30

3

............................................2 2...........................................2....................................2.2...................................2.3.............................. 2.3......................................... 3.3...................................... 3.3.2..................................... 3.3.3.......................................... 4.4....................................... 4 2 6 2........................................ 6 2.2.............................................. 6 2.3................................................ 7 2.3....................................... 8 2.4 n.......................................... 9 3 3........................................ 3........................................ 3..2................................... 3.2..................................... 3.3....................................... 3.3......................................... 3.3.2 Liouville...................................... 3.4.......................................... 2 3.5....................................... 2 3.6......................................... 2 3.7.......................................... 3 4 4 4............................................. 4 4.2..................................... 4 4.2............................................. 4 4.2.2..................................... 4 4.2.3....................................... 5 4.3......................................... 7 4.3.................................... 7 4.3.2................................. 7 4.4............................... 2 4.4....................................... 2 4.4.2...................................... 2 4.4.3..................................... 22 i

5 25 5.......................................... 25 5.2.......................................... 26 5.3....................................... 27 5.4....................................... 28 5.5......................................... 28 5.6....................................... 29 5.7 3.......................................... 3 6 32 6.......................................... 32 6.2......................................... 33 6.3....................................... 34 6.4.................................... 35 6.4....................................... 35 6.4.2....................................... 35 6.4.3........................................ 36 6.4.4.......................................... 37 7 ( ) 38 7............................................... 38 7.2........................... 38 7.3...................................... 4 7.4................................. 4 7.5.............................................. 4 8 43 8..................................... 43 8.2....................................... 43 8.3............................................ 45 8.4............................................ 46 8.5.............................................. 47 9 49 9............................. 49 9.2............................................ 49 5............................................. 5.2 ( ).......................... 5.3............................................ 52.4......................................... 52 ii

. dq dw du Á U d'w d'q du = dq + dw (.) ( ) dw = pdv du = dq pdv (.2) ( ) pdv ( 2) dq dw d Q d W.2 2.2. 2 A B A B 2 ( ) ( ) 2 ( ).2.2 R ( T ) R 2 ( T 2 ) R Q R 2 Q 2 Q T + Q 2 T 2 (.3)

η η T T 2 T (.4) ( ) n (.3) n i= Q i T i (.5) T 2 lr 2 dq T (.6) T T T Q Q 2 lr (T ) W.2.3 L L 2 2 L dq/dt S() L L 2 dq/dt S(2) : Ô dq = S() L T dq = S(2) L 2 T 2 L d Q/dT dq = S(2) S() T L 2 (.7) dq T L 2 L 2 : Ô = ds (.7) S (.6) dq T ds (.8) (dq = ) ds (.9) 2 ( ) dq = T ds 2

du = pdv + T ds (.).3 (.) U V S H F G.3. L dl dl = Xdx + Y dy+,... (x, y,...) (X, y,...) L = L Xx L d L = dl Xdx xdx = xdx + Y dy+,... (.) d L (X, y,...).3.2 H (V, S) (S, p) H = U + pv (.2) dh = du + pdv + V dp = T ds + V dp (.3) F (V, S) (V, T ) 3

F = U T S (.4) df = du T ds SdT = pdv SdT (.5) G (V, T ) (T, p) G = F + pv = H T S = U + pv T S (.6) dg = V dp SdT (.7).3.3 δs =, δ 2 S < (.8) ( ) d Q T ds d Q T ds d Q = T ds T ds δf =, δ 2 F > (.9) ( ) d Q T ds d Q = du d W = du + pdv du T ds + pdv df = d(u T S) = du T ds SdT SdT = pdv = df = du T ds δg =, δ 2 G > (.2).4 < ϵ > < ϵ >= 2 m < v2 >= 2 m < v2 x + v 2 y + v 2 z >= 2 m < v2 x > + 2 m < v2 y > + 2 m < v2 y > (.2) 4

2 m < v2 x >= 2 m < v2 y >= 2 m < v2 z >= kt 2 (.22) 2kT k R N A k = R/N A (.23) k B < ϵ >= 3 kt (.24) 2 U = N < ϵ >= 3 NkT (.25) 2 C V = ( ) U T V = 3 Nk (.26) 2 2 3 2 5 < ϵ >= 5 kt (.27) 2 U = N < ϵ >= 5 NkT (.28) 2 C V = ( ) U T V = 5 Nk (.29) 2 5

2 2. n f(x, x 2,, x n ) m g i (x, x 2,, x n ) =, {x i } (i =, 2,, m) m L({x i }, {α i }) = f(x, x 2,, x n ) α i g i (x, x 2,, x n ) (2.) i= {α i } n + m L =, x i (i =, 2,, n) (2.2) L =, α i (i =, 2,, m) (2.3) (2.4) {x i } {α i } [ 2. ] g(x, y, z) = x 2 + y 2 + z 2 = f(x, y, z) = x + y + z 2.2 [ 2.2 ] () 5 5 (2) 5 3 (3) 5 3 n m np m = n! (n m)! (2.5) n m n m (2.6) [ 2.3 ] () 6 2 4 2 6

(2) 2 4 4 3 n m n m nc m = n! m!(n m)! (2.7) n n n 2 n m ( n + n 2 + + n m = n) m n! n!n 2! n m! (2.8) [ 2.4 ] 3 5 4 2 3 3 3 2.3 [ 2.5 ] 6 () 3 (2) [ 2.6 ] 6 2 () 6 (2) [ 2.7 ] i p i p : p 2 : p 3 : p 4 : p 5 : p 6 = : : 2 : 2 : 3 : 3 () 3 (2) m x x 2 x 3 x m n n 2 n 3 n m x i p i p i = n i (2.9) m n i i= m p i = (2.) i= 7

m x i x f(x) x x + dx f(x)dx x x + dx p(x)dx = f(x)dx f(x)dx (2.) p(x) p(x)dx = (2.2) 2.3. x i A i A m A = p i A i = i= m n i A i i= (2.3) m n i i= x A(x) A A = p(x)a(x)dx = f(x)a(x)dx f(x)dx (2.4) ( ) 2 = (x x ) 2 σ 2 σ 2 = p(x)(x x ) 2 dx = p(x)(x 2 2x x + x 2 )dx = x 2 2 x 2 + x 2 = x 2 x 2 σ 2 = x 2 x 2 (2.5) 8

σ σ x x σ f(x) = ] [ σ 2π exp (x x )2 2σ 2 (2.6) 2.4 n [ 2.8 ] Γ(s) = () Γ(s + ) = sγ(s) ( ) Γ(S + ) = (2) Γ(n) = (n )! (3) Γ( 2 ) = π x s e x dx Γ(s) e x2 dx = π/2 x s e x dx = [ 2.9 ] r n γ n γ n = γ n = x s ( e x ) dx... x 2 +x2 2 +...+x2 n r2 dx dx 2... dx n γ n πn/2 Γ( n 2 + )rn n r γ n = C n r n σ n σ n = d dr γ n = nc n r n I n =... exp{ a(x 2 + x 2 2 +... + x 2 n)}dx dx 2... dx n { } n () I n = exp( ax 2 )dx I n (2) I n = exp( ar 2 )σ n dr I n C n (3) 2 γ n = πn/2 Γ( n 2 + )rn r n V V = πn/2 Γ( n 2 + )rn (2.7) 9

3 3. 3.. x i I i p i I i = I(p i ) = log p i (3.) x i x j I I(p i ) > I(p j ), p i < p j I(p i p j ) = I(p i ) + I(p j ) (3.) 3..2 S = p i I(p i ) = i p i log p i (3.2) [ 3. ] W p i = /W ( ) 3.2 (3.2) (.8) ( / ) R N A k = R/N A (3.3) S = k i p i log p i (3.4)

3.3 3.3. p q Hamiltonian H(p, q) Hamiltonian H = N i= 2m i p 2 i + Φ(r, r 2,..., r N ) (3.5) dq i dt = H p i, dp i dt = H q i (3.6) Newton dp dt = Φ r (3.7) p q 6N ( ) k m x Hamiltonian H = 2m p2 x + 2 kx2 px 2mE x 2E k 3.3.2 Liouville t (q, p) dqdp t (q, p ) dq dp q i = q i + dq i dt t = q i + H dq dp t p i (q,p) p i = p i + dp i dt t = p i H dq' t dp' q i (q',p') (q, p) (q, p ) p dqdp dq dp Jacobi dq dp = (q, p ) (q, p) dqdp dqdp (3.8) Liouville q

dq dp = dqdp (3.9) ( ) (q, p ) (q, p) = + 2 H q p t 2 H q 2 t 2 H p 2 2 H p q ( 2 ) H + q p 2 H = (3.) p q 3.4 2 Schrödinger Schrödinger p iħ (3.) x p x x p x h (3.2) [x, p x ] = iħ (3.3) ( ) (3.) Schrödinger 3.5 ( ) (a priori) Newton ( ) 3.6 (q, bmp) (E = const.) { const. : (q, p) E = const. ρ = (3.4) : 2

A A Ā A A ( ) A A A ρ(q, p)dqdp ( A = A ) Liouville ( A = Ā ) ( Ā = Ā ) A A Ā 3.7 3

4 4. E N V (Micro Canonical Ensemble) 4.2 4.2. E N V t E t E h (4.) h t E E + δe W (E, N, V, δe) δe ( ) W (E, N, V ) = (4.2) E<E l <E+δE ( ) W (E, N, V ) = h 3N N! E<H(N,V )<E+δE dqdp (4.3). ( q p h) h f (f ) E E + δe q p = h 2. ( ) N! 4.2.2 (, E) ( ) Ω (E, N, V ) ( ) Ω (E, N, V ) = (4.4) <E l <E 4

l ( ) Ω (E, N, V ) = h 3N N! <H(N,V )<E dqdp (4.5) (E, E + δe) (, E) Ω(E, N, V ) Ω (E, N, V ) Ω(E, N, V ) = Ω (E, N, V ) = d de Ω (E, N, V ) (4.6) E ( ) Ω(E, N, V )de (4.7) E Ω(E, N, V )de W (E, N, V ) = E+δE E Ω(E )de ΩδE = d de Ω (E, N, V )δe (4.8) 4.2.3 ( ) Ω = h 3 <ϵ<e dqdp = h 3 ϵ = 2m p2 = 2m (p2 x + p 2 y + p 2 z) (4.9) dq < p < 2mE dp = h 3 dxdydz V 4π 3 ( 2mE) 3 dxdydz < p < 2mE < p < 2mE dp x dp y dp z (4.) dp x dp y dp z 2mE Ω = 4πV 3 ( ) 3/2 2mE (4.) h 2 5

Ω = d ( ) 3/2 2m de Ω = 2πV E (4.2) h 2 W (E, δe, N, V ) = ΩδE = d de Ω = 2πV ( ) 3/2 2m EδE (4.3) h 2 ( ) Schrödinger ( ħ2 2 2m x 2 + 2 y 2 + 2 z 2 ) ψ(x, y, z) = ϵψ(x, y, z) (4.4) L ψ k (x, y, z) = e ikr, k = (k x, k y, k z ), V = L 3 V ϵ k = ħ2 k 2 2m, k2 = k x 2 + k y 2 + k z 2 k x k y k z k x = 2π L n x (n x =, ±, ±2, ±3,...) k y = 2π L n y (n y =, ±, ±2, ±3,...) (4.5) (4.6) k z = 2π L n z (n z =, ±, ±2, ±3,...) (2π)3 < ϵ k < E V 2mE Ω (E) ħ Ω (E) = ( ) 3 = 4π 2mE / (2π)3 = 4πV ( ) 3/2 2mE 3 ħ V 3 h 2 (4.7) <ϵ k <E Ω(E) = d de Ω (E) = 2πV W (E, δe, N, V ) = 2πV ( 2m h 2 N ) 3/2 E (4.8) ( ) 3/2 2m EδE (4.9) h 2 N Ω Ω = h 3N dqdp (4.2) N! = H E h 3N N! dx dy dz dx 2 dy 2 dz 2 dx N dy N dz N (4.2) dp x dp y dp z dp x2 dp y2 dp z2 dp xn dp yn dp zn (4.22) H E 6

H E dx dy dz dx 2 dy 2 dz 2 dx N dy N dz N = dp x dp y dp z dp x2 dp y2 dp z2 dp xn dp yn dp zn dx dy dz dx 2 dy 2 dz 2 dx N dy N dz N = V N 2mE 3N Ω = V N (2πmE) 3N/2 N!h 3N Γ ( (4.23) 3N 2 + ) W (E, δe, N, V ) = 3NV N (2πm) 3N/2 E 3N 2 2N!h 3N Γ ( δe (4.24) 3N 2 + ) 4.3 W l p l l p l = (4.25) W S S = k log W (4.26) δe S(E, N, V ) = k log W (E, δe, N, V ) k log Ω (4.27) (4.27) 3 4.3. n log n! n log n n (4.28) 4.3.2 (4.23) (4.24) W = Ω 3NδE 2E k log W = k log Ω + k log 3NδE 2E (4.29) k log W (E, δe, N, V ) k log Ω (4.29) 2 log Ω log Ω = N log V + 3N 2 log E + N log(2πm)3/2 log N! N log h 3 log 3N 2! (4.3) 7

log Ω N log V + 3N 2 log E + N log(2πm)3/2 N log N + N N log h 3 3N 2 log 3N 2 + 3N 2 ( = N log V N + 3 2E log 2 3N + log (2πm)3/2 e 5/2 ) h 3 (4.3) E N V N (4.29) 2 N ( 24 ) S(E, N, V ) = k log W (E, δe, N, V ) k log Ω ( S = kn log V N + 3 2E log 2 3N + log (2πm)3/2 e 5/2 ) h 3 (4.32) S E = T 3 2 kn E = T E = 3 2 NkT k [ 4. ] ω N E T N ( p 2 () H = i 2m + ) 2 mω2 x 2 i Ω (E) i= (2) S k log Ω S (3) S E = E T T (4) [ 4.2 ] ω N E T ( 2 + m) ħω (m =,, 2, 3,...) () E = ( 2 N + M) ħω W M (M + N )! M!(N )! N ( ) i m i m i = M m i (2) S = k log W M (3) S E = E T T (4) E T [ 4.3 ] µ H µh µh N () MµH W M (2) M < S = k log W M (3) S E = E T T i= 8

(4) T [ 4.4 ] N a x a () x W (2) Stirling S x (3) F (4) X X = F X x (5) X x [ 4.5 ] N Frenkel N n n N n N, N w () N n N W (2) Stirling S (3) F (4) F = T n n 9

4.4 4.4. ε, ε 2,, ε j,, 2,, j, n, n 2,, n j, (n, n 2,, n j, ) {n j } N N = n j j (4.33) E = n j ε j j (4.34) ε j ε 2 ε n j j N {n j } W {n j } W {n j } = N! j n j! ( j ) n j (4.35) j W {n j } = j ( j ) n j n j! (4.36) W (E, N, V ) W (E, N, V ) = {n j } ( nj = N nj ε j = E W {n j } = ) {n j } ( nj = N nj ε j = E ( j ) nj j ) n j! (4.37) {n j } {n j } W (E, N, V ) W {n j } {n j } {n j } {n j } 4.4.2 (4.33) (4.34)W {n j } W {n j } {n j } W {n j} log W {n j } {n j } L({n j }, α, β) = log W {n j } α n j N β n j ε j E (4.38) j j α β log W {n j } = log j ( j ) n j n j! j {n j (log n j ) n j log j } (4.39) 2

log W {n j } j L({n j }, α, β) = j {n j (log n j ) n j log j } α j n j N β j n j ε j E (4.4) log W {n j } {n j } L = log n i + log i α βε i = (4.4) n i L α = n j N = (4.42) j L β = n j ε j E = (4.43) j (4.4) n i = i e α e βε i (4.44) Z = i i e βε i (4.45) (4.42) α e α = N Z (4.46) β (4.43) n i n j (4.4) log n j + log j = α + βε j n j j ( n j log n j n ) j log j = α n j + β n j ε j = αn + βe (4.47) j j j { n j (log n j ) n j log j } = α + βe + N (4.48) W (E, N, V ) W {n j } S = k log W k log W {n j } = k(αn + βe + N) (4.49) β S E = T S E = kβ = T (4.5) (4.5) β = kt (4.52) 2

β (4.52) j n j n j = N Z j e βε j (4.53) Z = j j e βεj (4.54) n l = N Z e βε l (4.55) Z = l e βε l (4.56) l (Maxwell ) Boltzmann Z (Maxwell ) Boltzmann l n l e βε l N n : n 2 : : n l : = e βε : e βε2 : : e βε l : (4.57) E = l n lε l E E = l ε l n l = l ε l N Z e βε l = N ε l e βε l (4.58) Z l E = N Z β Z (4.59) Boltzmann ( ) Boltzmann Bose(-Einstein) Fermi(-Dirac) Boltzmann Bose Fermi 4.4.3 m k k ε = ε k = ħ2 k 2, k = k (4.6) 2m 22

Z = l e βε l = k e βε k = V [ (2π) 3 exp ( βħ2 2 ) ]3 k x dk x 2m = V λ T 3, λ T 2 = βh2 2πm (4.6) α e α = N Z = N V λ T 3 (4.62) < ε > = ε l e βε l Z l = Z V (2π) 3 ħ 3 2m = 3 Z V (2π) 3 ħ 3 2m [ ] (kx 2 + ky 2 + kz) 2 exp β ħ2 (kx 2 + ky 2 + kz) 2 dk 2m [ ] kx 2 exp β ħ2 (kx 2 + ky 2 + kz) 2 dk = 3 V 2m 2 Z β λ 3 T = 3 2β (4.63) 3 2kT β = kt (4.49) S = kn + kαn + kβe (4.64) e α = λ 3 3 T N/V E = N < ε >= N 2β ( S = kn log V N + 3 2E log 2 3N + log (2πm)3/2 e 5/2 ) h 3 (4.65) (4.32) Helmholtz Helmholtz F = E T S ( ) F p = = NkT V V T,N (4.66) pv = NkT (4.67) [ 4.6 ] ω N Boltzmann E T ( 2 + m) ħω (m =,, 2, 3,...) () Boltzmann (2) 23

(3) ( 4.4) [ 4.7 ] µ H µh µh N Boltzmann () Boltzmann (2) (3) ( 4.5) 24

5 E N V 5. I II H I+II I+II ( ) H I+II H I+II = H I + H II + H (5.) H ) I II E I E II W I W II I+II E W E = E I + E II (5.2) W (E) = W I (E I ) W II (E II ) (5.3) I+II S I S I II S II S(E) = S I (E I ) + S II (E II ) (5.4) N V 2) I E I S(E) E I = E I E I = E E II E I = E II S(E) E S = (S I + S II ) = S I + S II E I E I E I E I = T = S I E I S II E II = (5.5) T I = S I E I = S II E II = T II (5.6) 2 2 25

5.2 I II I+II II I ( ) I I S E I E I = I I I I N V I E N l ε l p l p l I p l ( ) p l S = k l p l log p l (5.7) p l = (5.8) l E p l ε l E = (5.9) l L({p l }, α, β) = k ( ) ( ) p l log p l kα p l kβ p l ε l E (5.) l l l α β L = k log p i k(α + ) kβε i = p i (5.) L α = p l = l (5.2) L β = l p l ε l E = (5.3) (5.) p i = e (α+) e βεi (5.4) (5.2) e (α+) = /Z N Z N = i e βε i p i = Z N e βε i (5.5) β β (5.3) (5.5) (5.7) S = k i p i log p i = kβ i p i ε i + k log Z N = kβ E + k log Z N (5.6) 26

E E S E = T β kt i l l ( ) p l = Z N e βε l, β = kt Z N = l (5.7) e βε l (5.8) l e βε l p : p 2 : : p l : = e βε : e βε2 : : e βε l : (5.9) ( ) ρ(q, p)dqdp = Z N N!h 3N e βh dqdp (5.2) Z N = N!h 3N e βh dqdp (5.2) Canonical Ensemble Canonical Z N (4.56) N Z Z N 5.3 E E = l p l ε l (5.22) (5.7) p l E = ε l e βε l (5.23) Z N l E = Z N β Z N = β log Z N (5.24) 2 27

5.4 (5.8) (5.2) ε l E ε l = E W (E, N, V ) W (E, N, V ) Ω(E) Z N = l e βε l = E e βe W (E, N, V ) (5.25) Z N = Ω(E)e βe de (5.26) Z N W (E, N, V ) Ω(E) E (5.6) β = kt E E Z N F kt log Z N = E T S = F (5.27) W (E, N, V ) β = kt Z N S F S = k log W (E, N, V ) (5.28) F = kt log Z N (5.29) F = U T S U V S T V F ds = T du + P dv T (5.3) df = SdT P dv (5.3) 5.5 H = i () Z N = N!h 3N 2m p2 i dq dq 2 dq N dp dp 2 dp N exp { β i } 2m p2 i dq dq 2 dq N = V N (5.32) 28

dp dp 2 dp N exp { β i } 2m p2 i ( = { dp x dp y dp z exp β }) N ( ) 3N 2m p2 = 2πmkT (5.33) h2 λ 2 T = 2πmkT Z N = V N Z N!λ 3N T (5.34) Z N = N! ZN (5.35) N N! (2) E = β log Z N = 3N β log λ T = 3N h 2 λ T β 2πmkT = N 3 kt (5.36) 2 kt /β β = kt (3) (4) P = V ( kt log Z N ) = kt V log Z N = kt V N log V = kt N V (5.37) P V = NkT (5.38) ( ) F S = = k log Z N + kt T V Z N T Z N = k log V N ( ) V N ( ) V N N!λ 3N + kt T N!λ 3N T T N!λ 3N = k log V N T N! λ 3N T k {N log V N log N + N 3N log λ T + 32 } N [ { (2πmk = kn log V N + 3 ) }] 3/2 2 log T + log e 5/2 h 2 + k 3 2 N (5.39) Stirling log N! = N log N N 5.6 x ( x) 2 = (x x ) 2 ( x)2 = (x x ) 2 = x 2 ( x ) 2 (5.4) 29

( E) 2 = E 2 E 2 (5.4) ( E) 2 E = ε l e βε l (5.42) Z N T T = kt 2 β l C v = T E = kt 2 β E = { } kt 2 ε l e βε l β Z N kt 2 ε l e βε l Z N β l l { } = kt 2 ZN 2 ( ε l )e βε l ε l e βε l kt 2 ( ε 2 Z l)e βε l N l l l ( ) 2 ( ) = kt 2 ε l e βε l + Z N kt 2 ε 2 Z le βε l N l = kt 2 ( E2 E 2 ) (5.43) l ( E) 2 = kt 2 C v (5.44) K f E = 2 fkt, C v = fk (5.45) 2 ( E)2 = E 2 fk2 T 2 2 fkt f (5.46) mol f = 3N 24 ( E)2 2 (5.47) E 5.7 3 S = ( ) F T V = k log Z N + kt Z N T Z N (5.48) 3

Z N = E e βe W (E, N, V ) (5.49) E { } T e E/kT W (E, N, V ) S = k log e E/kT E W (E, N, V ) + e E/kT W (E, N, V ) E T E = E (5.5) W (, N, V ) S = k log W (, N, V ) (5.5), T = 3 T S ( ) [ 5. ] ξ ε = cξ 2 2kT [ 5.2 ] ω N N ( p 2 H = i 2m + ) 2 mω2 x 2 i i= () Z N Z Z N (2) < E >= Z N β Z N < E > (3) C v [ 5.3 ] ω N ( 2 + m) ħω (m =,, 2, 3,...) () Z N (2) < E > (3) C v [ 5.4 ] µ H µh µh N () Z N (2) < E > (3) C v (4) F (5) S 3

6 6. I II H I+II I+II ( ) H I+II H I+II = H I + H II + H (6.) H ) I II E I E II I II N I N II W I W II I+II E N W E = E I + E II (6.2) N = N I + N II (6.3) W (E, N) = W I (E I, N I ) W II (E II, N II ) (6.4) I+II S I S I II S II S(E, N) = S I (E I, N I ) + S II (E II, N II ) (6.5) V 2) I E I S E I E I = E II S = (S I + S II ) = S I + S II E I E I E I E I S E = T = E I E I = E E II = S I E I S II E II = (6.6) T I = S I E I = S II E II = T II (6.7) 2 2 I N I S N I = N I N I = N N II N I = N II S = (S I + S II ) = S I + S II N I N I N I N I 32 = S I N I S II N II = (6.8)

S N = µ kt µ I T I = S I N I = S II N II = µ II T II (6.9) T I = T II µ I = µ II 2 2 6.2 I II I+II II I ( ) I I S E I = E I S N I = N I I I I I V I E N N l ε l n l p l p l I p l ( ) p l S = k l p l log p l (6.) p l = (6.) l E p l ε l E = (6.2) l N p l n l N = (6.3) l L({p l }, α, β, γ) = k ( ) ( ) ( ) p l log p l kα p l kβ p l ε l E kγ p l n l N l l l l (6.4) 33

α β γ L = k log p i k(α + ) kβε i kγn i = p i (6.5) L α = p l = l (6.6) L β = l L γ = l p l ε l E = (6.7) p l n l n = (6.8) (6.5) p i = e (α+) e βε i γn i (6.9) (6.6) e (α+) = /Ξ Ξ = i e βεi γni p i = Ξ e βε i γn i (6.2) β γ (6.2) (6.) S = k i p i log p i = kβ i p i ε i + kγ i p i n i + k log Ξ = kβ E + kγ N + k log Ξ (6.2) E E N N S E = S T N = µ kt β = kt γ = µ kt i l l p l = Ξ e β(ε l µn i), β = kt Ξ = e β(ε l µn i) l (6.22) (6.23) l n l = N l N (6.22) (6.23) p l = Ξ e β(ε l(n) µn), β = kt Ξ = e β(ε l(n) µn) N l (6.24) (6.25) Ξ 6.3 (6.25) ε l (N) l N N N Ξ 34

Ξ = N e βεl(n) e βµn = N l Z N e βµn (6.26) Ξ Z N N (6.2) β = kt γ = µ kt E E N N F = E T S G = Nµ = F + pv Xi pv kt log Ξ = F G = pv (6.27) 6.4 6.4. Fermi Bose 6.4.2 k ϵ k n k (6.25) N N ε l N = k ε l = k n k (6.28) n k ϵ k (6.29) 35

Ξ Ξ = e βnµ Z N = e β k n kϵ k e βnµ N= N= {n k } = e β k (µ ϵ k)n k = e β(µ ϵk)nk (6.3) n k n k n 2 {n k } (6.28) {n k } N N = Ξ Ne β(εl(n) µn) = kt log Ξ (6.3) µ N l n k (6.3) Ξ = { } + e β(µ ϵ k) (6.32) k N N = kt µ log Ξ = k e β(ϵ k µ) + (6.33) (6.28) k n k n k = e β(ϵ k µ) + n k ϵ k ϵ (6.34) f(ϵ) = e β(ϵ µ) + (6.35) 6.4.3 n k (6.3) Ξ = { } e β(µ ϵk)nk = e β(µ ϵ k) k n k k N N = kt µ log Ξ = k e β(ϵ k µ) (6.28) k n k n k = e β(ϵ k µ) n k ϵ k ϵ (6.36) (6.37) (6.38) f(ϵ) = e β(ϵ µ) (6.39) f(ϵ) 36

6.4.4 n k = β(ϵ k µ) (6.4) e β(ϵ k µ) ± e β(ϵ k µ) (6.4) (6.4) (T ) (N/V ) (6.4) 3 37

7 ( ) 7. ( ) ħ2 2 2m x 2 + 2 y 2 + 2 z 2 ψ(x, y, z) = ϵψ(x, y, z) (7.) L ψ k (x, y, z) = e ikr, k = (k x, k y, k z ), V = L 3 V ϵ k = ħ2 k 2 2m, k2 = k x 2 + k y 2 + k z 2 k x k y k z k x = 2π L n x (n x =, ±, ±2, ±3,...) k y = 2π L n y (n y =, ±, ±2, ±3,...) (7.2) (7.3) k z = 2π L n z (n z =, ±, ±2, ±3,...) (2π)3 < ϵ k < E V 2mE Ω (E) ħ Ω (E) = ( ) 3 = 4π 2mE / (2π)3 = 4πV ( ) 3/2 2mE 3 ħ V 3 h 2 (7.4) <ϵ k <E Ω(E) = d de Ω (E) = 2πV ( 2m h 2 ) 3/2 E (7.5) k 2 2 ħ (= h/2π) n(e) = V (2m)3/2 2π 2 ħ 3 E (7.6) n(e) Ω(E) n(e) [ 7. ] 2 7.2 E f(e) (6-8) f(e) = e β(e µ) + (7.7) 38

f(e) µ N n(e) f(e) N = f(e)n(e)de (7.8) (7.8) (7.8) N = µ n(e)de (7.9) µ E F N = EF n(e)de (7.) N (7.) (7.6) N = V (2m)3/2 3π 2 ħ 3 E 3/2 F (7.) E F ( 3π 2 ) 2/3 N ħ 2 E F = V 2m (7.2) E F = k B T F T F [ 7.2 ].97g/cm 3 Na Na 23 ħ2 k 2 2m E F = ħ2 k 2 F 2m k F (7.2) (7.3) ( 3π 2 ) /3 N k F = (7.4) V 2π/L ( ) N k F k F k F 39

7.3 f(e) = e β(e µ) + e β(e µ) >> (7.5) f(e) = e β(e µ) T (T >> T F ) (7.5) (T << T F ) K ( 3K) g(e) f(e) I = ( ) g(e)f(e)de µ G(E) = g(e)de G() = f( ) = g(e)de + π2 6 (k BT ) 2 g (µ) (7.6) I = [G(E)f(E)] df(e) de G(E)dE = df(e) G(E)dE (7.7) de G(E) E = µ G(E) G(µ) + (E µ)g (µ) + 2 (E µ)2 G (µ) (7.8) (7.7) { I f (E) G(µ) + (E µ)g (µ) + } 2 (E µ)2 G (µ) = G(µ) f (E)dE G (µ) (E µ)f (E)dE 2 G (µ) (E µ) 2 f (E)dE (7.9) f() = G(µ) 2 E µ = x { { } (E µ) de = e β(e µ) + µ { } x e βx dx (7.2) + e βx +} (7.2) µ { } βx x e βx dx = + e βx dx (7.2) + 2 + e βx x (7.9) 2 (7.9) 3 β(e µ) = x (E µ) 2 f (E)dE (E µ) 2 f (E)dE = β 2 x 2 e x (e x + ) 2 dx = π2 6 (k BT ) 2 (7.22) 4

x 2 e x π2 (e x dx = + ) 2 3 (7.6) I G(µ) + π2 6 (k BT ) 2 G (µ) = µ g(e)de + π2 6 g (µ) (7.23) 7.4 N = = EF n(e)f(e)de µ µ n(e)de + π2 6 (k BT ) 2 n (µ) µ n(e)de + n(e)de + π2 E F 6 (k BT ) 2 n (µ) N + (µ E F )n(e F ) + π2 6 (k BT ) 2 n (µ) (7.24) µ = E F π2 6 (k BT ) 2 n (µ) n(e F ) E F π2 6 (k BT ) 2 n (E F ) n(e F ) (7.25) 3 n(e) = V (2m)3/2 2π 2 ħ 3 E n(e) = V (2m) 3/2 4π 2 ħ 3 E µ = E F π2 2E F (k B T ) 2 (7.26) [ 7.3 ] 2 7.5 (3k B /2) U = U µ EF En(E)dE + π2 6 (k BT ) 2 {n(µ) + µn (µ)} En(E)dE + µ En(E)f(E)dE (7.27) E F En(E)dE + π2 6 (k BT ) 2 {n(e F ) + E F n (E F )} U + E F n(e F )(µ E F ) + π2 6 (k BT ) 2 {n(e F ) + E F n (E F )} = U + π2 6 (k BT ) 2 n(e F ) (7.28) 3 4 (7.25) 4

C el = U E = π2 kb 2 n(e F ) T = γt, γ = π2 kb 2 n(e F ) 3 3 (7.29) 3 (7.6) (7.2) (7.4) γ = mπk2 B V 3ħ 2 ( ) /3 3N = mk2 B V πv 3ħ 2 k F (7.3) (C el = γt ) (γ n(e F )) << [ 7.4 ] 23.97g/cm 3 γ 3.5 4 cal mol K 2 42

8 8. N m u m M C Mü m = Cu m (8.) ω = C M (phonon ) n = U e ħω/k BT (8.2) (8.3) U = Nħω e ħω/k BT (8.4) C v T C v = U ( ) 2 ħω T = Nk e ħω/k BT B (8.5) k B T (e ħω/k BT ) 2 C v = Nk B (T ) (8.6) 3 3 C v = 3Nk B (T ) (8.7) Dulong Petit C v e ħω/kbt (T ) (8.8) Dulong Petit C v T 3 (8.9) 8.2 N C N 43

m m Mü m = C(u m+ u m ) C(u m u m ) (8.) Mü m = C(2u m u m+ u m ) (8.) u m = ũ m e iωt (8.2) (8.2) (8.) Mω 2 ũ m e iωt = C(2ũ m ũ m+ ũ m )e iωt (8.3) Mω 2 ũ m = C(2ũ m ũ m+ ũ m ) (8.4) ũ N+ = ũ, ũ = ũ N (8.5) (8.4) ũ m = ue imka, K = 2π Na (8.6) Mω 2 ue imka = C(2ue imka ue i(m+)ka ue i(m )Ka ) (8.7) Mω 2 = C(2 2 cos Ka) (8.8) ω 2 = 2C M ( cos Ka) = 4C M sin2 Ka (8.9) 2 ω = 4C M sin 2 Ka (8.2) 44

ω - π - π π π π a K π 2pi a a π a K π a K = ũ m = ue iωt (a) K = π a ũ m = ue i(ωt mπ) (b) (a) (b) 8.3 2 2 ( (a)) 2 ( (b)) (a) (b) 45

8.4 (8.6) 3 ũ m = ue ik x m (8.2) x m = (x m, y m, z m ), k = (k x, k y, k z ) (8.22) x m = L N m x ( m x < N) k x = 2π L n x ( N/2 n x < N/2) y m = L N m y ( m y < N) k y = 2π L n y ( N/2 n y < N/2) z m = L N m z ( m z < N) k z = 2π L n z ( N/2 n z < N/2) (2π/L) 3 k N = 4 3 πk3 (2π/L) 3 (8.23) k ω (8.2) ω = vk (8.24) v ω π/ (8.23) N V ω3 6π 2 v 3, V = L 3 (8.25) 3 ( ) D(ω) N D(ω) = dn dω = V ω2 2π 2 v 3 (8.26) (a) (b) (ω) () (a) ω ω (b) 46

N 3 ω N = V ω3 6π 2 v 3 (8.27) ( 6π 2 ) /3 N ω D = v (8.28) V 8.5 U U = 3 dωħωd(ω)f(ω) = 3 Θ D dω V ħω3 2π 2 v 3 e ħω/k BT = 3V ħ 2π 2 v 3 U = 3V ħ ωd 2π 2 v 3 x = ω 3 dω e ħω/k BT ħω k B T U = 3V k4 B T 4 ħωd /k B T 2π 2 v 3 ħ 3 ħω k B T = Θ T x3 dx e x ωd ω 3 dω e ħω/k BT (8.29) (8.3) (8.3) (8.32) ħω D = k B Θ D (8.33) ( ) 3 T ΘD /T U = 9Nk B T dx x3 e x x D = Θ D T ( ) 3 T xd U = 9Nk B T dx x3 Θ D e x Θ D /T 5 xd x3 Θ D dx e x () dx x3 e x = π4 5 (8.34) (8.35) (8.36) (8.37) 47

Θ D [K] T [K] Θ D /T 5 [ 8. ] x 3 e x U = 3Nk BT 4 π 4 5Θ 3 D (8.38) C v = U ( ) 3 T = 2π4 Nk B T T 3 (8.39) 5 Θ D x = 2 x D x D = Θ D /T U = 3NkT x 3 e x dx 48

9 9. ( ) Fe Co Ni ( - ) H = J i j σ i σ j µh i σ (9.) σ i - J J > J < 2 3 9.2 (9.) H = i J σ j µh σ i (9.2) j i j i σ j σ j i i σ j i σ j σ j Z σ (9.3) j i Z H H = µh eff σ i (9.4) i H eff = JZ µ σ + H (9.5) H eff Z = e βµh eff + e βµh eff = 2 cosh βµh eff (9.6) Nµ σ M = Nµ σ = Nµ tanh βµh eff (9.7) H eff M 49

{ M = Nµ tanh βµh + β JZ } Nµ M (9.8) ( ) H = M = H = X = βjz Nµ M kt X = tanh X (9.9) JZ Y = kt X, Y = tanh X (9.) JZ X = Y = tanh X kt JZ < X T C = JZ k (9.) T < T C Nµ T > T C 5

. 4 He 2.2K 2 ( ) 4 He 4 He E = ħ2 k 2 2m (.) N(E) = V (2m)2/3 4π 2 ħ 3 E (.2) k = n N N n N = n + N (.3) n = e α, α = βµ (.4) µ α n α = (.2) (.3) N = n + = n + V (2m)3/2 4π 2 ħ 3 f(e)n(e)de = n + N(E) e β(e µ) de E /2 de (.5) e β(e µ).2 ( ) F σ (α) = Γ(α) F σ (α) = Γ(α) y σ e y+α dy (.6) [ ] e (y+α) y σ + e (y+α) + e 2(y+α) +... dy = e α σ + e 2α 2 σ + e 3α 3 σ + = n= e nα n σ (.7) 5

e n(y+σ) y σ dy = Γ(σ) n σ e nσ (.8) () σ α e α α (2) (.7) α α α F σ () = n= = ζ(σ) (.9) nσ ζ(x) ζ(3/2) = 2.62 ζ(5/2) =.342.3 (.6) (.5) ( ) 3/2 N = n + N 2πmkB T = n + V F 3/2(α) (.) α = N ( ) 3/2 N max 2πmkB T = 2.62V h 2 h 2 N = n + N max (.) ( ) 3/2 N > N max 2πmkB T = 2.62V (.2) N = N h2 T C = 2πmk B h 2 ( ) 2/3 N (.3) 2.62V T < T C.4 2πV (2m)3/2 E = En(E)f(E)dE = h 3 e β(e µ) de = 3 ( ) 3/2 2 V k 2πmkB T BT F 5/2(α) (.4) h 2 E 3/2 T < T C ( ) α = E = 3 ( ) 3/2 2 V k 2πmkB T BT ζ(5/2) (.5) h 2 52

C = E T = 5 4 V k B ( ) 3/2 2πmkB T 3/2 (.6) T C T 3/2 T = T C 5ζ(5/2) 4ζ(3/2) R =.972R T > T C 3 2 h 2 53