stexb08.dvi

B 1 1.1 V N 1 H = p 2 i 2m i 1. Z = β =(k B T ) 1. 1 h 3N N! exp( βh)d p 1 d p N d x 1 x N 2. F ( F = k B T log Z ) 3. ( ) F p = V T 1.2 H μ μh μh N H T 1. Z Z 1 N Z 1 Z 2. F S ( ) F S = T 3. U = F + TS 4. M = ( F/ H) 5. C =( U/ T) H

1.3 N 1 H = N i=1 p 2 i 2m 1. Z N = 1 h 3N N! Ξ= λ N Z N N= exp( βh)d p 1 d p N d x 1 d x N β =(k B T ) 1. 2. N N = N= Nλ N Z N Ξ = λ λ ln Ξ 3. pv = k B T ln Ξ pv = N k B T

2 2.1 m ω H = p2 2m + 1 2 mω2 q 2 q p 1. Z cl = 1 h exp( βh)dqdp Z cl β =(k B T ) 1 2. ε n ( ε n = n + 1 ) hω 2 (n h = h/(2π)) Z = exp( βε n ) n= 3. Z 4. N ( ) Z N 5. U 6. C =( U/ T) N U = β log Z N 2.2 ( ) ω ω +dω g(ω)dω g(ω) ( = V 1 + 2 2π 2 c 3 c l 3 t ) ω 2 9N ω 3 D ω 2 ω ω D g(ω) = ω>ω D

(Debye ) c l c t ω D Debye g(ω)dω =3N( ) 1. ω j Z j 2. Z (Z = i Z i) F F = k B T i log Z i g(ω) F g(ω) 3. U [ ( )] F U = T 2 T T V,N 4. C V =( U/ T) V,N 5. C V

3 3.1 1, 2, 3,,l,, ε 1,ε 2, ε 3, ε l,, n 1,n 2,n 3,,n l,, n l n l =1 n l = N E N = l n l E = l n lε l 1. Z N Z N = {n l } exp( β l n l ε l ) (β =(k B T ) 1 ) N Z N Ξ(β,μ) Ξ(β,μ) = exp(βμn)z N 2. j n j N= n j = N n j exp( βe + βμn)/ξ {n l } ε j μ N = j n j 3.2 1. x = x = L L ( V =) φ(x) 2. φ(x =)=φ(x = L) ( ) k k = 2πn (n = L 3. N k F N k F 4. ɛ ɛ = k2 h 2 2m

N = 2L π 2mɛ F h 2 ɛ F ɛ F +dɛ F dn D(ɛ F )= dn dɛ F 5. ɛ F 6. D(ɛ F )= L3 2π 2 (2m h 2 )3/2 ɛ F 3.3 V Fermi ( ) p p +dp 2 V 4πp2 dp h 3 1. D(ε) D(ε) 2. N Fermi ε = p2 2m D(ε)dε =2V 4πp2 dp h 3 f(ε) = N = 1 e β(ε μ) +1 D(ε)f(ε)dε (T =) μ 3. (k B T 1) f(ε)d(ε)dε = μ D(ε)dε + π2 6 (k BT ) 2 D (μ)+ (1) μ(t ) (k B T )

4. E E = εd(ε)f(ε)dε (1) (k B T ) 2 E = μ εd(ε)dε + π2 6 (k BT ) 2 D(μ )+ 5. C V =( E/ T) V

4 Bose 4.1 Fermi Bose Fermi l, ε l, n l (l =1, 2, 3,... ) Fermi n l =1 n l = N N = l n l E = l ε ln l Ξ μ Ξ(μ, β) = exp( β ε l n l )exp(βμn) N= l {n l }: l n l=n = exp( β {n l } l (ε l μ)n l ) {n l } l n l = N l Ξ(μ, β) = exp( β(ε l μ)n l ) {n l } l Ξ(μ, β) = exp( β(ε l μ)n l ) l n l =1, Ξ(μ, β) = (1 + exp( β(ε l μ)) l k n k n k = N n k exp( β (ε l μ)n l )/Ξ(μ, β) {n l }: l n l=n l = β ε k log Ξ(μ, β) n k = 1 exp(β(ε k μ)+1 μ ρ = N /V ρ = 1 1 V exp(β(ε l μ)) + 1 l Bose Bose n l, 1, 2,..., Fermi 1. Bose Ξ(μ, β) 2. Bose k n k

4.2 Bose-Einstein Bose Bose μ( ) ρ = 1 V ( ) 2m 3/2 ε 1/2 n l =2π h 2 dε (2) l exp(β(ε μ)) 1 ε 1. (2) h(t,μ) ε = p2 2m ρ = h(t,μ), ( ) 2m 3/2 ε 1/2 h(t,μ) 2π h 2 dε (3) exp(β(ε μ)) 1 μ T ρ (μ = μ(t,ρ)) μ h(t,μ) h(t,) h(t,) T h(t,) x 1/2 π exp(x) 1 dx = 2 ζ(3/2) 2. ρ h(t,) ρ (3) μ (β 1) μ μ(t )=k B T {ln(ρλ 3 T ) ρ ( 2 h 2 2 λ3 T + }, λ T 2πmk B T ) 1/2 3. ρ = h(t c, ) T c? μ μ = μ = ε = n ε= μ = (ε =) ε = ρ T T c (2) ρ = ρ + h(t,)

T < T c μ = ε> ε = ρ 1 Bose-Einstein Bose-Einstein T c ρ =2.3 1 22 cm 3 4 He T c (ζ(3/2) 2.612 ) 4. T<T c C V = ( ) E T V,N (β 1) 5. Bose-Einstein

5 5.1 E rot I E rot = 1 1 2I p2 θ + 2I sin 2 θ p2 φ θ, φ p θ,p φ I 2r m I =2mr 2 Z rot Z rot = 1 h 2 π 2π dθ dφ dp θ dp φ exp( βe rot ) z x φ θ y 1. Z rot = 2Ik BT h 2 2. S rot C rot C rot = T S rot T, S rot = F rot T, F rot = k B T ln Z rot 5.2 J E rot = J 2 2I

1. j 2. 3. 2 4. Z QM rot = Z QM rot = (2l +1)exp( βl(l +1) h 2 /(2I)) l= (2l +1)exp( l(l +1) Θ rot T ), Θ rot = h2 2Ik l= B Θ rot Θ rot.741å 5. Z QM rot 5.3 E v (n) =(n + 1 2 ) hω v, n =, 1, 2,... (ω v ) 1. Z v Z v = exp( βe v (n)) n= 2. C v C v = T 2 F v T 2, F v = k B T ln Z v Θ v = hω v /k B 6K 2 (H 2 ) / singlet( ) l=even triplet( ) l=odd

6 6.1 1 f(ε) = exp(β(ε μ)) ± 1 ε + exp( μβ) 1 f(ε) exp( βε) 1. N μ N = i f(ε i ) exp( μβ) 1 N = i exp( βε i + βμ) =exp(βμ) i exp( βε i ) exp( βε i )= i D(ε)exp( βε)dε D(ε) =V 2π(2m)3/2 h 3 ε 1/2 ( ) 2πmkB T 3/2 exp(βμ) = N V h 2 2. exp( μβ) 1? 3. r ( N r V ) 1/3 λ D p 2 2m = 3 2 k BT p λ D = h/p λ D r?

6.2 3 He 3 He 3 He ( 1/2) 1 36.8cm 3 1. (ε F = h 2 kf 2 /(2m)) 2. (ε F ) 3. (T F = ε F /k B ) 6.3 3 He C v C v =24.1T [J/(mol K)] C v = γt, γ = k2 B mk F 3 h 2 V V k F 3 He / 1. 3 He 36.8cm 3 γ 3 He m m =4.98 1 27 kg 2. γ m m m? 6.4 ( ) n n A i, j (A) i,j a i,j [A, B] [A, B] =AB BA A (Tr A) TrA = 1. X, Y, Z n n n a i,i i=1 [X, [Y,Z]] + [Z, [X, Y ]] + [Y,[Z, X]]

2. n n A, B, H [A, H] =, [B, H] = [[A, B],H] 3. A, B n n Tr(AB) =Tr(BA) 4. A, L n n x e xl Ae xl = A + x[l, A]+ x2 2! e xl Ae xl x3 [L, [L, A]] + [L, [L, [L, A]]] + 3! X exp(x) e X = n= 1 n! Xn 5. ( i σ y = i ) ( cos φ sin φ exp( iφσ y )= sin φ cos φ ) 6. A(x) A(x) ( cos x sin x A(x) = sin x cos x A(x)A(y) = A(x + y) ) 7. B B = ( 1 2 2 1 exp(xb) ) x

8. n n X Y [[X, Y ],X]=, [[X, Y ],Y]= e X e Y =e X+Y + 1 2 [X,Y ]

7 7.1 ( ) (Zeeman) ε ε = p2 2m + σμ BH, μ B = e h 2m e c σ = ±1 σ =+1 σ = 1 1 μ e = μ B σ 1. μ 2μ B H N + N N + = V 4π 3h 3 (2m e(μ μ B H)) 3/2 N = V 4π 3h 3 (2m e(μ +μ B H)) 3/2 ε μ μbh μ H B 2. N μ (H) N = N + + N μ (H) 3. M = μ B (N + N ) M = χh + χ χ = 3 N 2 μ2 B μ (H =)

4. ( 19 K).8621[g/cm 3 ] 4s 1 χ[emu/g] 7.2 1. E N ε F E = 3 5 Nε F ( D(ε) E N ) E = εf D(ε) =V 4π(2m)3/2 h 3 ε 1/2 εd(ε)dε, N = εf D(ε)dε 2. P V E 3. B PV = 2 3 E B = V P V B = 5 3 P = 1E 9V 4. n ε F n =1.4 1 22 [cm 3 ] ε F =2.12[eV] 7.3 H( ) g E ψ g H(g) ψ(g) = E(g) ψ(g)

1. E(g) g = ψ(g) H(g) g ψ(g) 2. Z(g) = i exp( βe i(g)) g ln Z(g) g H(g) = β g A = TrA exp( βh) Z 3. H(ω) = p2 2m + mω2 x 2 2 ω g ω ln Z(ω) ω H(ω) ω x 2

8 8.1 1. 7 1 8 [m] 2 1 33 [g] N n[cm 3 ] 2. 2 1 9 [cm] ε F 1 2 (T F = ε F /k B ) 3. ε mc 2 p c ε = pc D(ε) D(ε) =V 8πε2 h 3 c 3 N N εf D(ε)dε ε F ε F 3 hcn 1/3 n 4. 1 1 12 [g/cm 3 ] ( )1 8 [ev] (.5[MeV] ) 5. β n p + e + ν.8[mev] β p e n p ( 1 )

8.2 ( ) R M (ρ = M/((4π/3)R 3 ) 1. U g G U g = 3 GM 2 5 R 2. He M p N e N e =2M/(4M p )=.5M/M p ( μ e N e = M/(μ e M p ) μ e =2 ) N e R U e m e h N e R ( ) 3. U = U e + U g R min 4. B 1.6 B 1 8.3 1. (ε pc) U e εf εd(ɛ)dε R M 2. U = U e + U g U e U g R M cr U e = U g M cr ( ) 2 1.1 2

9 μ (+) ( ) Ising( ) ±1 σ i (i =1,...,N, N = ) J J H = J σ i σ j <i,j> <i,j> H H = J σ i σ j μ H σ i <i,j> i J> 9.1 Ising ( ) N μ B N N H = J σ i σ i+1 μ B H σ i, σ i = ±1 i=1 i=1 Ising J> H σ N+1 = σ 1 N-1 N 1 2 3 4 1. Z N Z N =TrT N T ( ) e K+h e K T =, K = βj, h = βμ B H (4) e K e K h ( ) 2. T (N ) 1 f = k B T lim N N ln Z N

3. (H =) E 4. (H =) C V =( E/ T) V,N 5. m = μ B N i σ i = μ B N h ln Z N 6. (H 1) m = χ H χ 7. (N ) σ 1 σ m σ 1 σ m =e (m 1)/ξ ξ( ) 9.2 ( ) ( ) d H = J σ i σ j μ B H σ i, σ i = ±1 <i,j> i <i,j> i, j 1. σ i σ = m σ i m σ i σ j =((σ i m)+m)((σ j m)+m) m(σ i + σ j ) m 2 N H = JzN 2 m2 h eff ( H MF ) ( )h eff z z =2d 2. H MF σ i m ( ) m i σ i m =tanh(zkm + h), K = βj, h = βμ B H (5) H = m = (m, ) T c

3. f = F/N = k BT N ln Z MF, Z MF = exp( βh MF ) {σ i } (H =) f m T>T c T<T c 4. f m = (5) 5. (T >T c ) χ = m H H= (T T c ) 1 (Curie-Weiss) 6. (T = T c ) m H 1/3 7. (H =) T<T c m T c m (T c T ) 1/2 9.3 (Landau) F M H F = F + AM 2 + BM 4 + HM M H M F = F + AM 2 + BM 4 A, B B(T ) > 1. F (M) T >T c M = T<T c M = A?

2. F (T,M)=F (T )+a(t T c )M 2 + bm 4 a, b T >T c T <T c F M 3. F M = T <T c M 4. S = ( F/ T) ( F / T T c M ) 5. T = T c C = T ( ) S T 1 H = J <i,j> σ i σ j μ H i σ i H σ i = ±1 1. M = i σ i M = k BT μ 2. χ =( M / H) H= 3. E = H H log Z χ = μ k B T { M 2 M 2 } E = 1 log Z, β = β k B T 4. C = E/ T C = 1 k B T 2 { H2 H 2 }