβα β αβ v α β αβ v α β αβ v β α.4 αβ v β α = αβ v α β αβ v α β α, β, v, βα v α β = αβ v β α = αβ v α β, α, β V = αβ v α β a 4 αa β a β a α αβα β φr α u α r φr = α c α u α r.5 c α α c α = d 3 r φr u αr.6,.6.5 φr = u α r d 3 r φr u αr = α d 3 r φr α u α ru αr.5 c α a α ˆψr u α r u αr = δr r.7 α ˆψr = α u α r a α.8 ˆψr ˆ ˆψ ˆψ { ˆψr, ˆψ r } = αα u α ru α r { a α, a α } = α u α ru αr = δr r { ˆψr, ˆψr } = { ˆψ r, ˆψ r } = 0.6 a α = d 3 r ˆψr u αr

7 6 ˆψ, ˆψ T = α p β a α a β = α p β d 3 r m m ˆψ r u α r αβ αβ d 3 r ˆψr u βr α p β u α r u βr = ħ d 3 x u αx xu β x u α r u βr αβ αβ = ħ d 3 x u αxu α r x u β xu βr α β = ħ d 3 x δx r xδx r = ħ r δr r T = ħ d 3 r m ˆψ r r d 3 r ˆψr δr r = ħ d 3 r m ˆψ r ˆψr φr ħ d 3 r φ r φr m, φ ˆψ, φ ˆψ H = ħ d 3 r m ˆψ r ˆψr + d 3 r d 3 r ˆψ r ˆψ r vr, r ˆψr ˆψr ˆψr c α a α,, second quantization quantum field theory a α 0 u α x ˆψ r 0 = α u αr a α 0 φx φx = α u αr u α x = δx r ˆψ r r

8 7 N N H = ti + vi, j. i= i j, t, v, N α α α N α, α,, α N, N H,,,. a a a N 0,,,, N v αβ α β = αβ v α β = α β v, α β α β v, β α t αβ = α t β H = T + V, T = t αα a αa α, V = v αβ α β 4 a αa β a β a α. αα αβα β E HF = H, T = t αα a αa α αα α,, α i i =,,, N a α a i a α = a α a a a N 0 = N a a a N a α 0 = 0 α = i, a αa i i a i α α = i, α i, a αa i, α i a αa i = 0 a i = 0 a i a i = a i a i = a αa α = δ αα θ α, θ α = {, α =,,, N 0,.3

9 8 T = V T i, j =,,, N α i, β j a β a α = 0 V = 4 N i, j= αβ N i= t i i v αβ ij a αa β a ja i a αa β a ja i, α = i, β = j α = j, β = i V = 4 N i, j= v ij ij a i a j a ja i + v ji ij a j a i a ja i = 4 N i, j= v ij ij v ji ij a i a j a ja i a i a j a ja i = a i a j a j a i = a i a i a i a j δ ij a i a j a i a j = δ ij a j = 0 a i a j a ja i = δ ij v V = 4 H N v ij ij v ji ij = i, j= N E HF = H = t i i + i= N i, j= v ij ij N v ij ij.4 i, j=. F f[ϕ] = ϕ F ϕ ϕ ϕ ϕ ϕ δϕ f[ϕ] δf f, δϕ δf = 0 ϕ ϕ f = ϕ F ϕ δϕ ϕ + ϕ δϕ f + ϕ ϕ δf = δϕ F ϕ + ϕ F δϕ δf = 0 ϕ ϕ δf = δϕ F f ϕ + ϕ F f δϕ δϕ F f ϕ + ϕ F f δϕ = 0

10 9 δϕ i δϕ, δϕ i δϕ δϕ F f ϕ + ϕ F f δϕ = 0 δϕ F f ϕ = 0, ϕ F f δϕ = 0 F f,, δϕ δϕ, ϕ, δϕ δϕ,,, δϕ F f ϕ = 0 f F, F.3.4 N k E HF N, H E HF k, k k k =, ϵ k δe HF ϵ k k k = 0.5, k.4 δe HF = δϕ t k + N δϕ j v k j + j= N N i δϕ v i k = δϕ t k + δϕ i v k i i= i=,.5 N δϕ t k + δϕ i v k i = ϵ k δϕ k i= N N α u HF β = α i v β i, u HF i v, i.6 i= i=, v,, h HF = t + u HF δϕ h HF k = ϵ k δϕ k, h HF k = ϵ k k.7 h HF, ϵ k h HF.7 Hartree Fock equation. h HF

11 0 Hartree Fock approximation, h HF N HF, u HF.6 { k }, { k } u HF, h HF,, u HF h HF N u HF, h HF u HF u HF self consistent u HF HF Fermi energy, HF Fermi sea ϵ F, ϵ k < ϵ F a k, hole ϵ F, ϵ k ϵ F > 0 particle state, hole state b k b k = { a k, ϵ k > ϵ F θ k = 0 a k, ϵ k ϵ F θ k =, b k, k b k = 0 HF, particle hole vacuum, p, p, h, h, α, β α ψ α r, / ψ α r, ψ α r = fα r g α r, ψ αr = fαr gαr h HF h = t h + h h v, h h h h v, h h.8 h HF ψ h r = t ψ h r + h d 3 r ψ h r vr, r ψ h rψ h r ψ h rψ h r u H r = h d 3 r ψ h r vr, r ψ h r

12 ħ m + u H r ψ h r h d 3 r ψ h r vr, r ψ h r ψ h r = ϵ h ψ h r u H r r local,, 3 Fock term exchange term ψ h, Hartree Fock, N, v, ψ v u H r = d 3 r vr, r ρr, ρr = ψ h rψ hr h ρr r, u H v.6 ϵ α = α h HF α = t αα + h v αh αh E HF = H = h t hh + v hh hh = ϵ h v hh hh.9 hh hh h E HF ϵ h.4 ψ α r ψ α r = C expik r χ σ, α = {k, σ}.0 χ σ σ = ± / χ / = 0, χ / = 0, χ σχ σ = δ σσ.0, d 3 r ψ αr ψ α r = C d 3 r = C =, L, ψx, y, z = ψx + L, y, z = ψx, y + L, z = ψx, y, z + L expik x L = expik y L = expik z L =

13 , n x, n y, n z k x = π L n x, k y = π L n y, k z = π L n z. k, L C = / V V = L 3,, N N,, k k F k F, N = n x n y n z σ=±/ = n x k = kx + ky + kz / π = n L x + n y + n / z kf n y n z. n x, n y, n z k k x k x = π/l F k x = L k x F k x L L dk x F k x π π n x k x n y, n z, k F k = V π 3 d 3 k F k.3. N = V π 3 d 3 k = V k k F 3π k3 F ρ = N/V k ρ = k3 F 3π E E = ħ k m = V ħ π 3 d 3 k k = V ħ kf 5 m k k k F 5π m = 3 ħ kf 5 m N E/N 3/5, P ħ P = E V = 3 5 m N V 3π N V /3 = ħ kf 5 m ρ. ρr ρr = h ψ h rψ hr, ψ.0 ρ = k F /3π

14 3.5 Hartree Fock, Fock,,,,, vr, r = e r r,, MKS, CGS α, vr, r = α ħc r r, α = C = / V, U F = d 3 r ψ h r vr, r ψ h r ψ h r h = e V 3/ k σ χ σ χ σ χ σ = e V 3/ expik r χ σ k d 3 r r r exp i k r + k r + k r d 3 r r r exp ik k r r k k k F k.3 U F = e expik r π 3 V kf χ σ d 3 k 0 d 3 r expir r k k r r = u F k ψ h r u F k = e π 3 kf 0 d 3 k d 3 r expir r k k r r u F k r V,, r r u F k = e π 3 kf 0 d 3 k Ik k, Iq = d 3 r expir q r expir q = q expir q, r = 4π δr Iq = q d 3 r r expir q = q d 3 r expir q r = 4π q.4 u F k = e π kf 0 d 3 k k k

15 4 k k θ, t = cos θ u F k = e π = e π = e π kf k 0 kf 0 dk k dt [ k k k k + k kk t dk k log k + k log k k log k ] + k k k k + kk =k F k =0 = e π k FF k/k F.5 F x = + x x log + x x u F k,,,, 0 u F = e π k F dx x F x = kf 0 d 3 k F k/k F kf 0 d 3 k = e π k F dx x x log + x x = 3 + dx x F x 0 0 dx x dx x x log + x dx x x log + x = 4 x log x + + dx x + x 4 0 dx x F x =.6 u F = 3e π k F.7 r, k F local density approximation ρ = k 3 F /3π u F r = 3 e 3 π ρr /3 +Ze ħ m Ze + u H r + u F r ψ h r = ϵ h ψ h r r

16 5 u H r = e d 3 r ρr r r, ρr = h ψ h r ψ hr.7 E HF E HF, v hh hh = h h v, h h h h v, h h hh hh hh,.7 3/ u F r = e 3 π ρr /3.8 α u F r = α e 3 π ρr /3, α 3/ α Xα..8 α β v, α β = αβ v α β v.4,.5,.6. E HF kσ, k σ v k σ, kσ. E ex kσ, k σ v k σ 4πe, kσ = δ σσ V k k k k F k k F σσ kσ, k σ v k σ, kσ = 3 4 /3 3 e ρ 4/3 V π 3. ρ 4/3 V ρ 4/3 V = d 3 r ρ 4/3, ρ r ρr, h ψ h ψ h ψ h δψ, E ex δe ex δe ex = d 3 r δψ r u F r ψ h r u F

17 6.6, ABC Z AB Z, + normal product : ABC Z : NABC Z 0, α a α, : a pa h a h a p : = a pa h a h a p, HF,.8 b, b : a pa h a h a p : = : b pb h b h b p : = b pb h b h b p = a pa h a h a p, vac, c, c vac = 0, vac c = 0, 0 contraction AB AB = : AB : + AB AB : AB : AB, AB = : AB : + AB vac vac = AB : c α c β : + c α c β = c α c β + 0 = c α c β, : c α c β : + c α c β = c α c β + 0 = c α c β : c α c β : + c α c β = c α c β + 0 = c α c β, : c α c β : + c α c β = c β c α + δ αβ = c α c β ABC Z, ABC Z = : AB Z : + : AB C Z : + : ABC Z : + + : AB CD E Z : + : ABCD E Z : + + : AB CD EF G Z : +,, : ABC Z : = AC : B Z :, : ABCD Z : = AD : BC Z : : ABCD E Z : = AC : BD E Z : = AC BD : E Z :

18 7, 3 ABC = : ABC : + : ABC : + : ABC : + : ABC : = : ABC : + AB C AC B + BC A.9 c α, A A c α = 0 ABc α = : AB : + AB c α = : ABc α : + AB c α,.9 c α = : ABc α : + AB c α Ac α B + Bc α A Bc α = : Bc α : + Bc α = c αb + Bc α ABc α = Ac αb + A Bc α = c αa + Ac α B + A Bc α = c αab Ac α B + A Bc α AB = : AB : + AB, c α : AB : = : ABc α : ABc α = : ABc α : + AB c α Ac α B + A Bc α.9, n,, BC A : BC : = A BC BC = : ABC : + : ABC : + : ABC : + : ABC : A BC = : ABC : + : ABC : + : ABC : HF a αa β a β a α = : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : a αa β = a αa β = 0 + : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : + : a αa β a β a α : a αa β a β a α = : a αa β a β a α : a αa β : a β a α : + a αa α : a β a β : + a β a β : a αa α : a β a α : a αa β : + a β a β a αa α a β a α a αa β

19 8, v.3 4 αβα β v αβ α β a αa β a β a α = V res = v αβ α β 4 : a αa β a β a α :.0 αβα β αβα β v αβ α β = αβα β v αβ α β a β a β : a αa α : + a β a β a αa α + V res a β a β a αa α a β a β a αa α + V res = v αh α h a αa α v hh hh + V res αβh hh.6,. H = t αα + v αh α h a αa α v hh hh + V res αα h hh = α h HF α a αa α v hh hh + V res αα hh.7 H = ϵ α a αa α v hh hh + V res = ϵ α : a αa α : + V res + E HF. α hh α.8 b ϵ α : a αa α : = α p ϵ p : b pb p : + h ϵ h : b h b h : = p ϵ p b pb p h ϵ h b h b h ϵ α : a αa α : = 0 α, V res : a pa p a ha h : = b pb p b h b h V res = 0, V res H = E HF, HF H.9 E HF V res V res residual interaction.7.0, HF.8 b α V res v αβ α β k k = 0 4 V res = V 0 + V + V + V 3 + V 4, V pp = V 0 = 4 vpp p p : a p a p a p a p : = 4 vpp p p b p b p b p b p V hh = V 4 = 4 vhh h h : a h a h a h a h : = 4 vhh h h b h b h b h b h

20 9 V = v h p 4 p : p a h a p a p a p : + v p h p : p a p a h a p a p : + v p p h : p a p a p a p a h : + v p p p : h a p a p a h a p : V = 4 v h h p p : a h a h a p a p : + v h p h p : a h a p a p a h : + v h p p : h a h a p a h a p : + v p h h : p a p a h a p a h : + v p h p : h a p a h a h a p : + v p p h : h a p a p a h a h : V 3 = v ph 4 h : h a p a h a h a h : + v hp h : h a h a p a h a h : + v hh p : h a h a h a h a p : + v hh h : p a h a h a p a h : v h p p p = v p h p p, : a h a p a p a p : = : a p a h a p a p : V, V 3 4, V 5, V 3, V V = v p h p : p a p a h a p a p : + v p p p h : a p a p a h a p : V = v p h h p : a p a h a p a h : + v hh 4 p p : a h a h a p a p : + v pp h h : a p a p a h a h : V 3 = v p h h : h a p a h a h a h : + v h h p h : a h a h a h a p : V V ph h h V ph = v ph hp : a pa h a p a h : = v ph hp : b pb h b p b h : = v ph hp b pb h b h b p V V = v hh 4 p p : a h a h a p a p : + v pp h h : a p a p a h a h : = v h h 4 p p b h b h b p b p + v p p h h b p b p b h b h, V V 3 V Y = V + V 3 = v ph p : p a p a h a p a p : + v p p : ph a p a p a h a p : = = + v ph h : h a p a h a h a h : + v h h : ph a h a h a h a p : v p h p p b p b h b p b p + v p p p h b p b p b h b p + v p h h h b p b h b h b h + v h h p h b h b h b h b p v p h p p b p b h b p b p + v p h h h b p b h b h b h + h.c. h.c. V res = V pp + V hh + V ph + V V + V Y.

21 0 V pp = 4 vpp p p b p b p b p b p.3 V hh = 4 vh h h h b h b h b h b h.4 V ph = v ph hp b pb h b h b p.5 V V = 4 V Y = v h h p p b h b h b p b p + v p p h h b p b p b h b h v p h p p b p b h b p b p + v p h h h b p b h b h b h + h.c..6.7 V res,, v V pp p, p, p, p V pp : V hh : V ph : V V V Y : : p p h h h p p h p h h p p h h p p h p p h h p h p h p h V pp V hh V ph V V V Y V Y v αβ α β α, β, α, β,, v v αβ α β = α β v, α β α β v, β α v,, V ph = v ph hp b pb h b h b p = p h v, h p p h v, p h b pb h b h b p, p, h, p, h, p, p, h, h

22 h p p h p h = p h p h p h,, v, HF V res H H = H 0 + V res, H 0 = α ϵ α a αa α vhh hh H 0 = E HF H 0 a α a αh 0 = β ϵ β a β a βa α a αa β a β = β ϵ β a β δ αβ a β a αa β a αa β a β = ϵ α a α H 0 a α a α H 0 = ϵ α a α H 0 a αa β = ϵ α a α + a αh 0 aβ = ϵ α a αa β + a α ϵ β a β + a β H 0 = ϵ α ϵ β a αa β + a αa β H 0 H 0 a pa h = ϵ p ϵ h + E HF a pa h a pa h H 0 ϵ p ϵ h + E HF,, ϵ p ϵ h, n m H 0 n m H 0 a p a p n a h a hm = ϵ pi ϵ hi + E HF a p a p n a h a hm = i= i= + V res + E HF H 0 b α = 0, V res b α, V V vp p 4 h h b p b p b h b h = vp p 4 h h a p a p a h a h = + vp p 4 h h a p E HF H a p a h a h + 0

23 E HF H 0 a p a p a h a h = ϵ p + ϵ p ϵ h ϵ h a p a p a h a h a p E HF H a p a h a h = a p 0 ϵ p + ϵ p ϵ h ϵ a p a h a h h = 4 vpp h h ϵ p + ϵ p ϵ h ϵ h a p a p a h a h + a h a p = 0 a h a p = 0 a pa h α HF t + u HF α = ϵ α α u 0 µ t + u 0 µ = ϵ µ µ µ c µ µ HF = c c c N 0 H = µ h HF µ c µc µ v hh hh + V res, V res = v µν µ ν 4 : c µc νc ν c µ : µµ hh µνµ ν u = u HF u 0 h HF = t + u 0 + u H = α H ϵ µ c µc µ v hh hh + H, H = µ u µ c µc µ + V res hh µµ H = ph p u h c pc h + V res, = c c c N 0 u 0.8 Ψ ρ µν = µ ˆρ ν Ψ c νc µ Ψ density matrix µ, ν, c F = µν f µν c µc ν Ψ F Ψ = µν f µν Ψ c µc ν Ψ = µν f µν ρ νµ = Trfρ

24 3 Tr, Ψ HF µ ˆρ ν = c νc µ, = a a a N 0, HF α, β, p, p, h, h HF µ, ν µ = α α α µ c µ = α α µ a α.3 µ ˆρ ν = αβ α ν µ β a αa β = h µ h h ν ˆρ = h h h ˆρ = hh h h h h = hh h δ hh h = ˆρ Ψ ˆρ 0 ˆρ α = h h h α = θ α α, θ α = {, hole states 0, particle states, HF ˆρ H c, c, H = pq t µν c µc ν + µνµ ν v pq p q 4 : c µc νc ν c µ : + c νc ν : c µc µ : + c νc ν c µc µ E HF = H = pq = µν t µν c µc ν + v µν µ ν c νc ν c µc µ µνµ ν t µν ρ νµ + v µν µ ν ρ ν ν ρ µ µ.8 µνµ ν = Tr t ρ + Tr Tr v ρ ρ.9,, HF ρ αβ = θ α δ αβ.4

25 4 HF.9 ˆρ ˆρ ˆρ ˆρ = ˆρ 0 ˆρ = ˆρ 0 + δ ˆρ E HF, ˆρ 0 HF, ˆρ 0 + δ ˆρ δ ˆρ ˆρ 0 + δ ˆρ = ˆρ 0 + δ ˆρ δ ˆρ = ˆρ 0 δ ˆρ + δ ˆρ ˆρ 0, i.e. δρ αβ = θ α δρ αβ + δρ αβ θ β = θ α + θ β δρ αβ.30 δρ αβ 0 θ α + θ β =, θ α =, θ β = 0 θ α = 0, θ β =, δ ˆρ δρ ph, δρ hp 0 δe = E HF [ ˆρ 0 + δ ˆρ ] E HF [ ˆρ 0 ] = αβ h[ˆρ 0 ] αβ δρ βα + = ph h[ˆρ 0 ] ph δρ hp + h[ˆρ 0 ] hp δρ ph + h[ˆρ] αβ E HF ρ βα = t αβ + α β v αα ββ ρ β α, ˆρ = ˆρ 0 ρ βα = θ α δ αβ h[ˆρ 0 ] αβ = t αβ + h v αh βh = α h HF β.3 δρ hp, δρ ph δe = 0, h HF h ph = p h HF h = 0.3 h HF,, α h HF β = ϵ α δ αβ α [ h HF, ˆρ 0 ] β = α h HF ˆρ 0 ˆρ 0 h HF β = θ β θ α α h HF β,.3 α, β α [ h HF, ˆρ 0 ] β = 0,.3 [ h HF, ˆρ 0 ] = E HF, v ˆρ, HF v = v[ˆρ] h αβ = E HF ρ βα = t αβ + h v αh βh + hh v hh ρ βα hh, α h HF β h HF α α + β α d αβ β

26 5 d αβ c α = a α + β α d αβ a β, d Ψ = c c N 0 = a a N 0 + d β a β a a N 0 + d β a a β a 3 a N 0 + β β N = a a N 0 + d iβ a a i a β a i+ a N 0 i= β i β =,,, i, i +,, N a a i a β a i+ a N a β β a i 0 = 0 a i a i a i+ a N 0 = a i a i a i+ a N 0 = a i+ a N 0 a a i a β a i+ a N 0 = a a i a β a i a i a i+ a N 0 k i a k a β a i a a i a β a i+ a N 0 = a β a i a a N 0 β, i, p, h Ψ = + ph d hp a pa h, = a a N 0 Ψ H Ψ = H + ph d hp Ha pa h + ph d hp a h a ph d hp.3 a h a ph = 0.3. H a h a ph a h a ph = t αα a h a pa αa α + v αβ α 4 β a h a pa αa β a β a α αα αβα β a h a pa αa α a h a pa αa β a α a h a ph = t ph + h v ph hh = p h HF h

27 6.9 N ± ε/, c ± m, m =,,, N H 0 H 0 = εk 0, K 0 = N c +mc +m c mc m H int = V K + K + + K K, K + = m= N c +mc m, K = K + = m= N m= c mc +m, K ± + m m, HF, a 0m = D 0 c m + D +0 c +m, a m = D c m + D + c +m.34 0m m, HF a, a = D = D +0 D +/D 0.36 N a 0m 0 m= { a 0m, a 0m } = D 0 + D +0 =.35 { a m, a m } = D + D + =.36 { a 0m, a m } = D 0 D + D +0 D + = 0.37 D+0 D 0 + D + =, D + = D 0, D = D c m, c +m c m = κ D + a 0m D +0 a m, c +m = κ D 0 a m D a 0m.38 κ = D + D 0 D +0 D, κ =.35,.36,.37 κ = HF a m = 0, a 0m = 0.38 c +mc +m = κ D a 0m a 0m = D +0, c mc m = D + = D 0 K 0 = m D +0 D 0 = D +0 D 0 N

28 7, Wick K + K + = mm c +mc m c +m c m = c +mc m c +m c m c +mc m c +m c m mm.38 c +mc m = D D +δ mm = q δ mm, q D D +.39 K + K + = q N N, K K = q N N E HF = H = εn D+0 D 0 V N N q + q.37 D + D + = D 0 q = D 0D +0 E HF E HF D ±0 D 0 + D +0 = E = E HF λn D 0 + D +0 D D D E D 0 E D +0 = εn D 0 V N N qd +0 λnd 0 = 0 = εn D +0 V N N q D 0 λnd +0 = 0 M D 0 D +0 = λ ε D 0 D +0 / χq, M =, χ = V χq / ε N.40 D 0 D 0 = D +, D 0D +0 = D + D D+ M = λ D+ D ε D M D D + = λ ε D D + D ±0 λ/ε M, D ± λ/ε detm λ/ε = 0 λ = ε 4 + χ q

29 8 D 0 + D +0 = D +0 = ε χq λ ε/ D 0 D 0 = λ ε/ λ ε/ + ε χ q = λ ε/ λ ε/ + λ ε /4 = λ ε/ λ q λ D ±0, q q = D 0D +0, q D ±0, iteration n q q n q n+ = D 0D +0 = εχq n λ n ε/ D 0 = εχq n λ n D ±0 λ n = ε χ q n + /4 q n+ = χ q n + 4χ q n.4 q q = χ q + 4χ q 0, χ q = qχ, χ, q χ = χ χ.4 q = 0 λ = ϵ, D 0 =, D +0 = 0, q = χ /4χ λ = ϵχ, D 0 = χ + χ χ, D +0 = η χ, η = { +, q > 0, q < 0 HF.4 q E HF = εn D +0 D 0 χ q + q q = D 0 D +0 = D 0 D 0, q = 0 D 0 = D 0 = ± 4 q D 0 = + 4 q E HF = εn, D +0 = 4 q 4 q + χ q + q.43 HF.40 Im q = 0 de HF = εnq dq χ d E HF, 4q dq = εn 4q χ 3/

30 9.4 q E HF d E HF dq = εn χ q=0 χ > q = 0 E HF q = χ /4χ χ > d E HF dq = εnχ χ Im q = 0 E HF q Re q = 0 d E HF dq fq = εn q 4q = εn 4q + χ 3/ χ > q = 0 d E HF /dq > 0 q = 0 Im q = 0 χ = EHF/εN q HF q q n = r n e iθ n.4 r n+ = χ r n, θ n+ = θ n = n+ θ 0 + 4χ rn a n = /r n a n+ = χ a n + 4, a n+ 4χ χ = χ a n 4χ χ a n qχ = χ n a 0 χ qχ, q χ = χ

31 30 n 0 rn q χ χ n rn = qχ χ n + qχ/r 0 0, χ <, q n q χ e n iθ 0, χ > χ >, Im q 0 0, q n.4 q χ, q n q χ e iθ 0 q χ e iθ 0.4, q n+ = a χ q n + 4χ q n + a q n, 0 a < q n, a = q n.43 E HF.44 q n.4 Hartree-Fock, χ = Im qn Re q n χ = Im qn Re q n

32 3.4 H = H 0 + H int a, a, HF Wick H int = V c +mc +m c mc m + h.c. = : H int : + εn χ q + q εχ q c +mc m + q c mc +m :.. : a, a H = H 0 + H int = H HF + : H int : + εn χ q + q H HF H HF = ε m c m c +m M c m c +m M c m c +m = κ a 0m a m D, D = D + D +0 D DD = D D = H HF H HF = ε a a 0m 0m a m DMD m a m DMD D HF.40 D 0

33 3 RPA 3 3 RPA 3., h p HF one particle one hole ph ph, 3p3h,,, ph, 3p3h, λ λ = X ph a pa h 3. ph, H λ = E λ X E, 3., a pa h, λ, H λ = E λ a h a ph λ = E a h a p λ 3.. E HF a h a p H a p a h X p h = ħω a h a pa p a h X p h, ħω = E E HF 3. p h p h H = H 0 + V res, H 0 = ϵ α : a αa α :, V res = v αβ α 4 β : a αa β a β a α : 3.3 α αβα β, HF a αa β = a αa β = 0 α, β, a h a p = 0 p, h a h a pa p a h = a h a pa p a h = a h a h a pa p = δ hh δ pp 0 a h a p : a αa α : a p a h = a h a p : a αa α : a p a h + a h a p : a αa α : a p a h = δ hα δ pp δ h α + δ hh δ pα δ p α = δ hh δ pp δ αp δ αh a h a p H 0 a p a h = δ hh δ pp ϵ ph, ϵ ph = ϵ p ϵ h a h a p : a αa β a β a α : a p a h = a h a p : a αa β a β a α : a p a h + a h a p : a αa β a β a α : a p a h + a h a p : a αa β a β a α : a p a h + a h a p : a αa β a β a α : a p a h = δ αp δ βh δ αh δ βp δ α hδ β p δ β hδ α p 3.4

34 3 RPA 33 v a h a p V res a p a h = v ph hp A ph p h = a h a p H a p a h = ϵ ph δ hh δ pp + v ph hp A ph p h X p h = ħω X ph 3.6 p h Tamm Dancoff equation ph, AX = ħω X A, v ph hp V res..5 V ph X ph ħω v X ph = Xph 0 + Xph +, ω = ω 0 + ω + ϵ ph Xph 0 + ϵ ph Xph + + v ph hp X 0 p h + X p h + = ħω 0 + ħω + X ph 0 + Xph + p h v ϵ ph X 0 ph = ħω 0 X 0 ph, ϵ ph X ph + p h v ph hp X0 p h = ħω0 X ph + ħω X 0 ph v 0,, V res, ħω 0 = ϵ ph, X 0 p h = {, p = p, h = h 0, a pa h,, Xp 0 h p h ϵ p h X p h + v p h h p = ϵ ph X p h + ħω X 0 p h p = p, h = h ħω = v ph hp = a h a p V res a pa h,, V res p, h p, h X 0 p h = 0, Xp h = v p h h p = a h a ϵ ph ϵ p h ϵ ph ϵ p V res a pa h p h λ a pa h + p h a p a h ϵ ph ϵ p h a h a p V res a pa h ph = a pa h, h, ph ph

35 3 RPA 34 H 0 p h = ϵ p h p h h p λ ph + p h p h p h λ ph + n ϵ ph H 0 V res ph n n ϵ ph H 0 V res ph, n ph, ph V res..7 V Y ϵ ph H 0 ϵ ph ϵ ph ϵ p h ϵ ph ϵ p h ϵ ph ϵ ph ϵ p h ϵ ph ϵ p h ph,, TDA ph n n p h p h p h n p p h + p h h h p h, V res TDA p h p h p h p h f v αβ α β = κ f αα f ββ, f αβ = α f β 3.7 κ A ph p h X p h = ϵ phx ph + κ f ph fp h X p h = ħωx ph p h p h C κ fp h X p h p h 3.6 f ph X ph = C ħω ϵ ph

36 3 RPA 35 C Dħω = ph f ph ħω ϵ ph = κ 3.8, ω, Dħω ħω, y = /κ Dħω κ > 0 O ħω κ < 0 ph ϵ ph, ϵ ph ħω 0, ħω 0, ħω 0 4/A /3 MeV, ϵ ph ħω ϵ ph, X ph λ a pa h ph, κ < 0 ϵ ph, κ > 0 ph collective state κ κ TDA = D0 = f ph ϵ ph, F = f αβ a αa β ph 3.9 λ F = ph αβ Xphf αβ a h a pa αa β = Xphf ph = C ph ph f ph = C ħω ϵ ph κ C λ λ = X ph X p h a h a pa p a h = X ph = php h ph C = ph f ph ħω ϵ ph = ddħω dħω 3.0

37 3 RPA 36 λ F = C κ = κ ph f ph ħω ϵ ph = κ ddħω d ħω Dħω 0 ϵ ph, ϵ, TDA ħω = ϵ ħω = ϵ + κ ph f ph λ F = κ ħω ϵ f ph ph = ph f ph = ph ph F, 3. RPA TDA, HF, HF, TDA RPA random phase approximation, H gs λ H gs = E 0 gs, H λ = E λ 3. λ = O λ gs, O λ gs = 0 O λ, O λ O λ = λ gs 3. [ H, O λ ] gs = E E 0 O λ gs = ħω O λ gs gs δo [ ] gs δo, [H, O λ ] gs = ħω gs [ δo, O λ ] gs 3. gs O λ = 0, gs HO λ = E 0 gs O λ = 0 3., 3., 3. TDA HF, O ph O λ = ph X ph a pa h, δo = a pa h 3.3

38 3 RPA [ ] a h a p, [H, a p a h ] X p h = ħω [ a h a p, a p a h ] X p h 3.4 p h p h, H E HF 3.3 a h a pa p a h = δ pp a h a h a p a h a h a p = δ pp δ hh δ pp a h a h a p a h a h a p a h = a p = 0 a h a pa p a h = δ pp δ hh [ ] a h a p, [H, a p a h ] = a h a p [H, a p a h ] = a h a p H a p a h a h a pa p a h H = a h a p H a p a h δ pp δ hh H H H = 0 [ ] a h a p, [H, a p a h ] = a h a p H a p a h TDA 3.5 [ a h a p, a p a h ] = δ hh δ pp, TDA gs HF, a h a p = 0 a h a p gs O λ = ph X ph a pa h Y ph a h a p, δo = a pa h, a h a p 3.5 RPA random phase approximation RPA 0 O λ 0 = 0 3. [ ] 0 a h a p, [H, O λ ] 0 = ħω 0 [ a h a p, O λ ] 0 [ ] a pa h, [H, O λ ] 0 = ħω 0 [ a pa h, O λ ] 0 RPA, RPA, RPA HF, RPA HF p h 0 [ a h a p, a p a h ] 0 [ a h a p, a p a h ] = δ pp δ hh O λ [ ] [ ] a h a p, [H, a p a h ] X p h a h a p, [H, a h a p ] Y p h = ħωx ph p h [ ] [ ] a pa h, [H, a p a h ] X p h a pa h, [H, a h a p ] Y p h = ħωy ph

39 3 RPA 38 A ph p h [ a h a p, [H, a p a h ] ] = a h a p H a p a h [ ] B ph p h a h a p, [H, a h a p ] = a h a pa h a p H P, Q [ P, Q ] = [ Q, P ] A ph p h = [ a h a p, [H, a p a h ] ] = [ [ a h a p, H ], a pa h ] [ ] = a pa h, [H, a h a p ] B ph p h = [ a pa h, [H, a p a h ] ] Aph p h X p h + B ph p h Y p h = ħω Xph p h B ph p h X p h + A ph p h Y 3.8 p h = ħω Yph p h 3.8 RPA RPA equation A B X X = ħω 3.9 B A Y Y A, B, X, Y RPA A, B TDA 3.5, B ph p h A ph p h = a h a p H a p a h = ϵ ph δ hh δ pp + v ph hp 3.0 TDA v a h a pa h a p H 0 = α ϵ α a h a pa h a p : a αa α : = 0, B V res 3.4 a h a pa h a p : a αa β a β a α : = a h a pa h a p : a αa β a β a α : + a h a pa h a p : a αa β a β a α : + a h a pa h a p : a αa β a β a α : + a h a pa h a p : a αa β a β a α : = δ αp δ βp δ αp δ βp δ α hδ β h δ β hδ α h 3. B ph p h Bph p h B ph p h = a h a pa h a p V res = v pp hh 3. V res..6 V V p h B ph p h = a h a pa h a p V res = V res a p a h a pa h = v pp hh = v hh pp

40 3 RPA 39 p h p h, O λ 0 = 0 RPA 0 0 = + p h + 4p 4h + X ph, Y ph 0 O λ = 0 [ ] [ ] 0 a h a p λ = 0 a h a p, O λ 0 a h a p, O λ = X ph [ ] [ ] a pa h λ = 0 a pa h, O λ 0 a pa h, O λ = Y ph X, Y ρ λ αβ = 0 a β a α λ, RPA ρ λ, 0 TDA, 3.7, RPA 3.8 ϵ ph X ph + κf ph f p h X p h + f p h Y p h = ħωxph p h ϵ ph Y ph + κfph f p h X p h + f p h Y p h = ħωyph p h C = κ p h f p h X p h + f p h Y p h 3.4 X ph = C Cf ph ħω ϵ ph, Y ph = Cf ph ħω + ϵ ph 3.5 D RPA ħω = ph ϵ ph f ph ħω ϵ = ph κ 3.6 D RPA ħω κ > 0 O ħω κ < 0

41 3 RPA 40 TDA, 3.9, TDA κ κ TDA, RPA κ κ RPA = D RPA 0 = f ph ϵ ph ph = κ TDA, ϵ ph ϵ, 3.6 ħω λ = ϵ + κ f ph ϵ ph κ f ph ϵ ph ħω λ ϵ + κ f ph ph TDA, ωλ < 0, ω λ 3.3, RPA,,,.0 ψ k σ r = V expik r χ σ, χ σχ σ = δ σσ, σ, σ = ± / k F, k k F 3.5,,, q O = + X k,σ a k+q,σ a k,σ Y k,σ a k,σ a k q,σ 3.7 k σ, ± k k F, k ± q > k F k, k k k + q, k + q k = q 3.8 h kσ, X p k+q σ, Y p k q σ ħω E k+q + E k X kσ = + k σ C kσ,k σ qx k σ + k σ B kσ,k σ qy k σ 3.8 ħω + E k q E k Y kσ = + k σ Bkσ,k σ q X k σ + k σ Ckσ,k σ q Y k σ 3.9 k σ

42 3 RPA 4, m E k = ħ k /m v k σ, k σ v k σ, k σ = δ σσ δ σ σ,.4 C kσ,k σ q = k+q σ, k σ v k σ, k +q σ B kσ,k σ q = k+q σ, k q σ v k σ, k σ = δ σ σ δ σ σ v, = e V e V exp d 3 r e r r exp ik d 3 r d 3 k r + ik k r r r r d 3 r exp ik k + k k r ik k r = δ σ σ δ σ σ δ k +k, k +k C kσ,k σ q = k+q σ, k σ v k σ, k +q σ r e V k k = k+q σ, k σ v k σ, k +q σ k+q σ, k σ v k +q σ, k σ = 4πe V q δ σσ k k B kσ,k σ q = k+q σ, k q σ v k σ, k σ = k+q σ, k q σ v k σ, k σ k+q σ, k q σ v k σ,, k σ = 4πe V q δ σσ k k + q q = q k k F, k ± q > k F k k k F, k k k F, C, B /q C kσ,k σ q = B vq 4πe kσ,k σ q =, vq = V q 3.8, 3.9 ħω E k+q + E k X kσ = vq + X k σ V + Y k σ k σ k σ ħω + E k q E k Y kσ = vq + X k V σ + Y k σ k σ k σ N = + X kσ + kσ kσ Y kσ

43 3 RPA 4 X kσ = vq V N F ω, q = vq V N, ħω E k+q + E k + k Y kσ = vq V ħω E k+q + E k σ {, k kf θ k = 0, k > k F F ω, q F ω, q = vq V k k N ħω + E k q E k ħω + E k q E k θk θ k+q θ k θ k q ħω E k+q + E k ħω + E k q E k k k θ k = θ k, E k = E k F ω, q = vq θk θ k+q θ k θ k+q V ħω ω k,q ħω + ω k,q = vq V k = 3.30 ħω k,q θ k θ k+q ħω ħω k,q, ħω k,q = E k+q E k 3.3 k F ω, q ω F ω, q ω = ± ω k,q, ω /ω V k ω k,q ω = ω k,q = ω F ω, q = F ω, q ħ k q + q, k k F, k + q > k F 3.3 m 0 ω V, ω k,q ω k k F ω max 3.3 ω k,q = ħ k q + q m ħ kq + q m ħ k F q + q = ω max q m

44 3 RPA 43, /ω,, ω 3.3 S = k k = k + q ħω k,q θ k θ k+q ħω ħω k,q = k θ k θ k+q E k+q E k ħω E k+q E k S = E k E k q θ k q θ k ħω E k k E k q k = k S = k θ k+q θ k E k E k+q ħω E k E k+q = S S = 0 F ω, q = vq V k fω, q = V V π 3 k kf 0 ħω k,q θ k ħω ħω k,q = vq V d 3 k fω, q = V k θ k ħω ħω k,q = vq fω, q + f ω, q k = ħω ħω k,q π ħ, t = cos θ, θ q k fω, q = θ k ħω ħω k,q kf k = k F k, x = m ħk F q mk F π ħ q = mk F 4π ħ q F ω, q = vq mk F 4π ħ q 0 = mk F 4π ħ q gx, 0 θ k ħω + ħω k,q dk k dt ω ħkqt + q /m ω ħq m dk k log k + x log k x [ xk + k x log k ] + x k = k x k =0 gx = x + x log + x x gx + + gx, x ± = m ħk F q vq mk F 4π ħ q = e mkf πħ q 3 ± ω ħq m 3.33 x ±, CGS, α α = e /ħc e mk F πħ q 3 = α π mc kf [ ][ ] ħc q 3 = = [ ][ ] 3

45 3 RPA 44 q 0 F ω, q = ω 0 q 0 x ± ± x ± gx = x + x x + 3x + 5x 4 + = 4 + 3x 5x + F ω, q = vq mk F 3π ħ + + q x + x 5 x x ħk Fq ħq x + x mω mω, x ħkf q ħq + x 3 3 mω mω k 3 F q F ω, q = vq 3π mω ħkf q + mω 3.34 ω p = F ω, q = ω p ω 4πe m k 3 F 3π = + 3 vf q 5 ω + 4πe ρ m, ρ = k3 F 3π, v F = ħk F m =, ω ω p vf q ωp 3.35 ω p v F 3.33 F ω, q = 3.35 ρ ρ 0 3 cm 3 = 00 nm 3 ω p 4παħc3 ρ ħω p = mc mc ev, ħc 00 ev nm, α = e ħc 37 4π π ev = 80 ev = ev 37 ω max ωq = ω max q q, 0 ħω ev ħω p 0 ħω max q k F k F q nm

46 3 RPA 45 ρ 0, ρ B ρ B = ρ 0, ρ 0 ρ = ρ 0 + δρr, t, E = 4πe ρ B ρ 0 + δρr, t = 4πe δρr, t E vr, t, P ρ, mρ 0 v t = P ρ 0 + δρ eρ 0 E = κ δρ eρ 0 E,, δρ t ρ δρ + ρ v = t t + ρ 0 v = 0 κ = P ρ ρ=ρ0 = ρ 0 v t = κ m δρ + eρ 0 m E = κ m δρ 4πe ρ 0 δρ 3.36 m q 0 δρ t = 4πe ρ 0 m δρ δρ 4πe ρ 0 m 3.4, /r,, vq /q q = 0, q 0 ω, v, = f 0 δr r k σ, k σ v k σ, k σ = f 0 δ σ σ δ σ σ V d 3 r e ik +k k k r = f 0 δ σσ δ σ σ V δ k +k, k +k 3.8, 3.9 C kσ,k σ = B kσ,k σ = f 0 δ σσ V N σ = + X kσ + k k Y kσ

47 3 RPA 46 X kσ = f 0 V N + + N = + X kσ + kσ N + + N N σ, Y kσ = f 0 ħω E k+q + E k V = N + + N f 0 V Y kσ kσ + k ħω E k+q + E k N + + N N σ ħω + E k q E k k ħω + E k q E k 3.30, vq = 4πe /q vq = f 0 /, RPA ω 3.33 F ω, q = f 0 q 0 ω F ω, q = f 0 mkf 8π ħ gx + + gx = 3.37 q k 3 F q 6π mω + 0 ω 0 ω q q 0 x ± = m ± ω ħq = ± x, x = mω ħk F q m ħk F q = ω v F q, = q 0 k F, gx ± ± x gx x 3.37 gx + + gx = gx g x + g x g x = g x = 8 x log + x x F x = x log + x x =, F 0, ω max q 0 ω max = ħ k F q + q ħk Fq m m = v Fq F 0 = mk F π ħ f ω ω > ω max F x, x >, f 0 > 0 x > xf 0 /F 0 ω = v F xf 0 q 0 x ω q, = 0 zero sound v F xf 0, v F

48 3 RPA 47 first sound 3.36 e = 0 c δρ t c = = κ m δρ κ m P = ħ kf 5 m ρ = 5 3π /3 ħ m ρ5/3 κ = ħ kf 3 m, c = ħk F 3 m = v F < v F e /F 0, F x F0 /3, F RPA RPA, ħω X ph, Y ph λ ħω λ, Xph λ, Y ph λ, Xph λ Y ph λ Xλ Y λ X λ Y λ X λ, Y λ X λ X λ, Y λ, X λ, Y λ,. F λ F 0 = 0 O λ F 0 = 0 [ O λ, F ] 0 [ O λ, F ] Y λ [ a α a β, a αa β ] = δ αβ a α a β δ βα a αa β [ O λ, F ] = ph = ph αβ f αβ X ph λ [ a h a p, a αa β ] Yph λ [ a p a h, a αa β ] Xph λ fph + Yph λ fhp λ F 0 [ O λ, F ] = ph Xph λ fph + Yph λ fhp = X λ Y λ f f 3.39

49 3 RPA 48 f ph f hp RPA,, F F = f ph a pa h + f hp a h a p ph ω λ 3.9 A X λ + B Y λ = ħω λ X λ, BX λ + AY λ = ħω λ Y λ A B B A Y λ X λ = ħω λ Y λ X λ 3.4 O λ ω λ O λ = Yph λ a p a h Xph λ a h a p ph ħω λ X λ Y λ A B X ρ = ħω B A Y ρ ρ X λ Y λ X ρ Y ρ λ ρ X λ Y λ A B X ρ = ħω B A Y ρ λ X λ Y λ X ρ Y ρ A ph p h = A p h ph = ϵ p ϵ h δ hh δ pp + v p h h p = ϵ p ϵ h δ hh δ pp + v h p p h = A ph p h B ph p h = B p h ph = vp p h h = vpp hh = B ph p h, A = A, B = B 3.43 X λ Y λ A B X ρ = ħω λ X λ Y λ X ρ Y ρ B A Y ρ 3.4 ħω λ ħω ρ X λ Y λ X ρ = 0 Y ρ ω λ ω ρ X λ Y λ X ρ Y ρ = 0

50 3 RPA 49 RPA λ ρ = 0 O λ O ρ 0 = 0 [ O λ, O ρ ] [ 0 Oλ, O ρ ] = X λ ph X ρ ph Y ph λ ph = X λ Y λ X ρ Y ρ Y ρ ph, ω λ ω ρ λ ρ = 0, λ λ =, [ O λ, O ρ ] = X λ Y λ X ρ = δ Y ρ λρ Y λ X λ A B = ħω λ Y λ X λ B A Y λ X λ A B X ρ B A Y ρ Y λ X λ A B X ρ B A Y ρ = ħω λ Y λ X λ X ρ Y ρ = ħω λ Y λ X λ X ρ Y ρ = ħω ρ Y λ X λ X ρ Y ρ ħω λ + ħω ρ Y λ X λ X ρ = 0 Y ρ ħω λ + ħω ρ 0 Y λ X λ X ρ = Y ρ ph YphX λ ρ ph Xλ phy ρ ph = [ ] [ O λ, O ρ ] = O λ, O ρ = , O O,, 4. O λ O λ RPA, ω λ 0, RPA O λ O λ a pa h, a h a p RPA F = ph f ph a pa h + f hp a h a p

51 3 RPA 50 a pa h, a h a p O λ, O λ F = u λ O λ v λ O λ λ , 3.46 [ ] u λ = [ O λ, F ], v λ = O λ, F = [ O λ, F ] 3.39 u λ = X λ Y λ f f f αβ = f βα v λ = X λ Y λ f = Y λ X λ f f f 3.47 O λ = ph Xph λ a pa h Yph λ a h a p F = λ Xphu λ λ + Yph λ ph vλ a pa h Yphu λ λ + Xph λ vλ a h a p f ph = λ X λ phu λ + Y λ ph vλ, f hp = λ Yphu λ λ + Xph λ vλ f f = λ = λ X λ Y λ v λ u Y λ λ + X λ X λ Y λ X λ Y λ Y λ X λ Y λ X λ f f, X λ Y λ λ X λ Y λ Y λ X λ Y λ X λ 0 = = X λ ph X λ λ p h Y λ ph Y λ p h = δpp δ hh, λ X λ ph Y λ p h Y λ ph X λ p h = , 3.45 Xph λ X λ ph Yph λ Y λ ph = δ λλ, YphX λ ph λ XphY λ ph λ = 0 ph ph

52 3 RPA X λ ph = Cf ph ħω λ ϵ ph, Y λ ph = Cf ph ħω λ + ϵ ph C Xph λ, Y ph λ 3.39 λ F 0 = 4κ ħω λ ph ϵ ph f ph ħω λ ϵ ph = D RPA ħω λ 3.6. ϵ ph ϵ λ F 0 = ϵ f ph ħω λ ph κ dd RPAħω λ 3.49 dħω λ 3.6 RPA gt = fe iωt + f e iωt, f =, αt, time dependent Hartree Fock equation, TDHF equation iħ h t αt = HF + gt αt N ˆρt = it it i= iħ t ˆρt = N i= i= iħ it it + t N i= it iħ it t N N = h HF + gt it it it it h HF + gt = [ ] h HF + gt, ˆρt i= 3.50 gt = 0 ˆρ 0 HF.33 [ h 0, ˆρ 0 ] = 0, h 0 h HF [ˆρ 0 ] 3.5 gt, ˆρt ˆρ 0 ˆρt = ˆρ 0 + δ ˆρt

53 3 RPA 5 h 0 ˆρ 0 HF { for hole states α h 0 β = ϵ α δ αβ, α ˆρ 0 β = θ α δ αβ, θ α = 0 for particle states.30, ˆρt = ˆρt, δ ˆρ ph 0 h HF ˆρ = ˆρ 0 h HF [ˆρt] = h 0 + δh δh = αβ h HF δρ αβ = h HF δρ p h ρ αβ ρ ˆρ=ˆρ0 p h p h ˆρ=ˆρ0 + h HF δρ h p ρ h p ˆρ=ˆρ0 3.5, ph iħ [ ] [ ] t δ ˆρ = h 0, δ ˆρ + δh + g, ˆρ 0 iħ t δρ ph = ϵ ph δρ ph + δh ph + g ph 3.53 gt ˆρ δ ˆρt = ˆρ exp iωt + ˆρ expiωt δρ αβ = ρ αβ exp iωt + ρ βα expiωt ϵ ph δρ ph + δh ph = p h A ph p h ρ p h + B ph p h ρ h p exp iωt + p h Aph p h ρ h p + B ph p h ρ p h expiωt A ph p h = ϵ ph δ pp δ hh + h HF ph ρ p h, B ph p h ˆρ=ˆρ0 = h HF ph ρ h p ˆρ=ˆρ exp±iωt ħω ρ ph = p h A ph p h ρ p h + B ph p h ρ h p + f ph ħω ρ hp = p h A ph p h ρ h p + B ph p h ρ p h + f hp [ A B B ħω A 0 0 ] ρ ρ = f f 3.55

54 3 RPA 53 [ A B B ħω A 0 0 ] ρ ρ = 0 ρ = ρ = 0,, ω RPA ω λ, ρ = X λ, ρ = Y λ 0, ω = ω λ RPA RPA h HF αβ = E HF ρ βα ṽ αα ββ, 3.54 A, B = h HF αβ ρ β α ˆρ=ˆρ0 = E HF ρ βα ρ β α ˆρ=ˆρ A ph, p h = ϵ ph δ pp δ hh + ṽ ph hp, B ph p h = ṽ pp hh v ˆρ.8 ṽ αα ββ = v αα ββ, A B 3.0, 3., v ˆρ ṽ αα ββ v αα ββ, RPA v ṽ 3.55 ρ RPA ω λ X λ, Y λ R αβ, α β ω R αβ, α β ω = 0 a β a α λ λ a α a β 0 0 a α a β λ λ a β a α 0, ε ħω ħω λ + iε ħω + ħω λ + iε λ, αβ, α β ph hp iε ω = ± ω λ, g αβ = α β R αβ, α β ωf α β = p h R αβ, p h ωf p h + R αβ, h p ωf h p, 3.3 g ph = λ λ = λ X λ ph ħω ħω λ + iε p h Y λ ph ħω + ħω λ + iε X λ p h f p h + Y λ p h f h p p h Z λ ħω ħω λ + iε Xλ ph Y λ p h f p h + Xλ p h f h p Z λ ħω + ħω λ + iε Y ph λ 3.58 Z λ = ph Z λ = ph Xph λ f ph + Yph λ f hp = X λ Y λ f f Yphf λ ph + Xphf λ hp = Y λ X λ f f

55 3 RPA 54 g hp = λ Z λ ħω ħω λ + iε Y ph λ Z λ ħω + ħω λ + iε Xλ ph g = g λ Z λ ħω ħω λ + iε X λ Y λ Z λ ħω + ħω λ + iε Y λ X λ RPA A M = B B ħω A 0 0 M X λ Y λ = ħω λ ħω X λ Y λ g M = Z λ g ħω ħω λ + iε M λ [ = λ = λ X λ, M X λ Y λ Z Y λ λ X λ [ X λ Y λ X λ Y λ Y λ Y λ X λ = ħω λ + ħω Z λ ħω + ħω λ + iε M Z λ ] Y λ X λ, 3.48, 3.55 ρ Y λ X λ Y λ X λ Y λ X λ ] f f = f f ρ αβ = g αβ = α β R αβ, α β ωf α β ω λ X λ, Y λ, R, H + V t, V t = e εt F exp iωt + F expiωt, F = f αβ a αa β αβ t = H 0 = E 0 0 0, t φt t = H t = 0 e εt, ε + 0 t e εt =, t = e εt 0 V 0, φt = e iht/ħ ϕt iħ H t φt = + V t φt iħ t ϕt = V It ϕt, V I t = e iht/ħ V te iht/ħ

56 3 RPA 55 t = ϕt = 0, ϕt ϕt = 0 + t dt V I t ϕt iħ ϕt = 0 + t dt V I t 0 + t t iħ iħ dt dt V I t V I t ϕt ϕt = + Ut 0 Ut = t dt V I t + + t iħ iħ n dt V Ut 0 = iħ t dt e iht /ħ V t e iht /ħ 0 tn dt n V I t V I t n + = t dt e ih E 0 iεt /ħ F exp iωt + F expiωt iħ = e ih E 0t/ħ e iωt ħω H E 0 + iε F 0 e iωt ħω + H E 0 iε F 0 φt = e iht/ħ + Ut 0 = e ie 0t/ħ + W t 0 W t = e iωt ħω H E 0 + iε F e iωt ħω + H E 0 iε F φt a β a α φt = 0 a β a α a β a αw + W a β a α 0 = 0 a β a α 0 + ρ αβ e iωt + ρ βαe iωt ρ αβ = 0 a β a α ħω H E 0 + iε F F ħω + H E 0 + iε a β a α 0 R αβ α β a ω = 0 β a α ħω H E 0 + iε a α a β a α a β ħω + H E 0 + iε a β a α ρ αβ = α β R αβ α β ωf α β 3.60 R αβ α β ω response function

57 3 RPA 56 H n, H n = E 0 + ħω n n 3.59 R αβ α β ω = 0 a β a α n n a α a β 0 0 a α a β n n a β a α 0 ħ ω ω n + iε ω + ω n + iε n 3.6 n = 0, RPA 3.57, H RPA HF 3.59, HF H = α ϵ α a αa α 3.6 Rαβ, 0 α β ω = a β a α ph ph a α a β a α a β ph ph a β a α ħω ϵ ph + iε ħω + ϵ ph + iε ph ph = a pa h a β a α ph = δ αp δ βh Rαβ, 0 α β ω = δ αα δ ββ ph δ αp δ βh ħω ϵ ph + iε δ αh δ βp ħω + ϵ ph + iε = δ αα δ ββ θ α θ β θ α θ β ħω ϵ αβ + iε = δ αα δ ββ θ β θ α ħω ϵ αβ + iε 3.6, S f ω = n n F 0 δħω ħω n = n F 0 δω ω n, ω > 0 ħ 0 R f ω = f Rf = fαβ R αβ α β ω f α β = 0 F n n F 0 0 F n n F 0 ħ ω ω n n + iε ω + ω n + iε = n F 0 ħ ω ω n + iε n F ω + ω n + iε, ε + 0 Im R f ω = π ħ n x ± iε = P x iπδx n n n F 0 δω ω n n F 0 δω + ω n 0 ω > 0 ω + ω n > 0 δω + ω n = 0 S f ω = n n F 0 δħω ħω n = π Im R f ω 3.64

58 3 RPA HF RPA RPA 3.3 ħω λ X λ ph = p h Aph p h Xλ p h + B ph p h Y λ p h = ϵ ph X λ ph + p h ṽph hp Xλ p h + ṽ pp hh Y λ p h = ϵ ph X λ ph + p h ṽ ph hp 0 a h a p λ + ṽ pp hh 0 a p a h λ RPA 0 a pa p λ = 0 a h a h λ = 0 ħω λ X λ ph = ϵ ph X λ ph + 0 α β ṽ pα hβ a α a β λ 0 α β ṽ pα hβ a α a β λ = ħω λ ϵ ph 0 a h a p λ B X λ + A Y λ = ħω λ Y λ 0 α β ṽ hα pβ a α a β λ = ħω λ ϵ ph 0 a pa h λ θ β θ α 0 α β ṽ αα ββ a α a β λ = ħω λ ϵ αβ 0 a β a α λ θ β θ α λ α β ṽ αα ββ a α a β 0 = ħω λ + ϵ αβ λ a β a α 0, αβ µνṽ µν νµ R µ ν α β = = R µν µ ν θ β θ α ħω ϵ αβ + iε µ ν ħω ϵ αβ + iε = λ λ 0 a ν ṽ a µ λ λ a α a β 0 αν βµ 0 a α a β λ λ a ν a µ 0 ħω ħω λ + iε ħω + ħω λ + iε 0 a β a α λ λ a α a β 0 ħω λ ϵ αβ ħω ħω λ + iε λ + 0 a α a β λ λ a β a ħω λ + ϵ αβ α 0 ħω + ħω λ + iε [ 0 a β a α λ λ a α a β 0 ħω ħω λ + iε ħω ϵ αβ + iε ] 0 a α a β λ λ a β a α 0 ħω + ħω λ + iε ħω ϵ αβ + iε = R αβ α β S αβ α β ħω ϵ αβ + iε

59 3 RPA 58 S αβ α β = λ 0 a β a α λ λ a α a β 0 0 a α a β λ λ a β a α S ph p h = λ X λ ph X λ p h Y λ ph Y λ p h = δpp δ hh, S ph h p = λ X λ ph Y λ p h Y λ ph X λ p h = 0, S S αβ α β = S βα β α S αβ α β = δ αα δ ββ θ β θ α, R S αβ α β ħω ϵ αβ + iε = R0 αβ α β R αβ α β = R0 αβ α β + 0 R αβ µνṽ µν νµ R µ ν α β 3.65 µν µ ν Bethe Salpeter equation RPA R Ṽ µν µ ν = ṽ µν νµ 3.65 R = R 0 + R 0 Ṽ R = R R 0 0 Ṽ R 0, R, Ṽ RPA R det R 0 ωṽ = 0 R = R 0 + R 0 Ṽ R = R 0 + R 0 Ṽ R 0 + R 0 Ṽ R 0 Ṽ R , Ṽ ṽ αβ α β = κf αα f ββ, RPA R αβ α β R 0 f ω = αβ α β f αβ R 0 αβ α β ω f α β, R f ω = αβ α β f αβ R αβ α β ω f α β R f ω = Rf 0 ω + κ Rf 0 Rf 0 ωr f ω = ω κ Rf 0ω 3.6 R 0 f ω = αβ f αβ θ β θ α ħω ϵ αβ + iε = ph f ph ħω ϵ ph + iε ħω + ϵ ph + iε 3.67 ħω ± ϵ ph ε = 0 R 0 f ω = ph ϵ ph f ph ħω ϵ ph

60 3 RPA 59 RPA κ Rf 0 ω = RPA 3.57 R f ω = λ F 0 ħ ω ω λ + iε λ F 0 ω + ω λ + iε λ >0 κr 0 f ω λ = 0 ω = ω λ λ F 0 = lim ω ω λ 3.49 lim ħω ħω λ R f ω = λ F 0 ω ω λ κ R 0 f ω = κ dr0 f ω λ dω λ ω ω λ + ħω ħω λ R 0 f ω κ dr0 f ω λ dω λ ω ω λ + = κ dr0 f ω λ d ħω λ 3.8 Sum Rule F energy weighted sum S = n ħω n n F 0 = n ħω n 0 F n n F 0 0, n H, : H 0 = E 0 0, H n = E 0 + ħω n n n [ H, F ] 0 = n HF F H 0 = E 0 + ħω n E 0 n F 0 = ħω n n F 0 S = n 0 F n n [ H, F ] 0 = 0 F [ H, F ] 0 S = n 0 [ H, F ] n = ħω n 0 F n 0 [ H, F ] n n F 0 = 0 [ H, F ] F 0 S = 0 F [ H, F ] 0 0 [ H, F ] F 0 = 0 [ F, [ H, F ] ]

61 3 RPA 60 ħω n n F 0, energy weighted sum 3.68, H H = T + V, T = ħ m N i, V = i= N vi, j i,j=, F r i F = N fr i, vi, j [ V, F ] = 0 [ H, F ] = [ T, F ] = ħ m N i= i= [ i, fr i ] = ħ m N i [ i, fr i ] + [ i, fr i ] i i= i fr i = i fr i + fr i i, i fr i [ H, F ] = ħ m N i i fr i + i fr i i i= [ F, [ H, F ] ] = ħ m = ħ m N i= N [ fr i, i ] i fr i + i fr i [ fr i, i ] i= i fr i = ħ d 3 r fr m N i= δr r i S = ħ m d 3 r ρr fr, ρr = 0 N δr r i 0 i= ρr d 3 r ρr = = N ρr energy weighted sum S fr = z f = 0, 0, S = ħ d 3 r ρr = ħ m m N energy weighted sum R f 3.64 S = 0 dω ħω n n F 0 δω ω n = ħ π 0 dω ω Im R f ω R f 3.63 F = F R f ω = R f ω Im R f ω = Im R f ω = Im R f ω

62 3 RPA 6 S = ħ π 0 dω ω Im R f ω = ħ π S = ħ π 0 dω ω Im R f ω dω ω Im R f ω 3.69 RPA energy weighted sum S RPA λ ħω λ λ F 0 ħω λ, λ RPA S RPA = [ F, [ H, F ] ] 3.70, HF, H BS 3.65 R f = R 0 f + f R 0 Ṽ R 0 f + f R 0 Ṽ R 0 Ṽ R 0 f RPA energy weighted sum S RPA S RPA = S n, n=0 S n = ħ π [ n ] dω ω Im f R 0 ω Ṽ R 0 ω f 3.67 S 0 S 0 = ħ ω Im π dω ph = ħ dω ω ph 3.6 S x = ħω I = f ph ħω ϵ ph + iε ħω + ϵ ph + iε f ph δħω ϵ ph δħω + ϵ ph = ϵ ph f ph 3.7 ph S = π Im f αβf α β ṽ αβ βα θ β θ α θ β θ α I 3.7 dx g x, g x =, S n I n = dx g n x, g n x = x x ϵ αβ + iεx ϵ α β + iε x x ϵ αβ + iεx ϵ α β + iε } {{ } n+

63 3 RPA 6 C, g n x dx g n x = 0 C x = Ke iθ, 0 θ π K dx g n x + π K 0 dθ ike iθ g n Ke iθ = 0 K ϵ αβ iε K K g n Ke iθ K n e inθ K π dx g n x + ik n dθ e i nθ = 0 K 0 K n 0 { iπ, n = I n = dx g n x = 0, n n S n = 0 I 3.7 S RPA = S 0 + S S = Re f αβf α β ṽ αβ βα θ β θ α θ β θ α S RPA = f αβ f α β ṽ αβ βα θ β θ α θ β θ α 3.73 [ F, [ H, F ] ] = S HF + S res S HF = [ F, [ H HF, F ] ], S res = [ F, [ V res, F ] ] H HF H = H HF + V res, H HF = ϵ α a αa α, V res = v αβ α β 4 : a αa β a β a α : α αβα β hh v hh hh [ H HF, F ] = αβ ϵ αβ f αβ a αa β, ϵ αβ = ϵ α ϵ β [ F, [ H HF, F ] ] = f αβ ϵ α β f α β [ a αa β, a α a β ] = f αβ ϵ α β f α β δ βα a αa β δ αβ a α a β S HF = ϵα β f αβ f α β δ βα δ αβ θ α θ β = ϵ βα f αβ θ α θ β αβ

64 3 RPA 63 θ α =, θ β = 0 θ α = 0, θ β = S HF = ph ϵ ph f ph 3.74 S 0 = S HF 3.75 S res = F V res F F V res V res F F V res F V res F = F V res F, F V res = V res F S = F V res F, S = F V res S res = S S + S S S = F V res F = 4 fαβ f α β ṽ µν µ ν a αa β : a µa νa ν a µ : a α a β 3.4, 3. a αa β : a µa νa ν a µ : a α a β = a αa β : a µa νa ν a µ : a α a β + a αa β : a µa νa ν a µ : a α a β + a αa β : a µa νa ν a µ : a α a β + a αa β : a µa νa ν a µ : a α a β = a αa µ a ν a α a β a µ a νa β a β a ν a µa β µ ν a αa β = θ α δ αβ, a α a β = θ αδ αβ a αa β : a µa νa ν a µ : a α a β = θ α θ β θ α θ β δ µ αδ ν α δ µ α δ ν α δ µβ δ νβ δ µβ δ νβ S = f αβ f α β ṽ ββ αα θ α θ β θ α θ β a αa β a α a β : a µa νa ν a µ : = a αa µ a α a ν a β a µ a β a ν a β a ν a β a µ = θ α θ α θ β θ β δ µ αδ ν α δ µ α δ ν α δ µβ δ νβ δ µβ δ νβ S = f αβ f α β ṽ ββ αα θ αθ α θ β θ β µ ν

65 3 RPA 64 S S = f αβ f α β ṽ ββ αα θ α θ β θ β θ α = f αβf α β ṽ αβ βα θ α θ β θ β θ α α β f βα = f αβ, α β, α β S res = S S = f αβf α β ṽ αβ βα θ β θ α θ β θ α S S + S S = f αβ f α β ṽ αβ βα θ β θ α θ β θ α S = S res S RPA = S 0 + S = S HF + S res = [ F, [ H, F ] ] ph [ F, [ H, F ] ] = f pha ph p h f p h f phb ph p h f p h f phb ph p h f p h + f ph A ph p h f p h = f A f B B, A, B 3.0, 3. A, F f αβ = f βα λ F 0 = X λ RPA 3.9 f f Y λ f, λ F 0 = f X λ f f Y λ 3.77 S RPA = λ = λ ħω λ λ F 0 f A f B B X λ X λ Y λ f A Y λ f S RPA = f A f λ B B A Y λ X λ Y λ X λ f f

66 3 RPA S RPA = f A f B B f A f

67 A 66 A A. i α i N α α α N α α α N N i i p N p = p p pn N P i α i pi, pi =,,, N, P α α α N αp N α p α pn N = α p α p αn p N A., p p, j = pi i = p j A. α α α N N P β β β N N = α j = p i i = pj = p α N α p i β i i= p αn p N β β N α α α N N P β β β N N = α i β pi α α α N N P β β β N N i= = α α α N = N α i β pi, P = P, P i= N β p β β N p βn pn N â a m a m : â a m = a m a m m â, i â âi P âip a a a N = P âi a N p a p an pn i a pi âi a pi P âip a a a N N = a pip a p a p an pn = a pi a a a N N

68 A 67 a a a N N pi âpi, P P âip = âpi, N F â,, ân P F â,, ânp = F âp,, âpn i= F â,, ân = N âi N N P F P = âpi = âi = F, [ P, F ] = 0 i= i= A., p p { p + p = p N u N! P P u = c p u A., u, P i j P ij P ij Pij = P ij +, c ij = ± ij ijji c ij = c i c j c c j c i = c jc jc = c c p c, c p c,, P A., c =, c = P u = u P u = p u A.3 A.4 u A.3 N, A.4, A.3 A.4 S = P, A = p P N! N! P P

69 A 68 P P S = P P N! P, P P = P P P S = P = S N! P A = p P P = p p+p P P = p p P = p A N! N! N! P P P P P P P P S = SP = S, P A = AP = p A A.5 P S = S = S = S N! N! P P p P A = A = p p A = A N! N! P P S = S, A = A A.6 P = P P P A = p P = p P = A, N! N! S = S A.7 N α P P α S S α, α A A α A.5 P α S = P S α = S α = α S, P α A = P A α = p A α = p α A, α S A.3, α A A.4, S, A, N, : N, A.3 A.4, α i j i j α P ij α = α

70 A 69 A.5 P ij AP ij = A, A A α = 0 α = C N A α α α N N C N A A = A = A α α = C N α α α N A α = C N N! = C N N! P P N α p α α α N N α N N P α α α N N p α p α p αn pn α α α N α i, pi = i 0 α α = C N /N! C N = N! P N N!A α α α N N = p P α α α N N N! α α α N = α α α N N!. α N α N α N N, A = a ij deta ij P p a p a p a NpN = P p a p a p a pnn A.3 N α α N A N!A α α N N, β β N A N!A β β N N A N α, β, A α α N β β N A = P = P = P p α α N N P β β N N p α p αn pn β β N N N p α pi β i i=

71 A 70, A α α N β β N A = detd ij, d ij α i β j = δ αi β j A.8 d ij D ij, i F = A α α N β β N A = A α α N β β N A = N N d ij D ij j= N d ij D ij = N i,j= N α i β j D ij A.9 i,j= N fi A i=, A A = A = A = A A α α N F β β N A = N! α α N N A F A β β N N = N! α α N N A F β β N N = = N p α α N N P fj β β N j= P N N p α pj f β j k j α pk β k A.0 j= P i = pj i,,, N P, i pj = i P, i A α α N F β β N A = ij N α i f β j ij A. i,j= p pk β k P k j α pj=i fi = A.9, ij = D ij Ψ A α α N Ψ = N β f β j β β j β β j+ β N A j= β N β f β j A α α N β β j β β j+ β N A j= j d ij = α i β = δ αi β β, j A α α N β β j β β j+ β N A = N δ αi β ij i=

72 A 7 A. N N A α α N Ψ = α i f β j ij i= j= N F β β N A = j= β β f β j β β j β β j+ β N A A. A.4 α a α a α { a α, a α } a αa α + a α a α = δ αα A.3 { a α, a α } a α a α + a α a α = 0 A.4 { a α, a α } a αa α + a α a α = 0 A.5 a α a α 0 = 0 0 N N α α N a a α a α N 0 = a α i 0 A.6 a αa α = 0,, a α A.5 α α N p p, α α N a A.8 a α α N β β N a = detd ij A.7 i= a α a α a β a β = δ αβ a α a β a α a β a α a β = δ α β a α a β δ α β a α a β + a β a α a α a β a α α β β a = 0 a α a α a β a β 0 = δ α β δ α β δ α β δ α β = detd ij, N = A.7, N = k A.7 N = k + A.3 a α a β a β n = δ αβ a β a β n a β a α a β a β n = δ αβ a β a β n δ αβ a β a β 3 a β n + a β a β a α a β 3 a β n

73 A 7 a α a β n n a α a β a β n = i+ δ αβi a β a β i a β i+ a β n + n a β a β n a α i= A.8 a α α k+ β β k+ a = 0 a αk+ a α a β a β k+ 0 k+ = i+ δ α β i 0 a αk+ a α a i= }{{} β a β i a β i+ a β k+ 0 }{{} k k k + D = d ij i, j k D ij, N = k A.7 0 a αk+ a α a β a β i a β i+ a β k+ 0 = detd i, i+j detd ij k+ a α α k+ β β k+ a = d i i+ detd i = detd ij A.7 N = k +, A.6 i= A.5 a α, a α F αβ α f β a αa β A.9 A.8 F β β N a = α f β a αa β a β a β N 0 αβ = N α f β a α i+ δ βiβa β a β i a β i+ a β N 0 αβ i= a α a β i N = α f β i a α i+ a β a β i a β i+ a β N 0 i= α i, i+ N F β β N a = α f β i a β a β i a αa β i+ a β N 0 A.0 i= α N = α f β i β β i αβ i+ β N a i= α N A., F = fi i= A.9