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あゆみふしはら
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1 I 1 Density Matrix 1.1 ( (Observable) Ô :ensemble ensemble average) Ô en =Tr ˆρ en Ô ˆρ en Tr Â n, n =, 1,, Tr Â = n n Â n Tr Â I w j j ( j =, 1,, ) ˆρ en j w j j ˆρ en = j w j j j Ô en = j w j j Ô j emsemble average j w j Tr ˆρ en = j w j =1 w j 1 k w k w k =1 k 1. Entropy ˆη ˆη = ln ˆρ = j ln w j j j 1

2 S ˆη S = ˆη (= Tr ˆρˆη) = ln ˆρ = Tr ˆρln ˆρ = j w j ln w j S = S.1 micro canonical ensemble, E,N,V V N E E E k Ω(E) k, k =1,, Ω(E) E δe (macro) δe Ω(E) Ô mc = 1 Ω(E) Ω(E) k Ô k Ω(E) k=1 Ô time average = Ô mc Ô mc(e) = Tr ˆρ mc (E)Ô ˆρ mc (E) = w j (E) j j j w j (E) = { 1 Ω(E), E = E j, otherwise E = E j E δe/ <E j <E+ δe/ Ω(E)

3 S = j=1 w j ln w j S mc (E) S mc (E) =lnω(e), Ω(E) =e Smc(E). canonical ensemble, T N,V (System) (Bath) (System+Bath) T 1 S T = E N,V T S,B System Bath T S = T B = T Bath T Ĥ E j j Ĥ j = E j j ˆρ c (T ) = 1 e βe j j j Z j = 1 Z e βĥ Z Z = j e βe j = Tr e βĥ e βf F (Helmholtz) ( β = 1 T ) T [K] k B [K] S(E) =k B lnω(e) S S k B T k B T S = ln ˆρ c c F = E TS E = Ĥ c df = 3

4 .3 grand canonical ensemble T µ V (System) (Bath) (System+Bath) T (Chemical Potential) μ μ S T = N E,V T S,B μ S,B System Bath T S = T B = T μ S = μ B T μ ˆN Ĥ [ Ĥ, ˆN ]= E j,n Ĥ E j,n = E j E j,n ˆN E j,n = N E j,n Ξ ˆρ gc (T ) = 1 e β(ej μn) E j,n E j,n Ξ j,n = 1 ˆN) e β(ĥ μ Ξ Ξ = j,n e β(e j μn) = Tr e β(ĥ μ ˆN) e βj J Ξ Z N N Ξ= Z N e βμn N S = ln ˆρ gc gc J = F μn = E TS μn E = Ĥ gc N = ˆN gc dj = 4

5 .4 T - p T -p ensemble T p N (System) (Bath) (System+Bath) T p p S T = V E,V T S,B p S,B System Bath T S = T B = T p S = p B T μ E j,v Ĥ E j,v = E j E j,v ˆV E j,v = V E j,v T - p ˆρ gc (T ) = 1 Y Y = j,v e β(e j+pv ) E j,v E j,v = 1 Y e β(ĥ+p ˆV ) j,v e β(e j+pv ) =Tre β(ĥ+p ˆV ) e βg G T - p T - p S = ln ˆρ Tp Tp G = F + pv = E TS + pv E = Ĥ Tp V = ˆV Tp dg =.5 E N V E, N, V, E V e, N V n, e, n : H H 1 V V O V 5

6 3 E,N,V E,N,V S S = S(E,N,V ) ds = S S S de + dn + dv E N,V N E,V V E,N 1 S T =, E N,V TdS = de μdn + pdv μ S T =, N E,V de = TdS + μdn pdv p S T = V E,N E = E(S, N, V ) S, N, V E E E T =, μ =, p = S N V S, N, V de = Legendre E F, (S T ) N,V S T F E TS S,V df = SdT + μdn pdv F F = F (T,N,V ) T,N,V F F F S =, μ =, p = T N V N,V T,V T,N,V df = ( ) F J, (N μ) N μ J F μn dj = SdT Ndμ pdv J J = J(T,μ,V ) T,μ,V J J J S =, N =, p = T μ V μ,v T,μ,V dj = ( ) J(T,μ,V ) V T μ J(T,μ,αV ) = αj(t,μ,v ) α α =1 V ( J V ) T,μ = J J = pv T,V S,N T,N T,μ 6

7 F G, (V p) V p G F + pv dg = SdT + μdn + Vdp G G = G(T,N,p) T,N,p G G G S =, μ =, V = T N p N,p T,N,p dg = ( T -p ) G(T,N,p) N T p G(T,αN,p) = αg(t,n,p) α α =1 N( G N ) T,p = G G = μn dg T,p SdT + Ndμ Vdp= Gibbs-Duhem E H, (V p) E = E(S, N, V ) V p H E + pv (enthaly) dh = TdS + μdn + Vdp H H = H(S, N, p) S, N, p H H H S =, μ =, V = S N p N,p S, N, p dh = S,p T,N S,N x, y, z x y z = = (x, z) (y,z) = (x,z) (x,y) (y,z) (x,y) x(y,z) y z(y,z) y = ( z,y ) x ( z x ) y x(y,z) z z(y,z) z da = Xdx + Ydy X y A x x u Y = (dda =Maxwell srelation) x y y = X + Y x u 7

8 II N N Φ( r 1,, r N ) (Bose) (Fermi) Φ(, r i,, r j, ) = +Φ(, r j,, r i, ) (Boson) (1) Φ(, r i,, r j, ) = Φ(, r j,, r i, ) (Fermion) () Fermi-Dirac (Fermi ) Bose-Einstein (Bose ) Fermi Bose r i = r j (i j) Φ(, r i,, r j, )= 4. Ĥ = k ɛ kˆn k ɛ k ˆn k k ˆN ˆN = k ˆn k ˆn k n k n k =, 1 n k =, 1,, 3,, n k =, 3, 8

9 p = h k ɛ k = p m = h k m (V = L 3 L ) φ( r) = 1 e i k r = 1 e i p r/ h V V k φ(x, y, z) = φ(x + L, y, z) =φ(x, y + L, z) =φ(x, y, z + L) k = (kx,k y,k z ) k α = π L n α, n α =,, 1,, 1,, (α = x, y, z) (3D: V = L 3 )(ɛ k = p m k = D(ɛ) = Vg 1/ π n x n y n z k= π L (nx,ny,nz) m 3/ 3 ɛ1/ h g dɛd(ɛ) (L ) 4.3 Ξ = Tr e β(ĥ μ ˆN) =Tre β k (ɛ k μ)ˆn k = ( ) e β k (ɛ k μ)n k = ( k n k =,1 k = (1 + e β(ɛk μ) ) k n k =,1 e β(ɛ k μ)n k ) k n k n k = 1 Ξ Tr ˆn ke β(ĥ μ ˆN) = 1 Ξ Tr ˆn ke β k (ɛ k μ)ˆn k = 1 ln Ξ β ɛ k 1 = e β(ɛk μ) +1 f(ɛ k) f(ɛ) 9

10 S f = f(ɛ) S = k ( ) fln f +(1 f)ln (1 f) 4.4 Ξ = Tr e β(ĥ μ ˆN) =Tre β k (ɛ k μ)ˆn k = ( ) e β k (ɛ k μ)n k = k n k =,1,, k = 1 1 e k β(ɛ k μ) ( ) e β(ɛ k μ)n k n k =,1,, k μ<ɛ k μ< k n k n B (ɛ) n k = 1 ln Ξ β ɛ k 1 = e β(ɛk μ) 1 n B(ɛ k ) S n B = n B (ɛ) S = ) ((1 + n B )ln (1 + n B ) n B ln n B k 5 D(ɛ) =V 1/ m 3/ π h ɛ 1/ N μ(t,n) 3 N = k ˆn k = k f(ɛ k )= dɛd(ɛ)f(ɛ) E = k ɛ k ˆn k = k ɛ k f(ɛ k )= dɛɛ k D(ɛ)f(ɛ) 1

11 5.1 T = ɛ F ɛ F k F ɛ F = h k F m k F = (3π n e ) 1 3 n e = N/V : electron density T F : k B T F = ɛ F 5. T T <<T F f(ɛ) ɛ μ k B T 1 T<<T F (ɛ μ T << T F dɛg(ɛ)f(ɛ) = μ dɛg(ɛ)+ π 6 g (μ)(k B T ) + O((k B T ) 4 ) μ(t,n)=ɛ F π 6 d dɛ ln D(ɛ) (k B T ) ɛ=ɛf ( ) C V = E T = π 3 D(ɛ F )(k B T ) 11

12 6 N V ρ = N V ) ( D(ɛ) = V m3/ 1/ π h ɛ 1/ ) 3 N = k f(ɛ k )= ρv Q (T ) = ζ 3/ (e βμ ) V Q (T ) = ( π h mk B T ) 3 ζ 3/ (λ) = π 1/ dɛd(ɛ)n B (ɛ) ( ) x 1/ λ 1 e x 1 = λ k k 3/ V Q (T,m) T μ< ζ 3/ (λ) λ ζ 3/ (e βμ ) < ζ 3/ (1) ρv Q (T C )=ζ 3/ (1) T C k=1 μ = k BT N (when V, N ) k B T C = π h ρ 3 m ζ 3/ (1) (N N ) V Q(T ) V = ζ 3/ (1) N ɛ = T C [ 3 ] T N = N 1 T C N O(N) ( ) (*) 7 k H = hω(ˆn i + 1 ) mode, i 1

13 ˆn i i (photon) hω 1 (Black Body Radiation) 7.1 F S =, = N N T,V μ = Ĥ wall N Ĥ + Ĥwall Z = Tr e β(ĥ+ĥwall) E,V N = ˆN = 1 β(ĥ+ĥwall) Tr ˆNe Z Tr Ĥ E i(n) N Ĥwall Z = e βe i(n) N i N = 1 Z N e βe i(n) i N 7. Planck n i = ˆn i = n(ω i )= 1 e β hω i 1 Planck ω i = ck i c ω k N(ω) N(ω) = 4 ) π 3 /( 3 πk3 L 13

14 L D = d dω N D(ω) =V ω π c 3 (V = L 3 : ) 7.3 E = hω i ˆn i = i = 1 V π c 3 h 3 (k BT ) 4 1 D(ω) hω e β hω 1 dxx 3 1 e x 1 (k B T ) 4 Stefan-Boltzmann I(ω) =D(ω) hωn B (ω) I(ω) ω 3 e hω k B T hω >> k B T, Wien ω k B T hω << k B T, Rayleigh Jeans (phonon) 3 a al n = a(n x,n y,z z ) n x,y,z =1,, 3,,L δ r n 1 T T = 1 (δ r n ) V V = V ({})+ δ r n V n δ r n δ rn= + 1 ( ) δ r n δ r m V δ r n δ r + m δ rn=δ r m= n m {δ r n = } = { } 3 3 C n {C nm } αβ V δrn δr α n β, α,β = x, y, z V () = V = 1 δ r n C nm δ t r m n m 14

15 V C nm = C( n m) C t ( n m) =C( m n) V (V ({δ r n + s}) =V ({δ r n }), s ) n C( n) = δ r n = 1 a( k)e i k n L 3 δ r (nx+l,ny,n z) = δ r (nx,ny,n z) k =(k x,k y,k z ) k α = π L l α, α = x, y, z l α =1,, 3,,L L 3 δ r n a( k) = a( k)( a( k) a( k) ) V = 1 C( k) = l k k a(k) C( k) a(k) e i k l C( l) C( k) C( k) = Uγ( k)u γ( k)=diag(γ 1,γ,γ 3 ) det ( C( k) γ j ( k)i) = ( ) V = 1 γ j ( k) b j ( k) k j=1,,3 b( k) = (b1 ( k),b ( k),b 3 ( k)) = a( k)u T = 1 k H = k ω j ( k) = γ j ( k) a(k) a(k) = 1 j=1,,3 k j=1,,3 ḃj( k) { 1 ḃj( k) + 1 ω j ( k) b j ( k) } k j k k H = ( hω j ( k) n j ( k)+ 1 ) k j=1,,3 15

16 hω 1 (*) C( k = ) = ω j ( ) = C( k)= C ( k) ω j ( k)=ω j ( k) (*) f( k, ω ) [ ω j ( f 1 1 k α k β ] f k)=k ω k k k α k β [...] α,β=x,y,z ω j ( k)=ck Debye 8. Debye ω j ( k)=ck 3/ 3L 3 =3V/a 3 Debye ω D { 3 D(ω) = Vω π c ω ω 3 D ω>ω D Debye ω D 3V = ω D = D(ω) = ( 6π c 3 a 3 dωd(ω) ) 1 3 9V ω a 3 ωd 3 ω>ω D E = dωd(ω) hωn B (ω) ωd = dω 9V ω 1 a 3 ωd 3 hω e β hω 1 = 9V (k B T ) 4 xd a 3 ( hω D ) 3 dx x3 e x 1 x D (T )= Θ D T = hω k B T Θ D Debye T<<Θ D x D (T ) T 3 C = E T ( T Θ D 16 ) 3

17 9 9.1 H N 1/ S 1,j=1,,N ( S j = S(S +1),S= 1/) M j = μs j M = N j=1 Mj N Ĥ = M H N = μ Ŝj H N = μh Ŝj z j=1 j=1 H z Z =Tre βh Tr S z 1,Sz,,Sz N Sz 1 Sz Sz N Ŝ z j Sz j = Sz j Sz j (Sz j = ±1 ) Ŝ z k Sz j = Sz j Ŝz k (k j) Z = S z 1 =± 1,,Sz N =± 1 e βμh N j=1 Sz j = S z 1 =± 1 N e βμhsz j SN z =± 1 j=1 = (e 1 βμh + e 1 βμh ) N =(cosh 1 βμh)n f (= F/N, Z = e βf ) f = 1 β ln cosh 1 βμh m = 1 N M z m = μ N N S z = μ 1 N Z Tr = 1 N Ising : j=1 N j=1 (βh) ln Z = 1 βμh μ tanh S z e βμh N j=1 Sz N 17

18 i, j JSi zsz j (Ising ) Ising 1 H = J N 1 j=1 S z j S z j+1 + H J> ( ) J < ( ) 18

5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E

5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N