Maxwell

I 2018 12 13

0 4 1 6 1.1............................ 6 1.2 Maxwell......................... 8 1.3.......................... 9 1.4..................... 11 1.5..................... 12 2 13 2.1................... 13 2.1.1 N................. 13 2.2......................... 15 2.3............................ 17 3 19 3.1..................... 19 3.2................ 20 3.3......................... 21 3.4......................... 21 3.5.......................... 22 3.6....................... 22 4 26 4.1......... 26 4.2.............................. 26 4.2.1.................... 27 4.3............. 29 5 31 5.1..................... 31 5.2....................... 32 5.3................... 34 5.4................ 34 5.5........ 35 6 37 6.1..................... 37 1

6.2..................... 37 6.3.......... 38 6.3.1.............. 41 7 45 7.1..................... 45 7.2.................... 47 7.2.1................ 48 7.2.2................. 48 7.2.3..... 49 7.3........................ 50 7.3.1.................... 51 7.4.............. 52 7.5.................. 52 7.6........................ 53 7.7........ 54 8 56 8.1..................... 56 8.2........... 57 8.2.1................ 57 8.2.2...... 59 8.3........................ 59 8.4................... 59 8.4.1 1.......... 60 9 61 9.1.................... 61 9.2............... 62 9.2.1..................... 64 9.2.2.................... 64 9.3.................... 64 9.4................ 66 9.5............................. 66 10 T P 68 10.1............... 68 10.1.1................ 69 10.1.2..................... 69 10.2 T P............................ 70 10.3................... 71 10.4......................... 72 10.5 T P................. 72 2

11 74 11.1..................... 74 11.1.1................ 74 11.1.2................... 75 11.2.................... 75 11.3.................. 76 11.3.1................ 76 11.3.2...................... 77 11.3.3................... 77 11.3.4............... 78 11.4.................. 78 11.4.1................. 79 11.4.2................. 79 11.4.3............... 80 12 82 12.1............................ 82 12.1.1........................ 82 12.1.2........................ 83 12.1.3 Liouville.................... 83 12.1.4........................ 84 12.2............................ 85 12.2.1.............. 85 12.2.2................. 86 12.2.3...................... 86 12.2.4............ 88 3

0 Feynman (Atomic Hypothesis) 1 1 I 4

I 5

1 1.1 V N P T P V = nrt (1.1) n R R = 8.31 J mol K (1.2) ( ) r i = (x i, y i, z i ), v i = (v xi, v yi, v zi ); i = 1, 2,, N. (1.3) N f(x, y, z, v x, v y, v z ) (1.4) (x, y, z) dx, dy, dz (v x, v y, v z ) dv x, dv y, dv z f(x, y, z, v x, v y, v z ) dxdydz dv x dv y dv z (1.5) 6

(x, y, z) (x, y, z) f(v x, v y, v z ). (1.6) N ( ) N = f(v x, v y, v z ) dxdydz dv x dv y dv z = V f(v x, v y, v z ) dv x dv y dv z (1.7) (x, y, z) (v x, v y, v x ) (, ) (x, y, z) V v x vx f(v x, v y, v z ) dxdydz dv x dv y dv z f(vx, v y, v z ) dxdydz dv x dv y dv z = V N v x f(v x, v y, v z ) dv x dv y dv z 1 v 2 v 2 x x f(v x, v y, v z ) dxdydz dv x dv y dv z f(vx, v y, v z ) dxdydz dv x dv y dv z = V v 2 N xf(v x, v y, v z ) dv x dv y dv z (1.8) 1.1 ( ) x f(x) x A(x) A A(x)f(x)dx A = f(x)dx A 2 A 2 1.2 ( ) x f(x) y = 2x y g(y) g(y) = 1 ( y ) 2 f 2 1 7

1.2 Maxwell f(v x, v y, v z ) v x, v y, v z v v 2 f(v x, v y, v z ) = h(v 2 ). (1.9) g(v x ) f(v x, v y, v z ) = g(v x )g(v y )g(v z ) (1.10) ( f g ) h(v 2 ) = g(v x )g(v y )g(v z ) v y = v z = 0 v 2 = v 2 x h(v 2 x) = g(0) 2 g(v x ) g(v x ) = h(v 2 x)/g(0) 2 (1.10) h(v 2 ) h(v 2 ) = 1 g(0) 6 h(v2 x)h(v 2 y)h(v 2 z) (1.11) (1.11) h(v 2 ) h(v 2 ) = Ae αv2 (1.12) A α a g(0) (1.11) h(x + y + z) = 1 a 6 h(x)h(y)h(z) x = y = z = 0 h(0) = a 3 y z y = z = 0 h (x) = 1 a 6 h(x)h (0)h (0) = α 2 h(x), (1.13) 8

α 2 = h (0) 2 /a 6 (α > 0) h(x) = Ae αx + Be +αx (1.14) f B = 0 f(v x, v y, v x ) = h(v 2 ) = A e α(v2 x+v 2 y+v 2 z) (1.15) (1.7) (1.12) A α N = V A e αv2 dv x dv y dv z [ ] 3 ( = V A e αv2 x π ) 3/2 dvx = V A (1.16) α 1.3 ( ) x y f(x, y) 2 f(x, y) g(x) h(y) f(x, y) g(x)h(y) 1.4 ( ) a b a b x = a + b y = ab x y 1.5 ( ) (1.13) (1.14) B 0 f 1.3 (1.12) P 2 9

x v = (v x, v y, v z ) v = ( v x, v y, v z ) x p x = 2mv x m (v x, v y, v z ) (v x > 0) t v x t S S v x t f(v x, v y, v z ) 1.1 0 dv x dv y dv z 2mv x S v x t f(v x, v y, v z ) 3 P S t P = 0 dv x dv y dv z 2mvx 2 f(v x, v y, v z ) (1.17) f (1.12) [ ] [ ] 2 P = 2mA dv x vx 2 e αv2 x dv y e αv2 y = ma ( π 3/2 0 2α α) (1.16) A P V = m 2α N (1.1) α = m N 2 nrt = m 1 2 k B T ( v, v, v ) x y z (v, v, v ) x y z v x t 1.1: x v x (> 0) v x t t 3 v x > 0 v x [ 0, ) 10

k B = R N/n = R N A 1.38 10 23 J/K Boltzmann N A N A 6.02 10 23 1/mol. f f(v x, v y, v z ) = N ( ) 3/2 ] m exp [ mv2 V 2πk B T 2k B T (1.18) Maxwell Maxwell-Boltzmann 1.6 ( ) e x2 dx = π I 1 = e αx2 dx, I 2 = x 2 e αx2 dx. 1.7 (Maxwell ) Maxwell (1.18) (1.7). 1.4 Maxwell (1.18) v = 0 2kB T m (1.19) Maxwell (1.18) v = vx 2 + vy 2 + vz 2 F (v) v v + dv F (v)dv f v v + dv 4πv 2 dv h(v 2 )4πv 2 dv (1.15) F (v) = N V F (v)dv = h(v 2 )4πv 2 dv ( ) 3/2 ] m 4πv 2 exp [ mv2 2πk B T 2k B T (1.20) v = 0 Maxwell (1.19) 1.8 () (1.20) (1.19) 11

1.5 Maxwell 1 2 m v 2 = 1 ( v 2 m ) 2 x + v 2 y + v 2 z = 3 2 m vx 2 Maxwell (1.18) (1.8) 1 2 m v 2 = 3 2 k BT (1.21) 1 2 m vx 2 1 = 2 m vy 2 1 = 2 m vy 2 1 = 2 k BT 1 2 k BT U C V U = N A 3 2 k BT C V = du dt = 3 2 N Ak B = 3 2 R (1.21) (rootmean-square speed) 3kB T v2 = (1.22) m 1.9 ( ) (1.22) 12

2 2.1 (Head) (Tail) 1/2 2 2 (1/2) 2 2 (1/2) 2 2 (1/2) 2 N n P (n; N) P (n; N) = N C n ( ) N 1 = 2 N! n!(n n)! ( ) N 1 (2.1) 2 (binomial distribution) 2.1.1 N N (Stirling s approximation) ln n! n(ln n 1) (2.2) ln P (n; N) n = N/2 2 ln P (n; N) 2 ( n 1 ) 2 N 2 N + P (n; N) exp [ 2 N (n 12 N ) 2 ] 1 P (n; N) = 1 [ 2N N π exp 2 (n 12 ) ] 2 N N (2.3) 13

P(n; N) 0.6 0.5 0.4 0.3 0.2 0.1 N = 2 P(n; N) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 N = 5 P(n; N) 0.25 0.2 0.15 0.1 0.05 N = 10 0 0 0.5 1 1.5 2 n 0 0 1 2 3 4 5 n 0 0 2 4 6 8 10 n 2.1: (2.1) (2.3) 1 2.1 (2.1) (2.3) n x = n/n x P G (x; N) (2.3) [ P G (x; N) = 2N π exp 2N ( x 1 ) ] 2 2 (2.4) (2.3) (2.4) (Gaussian distribution) (central limit theorem) (2.4) P G (x; N) dx = 1 x x (x x ) 2 x = (x x ) 2 = x P G (x; N) dx = 1 2, (2.5) ( x 1 ) 2 P G (x; N) dx = 1 2 4N (2.6) 2 (2.4) 1/ N 2.2 (2.4) x (2.3) 1/N 1 N 2 x 0 x 1 N (2.6) [ 0, 1] (, ) 14

2.3 (2.4) x 2.4 (2.5) (2.6) N n = xn n = 1 2 N, n 2 = N 2 x 2 = 1 4 N n = n n, x = x x N ( n2 = 1 2 N n2 = 1 0 (N ) n N N 1/ N (the law of large numbers) 2.5 (2.1) (2.3) ln P (n; N) n P (n; N) 2.2 V N 3 ( v) n P (n) V v N n ρ ρ = N/V n n n = vρ v p p = v/v P (n) P (n) = p n (1 p) N n NC n (2.7) 3 15

P (n) = p n (1 p) N n N! n! (N n)! = pn n! (1 N! p)n n (N n)! = nn n! = nn n! ( 1 n ) N n N(N 1)(N 2) (N n + 1) N N ( n 1 n ) (N/ n) n ( 1 n ) n N N N N N 1 N (N n + 1) N p v V = n N n N e n P n (n) lim P (n) = n n (2.8) N n! e = lim ϵ 0 (1 + ϵ) 1/ϵ (2.8) (Poisson distribution) (N) v v (n N) (the law of small numbers). (2.8) P n (n) = 1 (2.9) n=0 n = n P n (n) = n, (n n) 2 = n=0 n 2 P n (n) = n (2.10) n=0 0.4 0.3 <n> = 1 0.3 0.2 <n> = 2 0.15 <n> = 5 P 0.2 P P 0.1 0.1 0.1 0.05 0 0 2 4 6 8 10 n 0 0 2 4 6 8 10 n 0 0 2 4 6 8 10 12 14 n 2.2: (2.8) (2.11) 16

2.6 (2.8) (2.9) (2.10) 2.7 3 2.8 n =1, 2, 5, 10 2.9 n 1 (2.8) P n (n) 1 [ exp 1 ] (n n)2 (2.11) 2π n 2 n 2.3 2 N ( ) N L (N R ) (2.1) N L = N 2, (NL N L ) 2 = N 4 (NL N L ) 2 = 1 N L N N N N L N R N > δ N R ( N L ) y N L N R N = 2 N L N 1 P (y) (2.4) N P (y) = [ 2π exp N2 ] y2 (2.12) y > δ 1 δ δ P (y)dy = 2 δ 17 N [ 2π exp N2 ] y2 dy

N 10 24 δ (10 6 ) 10 1012 (equilibrium state) (non-equilibrium state) 4 (irreversible process) 2.10 N 10 24 δ 10 6 10 1012 4 18

3 3.1 N 1 ω n e n ( e n = n + 1 ) ħω ; (n = 0, 1, 2, 3, ) (3.1) 2 ħ h 2π ħ 1.05 10 34 J s n 1 n 2 N N n N (n 1, n 2, n 3,, n N ) (3.2) n i E E = N e ni = i=1 N i=1 ( n i + 1 ) ( ħω M + N ) ħω (3.3) 2 2 1 19

M = N n i (3.4) i=1 3.2 E W N (E) 2 n i (3.2) W N (E) E M (3.4) (3.2) (3.5) N M (3.6) M+N 1 C N 1 W N (E) W N (E) = M+N 1 C N 1 = (M + N 1)! M!(N 1)! (3.7) W N (E) N M 3.1 (3.5) (3.6) 3.2 (3.6) M+N 1 C N 1 3.3 N, M 1 (3.7) W N (E) ln W N (E) N [( 1 + M N ) ( ln 1 + M ) M N N ln M ] N (3.8) N M W N (E) 2 E 20

3.3 (3.2) (3.2) (3.2) (3.3) N E (3.7) W N (E) 3.4 3 (The principle of equal a priori probabilities) 4 3 4 21

3.5 ensemble E E E (micro canonical ensemble) 3.6 N A B 2 A N A B N B E A E B N A + N B = N (3.9) E A + E B = E. (3.10) (3.1) 1/2 E A = M A ħω, E B = M B ħω E W N (E) A E A B E B W NA (E A ) W NB (E B ) E A 22

E A B E B (= E E A ) P (E A, E B ) P (E A, E B ) = W N A (E A )W NB (E B ) W N (E) (3.11) N E A E A (3.11) ( ) Σ(E A, E B ) ln W NA (E A )W NB (E B ) (3.8) [( ) ( ) NA + M A NA + M A Σ(E A, E B ) = N A ln M A ln M ] A N A N A N A N [( ) ( ) A NB + M B NB + M B +N B ln M B ln M ] B N B N B N B N B P (E A, E B ) E A Σ(E A, E B ) d de A Σ(E A, E B ) = 0 (3.12) E A d Σ(E A, E B ) = Σ(E A, E B ) + de B Σ(E A, E B ) de A E A de A E B = Σ(E A, E B ) Σ(E A, E B ) E A E B E A = dm A de A = 1 M A ħω M A, = 1 E B ħω M B, P(E A, E B ) E * A E A 3.1: P (E A, E B ) E A 23

(3.12) [ ( ) NA + M A ln ln N A ( MA N A )] [ ln E A N A = E B N B = E N ( ) NB + M B ln N B ( MB N B )] = 0 (3.13) A B P (E A, E B ) E A E B (3.13) E A = N A N E, E B = N B N E Σ(E A, E B ) E A E A = EA (E A EA ) 2 Σ(E A, E B ) = Σ(EA, EB) + d Σ(E A, E B ) (E A E de A EA A) =EA + 1 d 2 Σ(E A, E B ) (E A E 2 A) 2 + (3.14) de 2 A EA =E A EA (3.12) d 2 ( N N Σ(E A, E B ) = + N ) (3.15) EA =EA E(E + Nħω) N A N B de 2 A (3.14) E A (3.11) E A [ P (E A, E B ) exp 1 ( N N + N ) (E A E A) ] 2 (3.16) 2 E(E + Nħω) N A N B N A, N B N O( N) (EA E A )2 E A ( ) 1 O N N 3.4 A B E A E B W A (E A ) W B (E B ) (3.11) 24

3.5 (3.14) 3.6 (3.15) 3.7 (3.16) 3.8 E A (3.16) E A 25

4 4.1 E W (E) S(E) S(E) = k B ln W (E) (4.1) k B R N A k B = R N A 1.38 10 23 J/K. (4.1) (Boltzmann s entropy formula) (4.1) 4.2 (4.1) ds = dq T (4.2) dq = de (4.2) 1 T = ds (4.3) de 26

(4.1) (4.1) (4.3) 2 2 4.2.1 A B E W (E) E A B E A E B W (E A, E B ) A B W A (E A ) W B (E B ) W (E A, E B ) = W A (E A )W B (E B ) E A E A B E B P (E A, E B ) P (E A, E B ) = W (E A, E B ) W (E) = W A(E A )W B (E B ) W (E) (4.4) E A + E B = E A B? E A E B P (E A, E B ) P (E A, E B ) (4.3) dp (E A, E B ) de A = 0 d ln P (E A, E B ) de A = 0 27

E (4.4) ds A (E A ) de A ds B(E B ) de B = 0 (4.5) S A (E A ) = k B ln W A (E A ) S B (E B ) = k B ln W B (E B ) T A T B (4.3) (4.5) 1 = 1 (4.6) T A T B (4.5) ds A (E A ) de A ds B(E B ) de B > 0 (4.7)? (4.3) 1 > 1 T A < T B T A T B (4.7) (4.4) d ln P (E A, E B ) de A > 0 dp (E A, E B ) de A > 0 P (E A, E B ) E A T A < T B B A T B > T A (4.3) 28

4.3 (4.1) S = S A + S B (4.8) S(E) = k B ln W (E) W (E) = W A (E A )W B (E B ) (4.9) E A +E B =E E A + E B = E E A E B A B E A E B S A (E A) = k B ln W A (E A), S B (E B) = k B ln W B (E B) W (E) = W A (E A )W B (E B ) > W A (EA)W B (EB) E A +E B =E k B S(E) > S A (E A) + S B (E B) (4.10) (4.8) (4.8) W(E A, E B ) E * A E A 4.1: E A N 29

EA E B (4.9) N W A (E A )W B (E B ) E A W A (E A )W B (E E A ) E A = EA N (4.9) W (E) N W A (E A)W B (E B) (4.11) N S(E) = k B ln W (E) S A (E A) + S B (E B) + k B ln N (4.12) (4.10) 2 S A (E A) O(N A ), S B (E B) O(N B ) ln N 4.1 (4.9) 4.2 (4.11) N (4.12) (4.11) N (4.12) 4.3 3 N S T 30

5 5.1 2 p p = h λ = ħk λ h ħ 2π k 2π/λ L L λ = L n λ n; n = 1, 2, 3, p = ħ 2πn L ħk n; n = 0, ±1, ±2, ±3, (5.1) n L 3 p = ħ 2π L (n x, n y, n z ) ħ 2π L n; n x, n y, n z = 0, ±1, ±2, (5.2) 3 n ϵ = p2 2m = ħ2 2m ( ) 2 2π ( ) n 2 x + n 2 y + n 2 z L 31

N n 1 n 2 N 3 N (n 1, n 2,, n N ) n N N i=1 ϵ i = (2πħ)2 2mL 2 N n 2 i (5.3) i=1 5.2 E W (E) E E E 1 Ω(E) E W (E) W (E) = Ω(E) Ω(E E) (5.4) Ω(E) (5.4) W (E) Ω(E) (5.3) E N i=1 n 2 i 2mL2 (2πħ) 2 E R2 (5.5) 3N (5.5) 2mL R = 2 (2πħ) E 2 3N (5.5) 3N d V d (R) = A d R d ; A d 2πd/2 d Γ(d/2) (5.6) E Ω(E) Ω(E) 1 ( ) L N! A 3N 3N 2mE (5.7) 2πħ 1 E E W (E) E S(E) E 32

Γ(n) Γ(n) 0 t n 1 e t dt (5.8) (n 1)! 1/N! N 2 d R S d (R, R) V d (R) V d (R R) (5.6) S d (R, R) V d (R) = 1 ( 1 R ) d R R/R 1 d d R/ R 2 1 S d (R, R) V d (R) (5.9) (5.4) (5.7) W (E) Ω(E) = 1 N! 2 π 3N/2 3N Γ(3N/2) V = L 3 V N (2πħ) 3N ( 2mE ) 3N/2 (5.10) 5.1 (5.5) n 1x, n 1y, n 1z, n 2x,, n Nz 3N R 5.2 R d (5.6) 5.3 (5.7) 1/N! 1/N! 5.4 ϵ 1 d ϵd 1 (1 ϵ) d 1 ϵ 1 (1 ϵ) 1/ϵ 1/e 2 N n i n i N! N! n i N! E n i E 33

5.3 S = k B ln W k B ln Ω = Nk B (ln N 1) + Nk B ( ln V 3 ln 2πħ + 3 2 ln π + 3 2 ln(2me) ) k B ln Γ ( ) 3 2 N k B ln 3 2 N (5.11) N ( ) 3 ln Γ 2 N 32 ( ( ) ) 3 N ln 2 N 1 [ ( 3 4πm S = Nk B 2 ln 3(2πħ) E ) + ln V 2 N N + 5 ] 2 (5.12) (5.13) (5.11) E 5.5 n ( Γ(n + 1) = n!, Γ n + 1 ) = (2n)! 2 2 2n n! N 1 (5.12) 5.4 T 1 T = ds de = Nk B 3 1 2 E (5.14) E = 3 2 Nk BT (5.15) (5.13) E V ( ) S = P (5.16) V E T 34

( ) P S T = = Nk B V E V (5.17) P V = Nk B T (5.18) 5.6 (5.13) (5.14) (5.17) 5.7 (5.16) de = T ds P dv 5.5 (5.10) P ex A m x V P ex V = Ax, P ex = mg/a g E mgx E 0 E + mgx = E 0 x 5.1: A m 35

E V x W (E, V ) = W (E 0 mgx, Ax) x S(E, V ) = k B ln W (E, V ) = S(E 0 mgx, Ax) (5.19) W S ( x ds dx = 0 (5.19) ( ) ( ) S S mg + A = 0 E V V E ( ) S = mg/a = P ex V E T T (5.16) P = P ex P P ex 36

6 N 6.1 N ϵ 1 ϵ 2 (ϵ 1 < ϵ 2 ) N 1 N 2 N E N = N 1 + N 2 (6.1) E = N 1 ϵ 1 + N 2 ϵ 2 (6.2) W W = N C N1 = N! N 1! N 2! (6.3) (6.3) N 1 N 2 (6.1) (6.2) N 1 N 2 N E W N E 6.1 (6.3) 6.2 (6.3) S = k B ln W ] = k B [N (ln N 1) N 1 (ln N 1 1) N 2 (ln N 2 1) ( ) ( ) ( ) ( )] N1 N1 N2 N2 = Nk B [ ln ln N N N N (6.4) 37

W S E N T 1 T = ds de = S dn 1 N 1 de + S dn 2 N 2 de = k B ϵ ln N 1 N 2 (6.5) ϵ = ϵ 2 ϵ 1 (6.5) N 1, N 2 N 1 = N 1 1 + e ϵ/k BT (6.6) e ϵ/k BT N 2 = N (6.7) 1 + e ϵ/k BT ( ) e ϵ/k BT E = N ϵ 1 + ϵ (6.8) 1 + e ϵ/k BT C C = de ( ) 2 ϵ dt = Nk e ϵ/k BT B ( k B T ) 1 + e ϵ/k B T 2 (6.9) (6.9) k B T ϵ/2 (Schottky type) 6.2 (6.9) 6.3 (6.9) k B T ϵ k B T ϵ 6.3 M (M ) N i ϵ i (i = 1, 2, 3,, M) 38

1 N 1, 2 N 2, i N i, N N i N = (6.10) M N i (6.11) E N i E = i=1 M ϵ i N i (6.12) i=1 N i (i = 1, 2,, M) W W(N 1, N 2,, N M ) = N! N 1!N 2! N M! (6.13) W N i (i = 1,, M) E (M 3) N E N i (i = 1, 2,, M) N E W (6.11) (6.12) N i (6.13) W = W(N 1, N 2,, N M ) (6.14) N= i N i, E= i ϵ in i W N W N i W cn a W(N 1,, N M) cn a W (6.15) c a 1/2 cn a W S = k B ln W = k B ( ln W + ln c + a ln N ) 39

W N S = k B ln W (6.16) N i W : N E (6.11) (6.12) N i (i = 1, 2,, M) (6.13) W? (6.11) (6.13) i p i N i N (6.17) 1 = N i N = i i E N = N i ϵ i N = i i 1 N ln W = (ln N 1) 1 N = i p i N (6.18) ϵ i p i E (6.19) N i (ln N i 1) = i i N i N ln N i N p i ln p i S (6.20) : (6.18) (6.19) p i (i = 1, 2,, M) (6.20) S? p i δp i p i p i + δp i S δs δp i δs = 1 N δ(ln W) = i (ln p i + 1) δp i (6.21) δp i δs S p i p i S S p i 40

p i δp i δp i (6.21) δp i ln p i + 1 = 0 p i = e 1 p i (6.18) (6.19) p i δ p i δ p i δ N = i δ p i = 0 (6.22) δ E = i ϵ i δ p i = 0 (6.23) S p i δ p i δ S = i (ln p i + 1) δ p i = 0 (6.24) : p i (6.18) (6.19) δ p i (6.22) (6.23) (6.24) p i δ p i (6.24) δ p i 6.4 N i (i = 1, 2,, M) (6.13) 6.3.1 (Lagrange s method of undetermined multiplier) (i) (6.18) (6.19) α β S S αn βe (6.25) 41

(ii) S p i δp i δ S = δs αδn βδe = i (ln p i + α + βϵ i ) δp i = 0 (6.26) p i α + 1 α p i α β (iii) α β (6.18) (6.19) p i p i α β δ p i δ S = 0 p i (6.26) S δs = 0 δ p i δ S = δ S αδ N βδ E = 0 (6.27) δ p i (6.22) (6.23) δ N = 0 δ E = 0 (6.27) δ S = 0 (6.26) δp i δp i ln p i + α + βϵ i = 0 (6.28) p i = e α βϵ i. (6.29) p i α β p i (6.18) (6.19) 1 = i E N = i p i = i ϵ i p i = i e α βϵ i (6.30) ϵ i e α βϵ i (6.31) α β α (6.30) e α = i e βϵ i f (6.32) 42

β β (6.31) E N = 1 ϵ i e βϵ i (6.33) f (6.29) S p i i p i N i N = e βϵi f (6.34) β β β = 1 k B T S (6.16) (6.20) p i S k B ln W = k B N i W p i (6.29) p i = e α βϵ i p i ln p i (6.36) α β 1 = p E i, N = p i ϵ i (6.37) i i N E p i N E (6.36) 1 T = ds de = k BN i ds dp i dp i de (6.35) = k B N i = k B N i (ln p i + 1) dp i de ( α βϵ i + 1) dp i de = k B N ( α + 1) i dp i de + k BNβ i ϵ i dp i de (6.38) (6.37) E 0 = i dp i de, 1 N = d(e/n) de = i dp i de ϵ i 1 T = k Bβ β = 1 k B T 43

β 1/k B T N i S (N 1, N 2, ) N i S (N 1, N 2, ) N E N i N = N (N 1, N 2, ), E = E (N 1, N 2, ) (6.39) α β N i S (N 1, N 2, ) k B αn (N 1, N 2, ) βe (N 1, N 2, ) (6.40) N i N i + δn i (6.41) 1 k B δs αδn βδe = 0 (6.42) Ni α β α β (6.39) N E Ni δn i N E S E (6.42) δn = 0 1 δs βδe = 0 (6.43) k B ( ) S E N = k B β (6.44) 44

7 A B B A A B B (heat bath) (environment) A 7.1 A B E T A n P n A n A E n B E T E n P n A n A n B W B (E T E n )P n P n W B (E T E n ) (7.1) S B [ ] 1 P n W B (E T E n ) = exp S B (E T E n ) k B E n E T E n [ ( 1 P n exp S B (E T ) ds )] B E n k B de E=ET [ ( 1 = exp S B (E T ) E )] n k B T [ exp E ] n (7.2) k B T 45

T 1 1 T = ds B. (7.3) de E=ET (7.2) P n (Boltzmann factor) A Z = e En/k BT (7.4) n (partition function) (sum over states) 2 (7.2) T n P n E n P n = e En/k BT Z (7.5) 3 (7.5) (Gibbs distribution) (Boltzmann distribution) n (7.5) (7.5) T (canonical ensemble) 7.1 (7.1) A B 7.2 P n (7.5) E = n E n P n E = d ln Z (7.6) dβ β = 1/k B T 1 A A (4.6) 2 n n E n n n 3 P n n E n 46

7.2 (7.5) (7.5) 2 A E P A (E) A E 1 Z e E/k BT A E A E W A (E) P A (E) = 1 Z W A(E)e E/k BT A S A P A (E) = 1 [ Z exp 1 ( ) ] E T S A (E) = 1 k B T Z e F A(E)/k B T (7.7) F A (E) E T S A (E) (7.8) A F A T S A E 4 A (7.7) E 0 P A (E) E 0 E 7.1: A 4 (7.8) E T 47

7.2.1 E 0 df A (E) de = d ( ) E T S A (E) = 0 (7.9) E=E0 de E=E0 ds A de = 1 E=E0 T (7.10) A A (7.9) (7.9) (7.10) E 0 E 0 (T ) (7.8) T A F A (T ) F A (T ) = F A ( E0 (T ) ) = E 0 (T ) T S A ( E0 (T ) ). (7.11) 7.2.2 P A (E) (7.7) E = E 0 E T S A (E) = E 0 T S A (E 0 ) + d ( 0 E T S A (E)) (E E 0 ) de + 1 d 2 ( 0 E T S 2 de 2 A (E)) (E E 0 ) 2 + = F A (T ) 1 2 T d2 S A de 2 (E E 0 ) 2 + (7.12) 0 0 E=E0 (7.9) 2 d 2 S A de 2 = 0 d de 1 T A (E) = 1 0 5 A T 2 A dt A de C A de 0 dt = 1 0 T 2 ( ) 1 de0 1 dt T 2 C A 5 T A (E 0 ) = T (7.10) 48

(7.12) E T S A (E) = F A (T ) 1 2T C A (E E 0 ) 2 + (7.13) (7.7) [ ] 1 P A (E) exp (E E 2k B T 2 0 ) 2 (7.14) C A E 0 (E E0 ) 2 = k B T 2 C A (7.15) 6 (E E0 ) 2 E = kb T 2 C A E 0 1 N (7.16) C A E 0 N 7.3 (7.14) (7.15) 7.2.3 (7.15) (E ) 2 E = E 2 E 2 (7.17) Z = e Enβ, E = 1 E n e Enβ, Z n n β = 1/k B T. d E (E dβ = ) 2 E E 2 = 1 E 2 Z ne Enβ n (7.18) (E E ) 2 = d E dβ = k BT 2 C (7.19) 7.4 (7.17) (7.19) 6 49

7.3 T F Z F (T ) = k B T ln Z(T ) (7.20) 7 (7.4) (E E, E] W (E) Z(T ) = n = E E e En/k BT = E E W (E) e E/k BT [ exp 1 ( ) ] E T S(E). k B T (7.7) E = E 0 (7.13) Z(T ) E E 1 E = 1 E exp [ E E exp 1 de exp [ F (T ) k B T E E [ F (T ) ] 2k B T 2 C (E E 0) 2 k B T F (T ) 1 ] k B T 1 2k B T 2 C (E E 0) 2 ] 2πkB T 2 C. E de E/ E k B T [ ] 2πkB T k B T ln Z(T ) = F (T ) k B T ln 2 C E (7.20) 2 1 O(N) E N ln N N (7.20) 7 A A 50

(7.20) 7.3.1 (7.20) df = SdT P dv (7.21) ( ) ( ) F F S =, P = (7.22) T V V T (7.20) T V (7.4) E n (7.20) Z(T, V ) = n e En(V )/k BT, F (T, V ) = k B T ln Z(T, V ). (7.23) F ( ) F = [ kb T ln Z(T, V ) ] T T V = k B ln Z k B T ln Z β (7.6) ( ) F = 1 ( ) F E = S T T V dβ dt ( ) F = [ kb T ln Z(T, V ) ] = 1 E n V T V Z V e Enβ n E n / V n n p n E n V p n(v ). ( ) F = 1 ( pn (V ) ) e Enβ = P (T, V ) V Z T n 51

7.4 I II 2 I II I II T I m Em I II n E II n (m, n) E (m,n) Z E (m,n) = E I m + E II n Z = (m,n) e E (m,n)β Z = e (EI m+en II)β = m n m e EI mβ n e EII n β = Z I Z II Z I Z II F = k B T ln Z = k B T ln ( Z I Z II ) = k B T ln Z I k B T ln Z II = F I + F II F I F II 7.5 T N i n i ε i (n i ) ( ε i (n i ) = n i + 1 ) ħω 2 N (n 1, n 2,, n N ) E(n 1,, n N ) E(n 1, n 2,, n N ) = i ε i (n i ). 52

Z Z = = = n 1,,n N =0 n 1 =0 e E(n 1,,n N )/k B T n N =0 e ε 1(n 1 )/k BT n 1 =0 e (ε 1(n 1 )+ +ε N (n N ))/k B T n N =0 e ε N (n N )/k B T ( ) N ( = e (n+1/2)ħω/k BT e ħω/2k B T = 1 e ħω/k BT n=0 [ ( )] N ħω = 2 sinh 2k B T F = k B T ln Z = Nk B T ln ) N [ ( )] ħω 2 sinh 2k B T 7.5 4.3 7.6 N 7.6 T N W (E) (5.10) Z(T, V ) = E = 1 N! E W (E) e E/kBT = 1 E V N (2mπ) 3N/2 (2πħ) 3N Γ(3N/2) 0 de W (E) e E/k BT de E 3N/2 1 e E/k BT t E/k B T 0 de E 3N/2 1 e E/kBT = (k B T ) 3N/2 dt t 3N/2 1 e t = (k B T ) 3N/2 Γ(3N/2) 0 53

Z(T, V ) = 1 N! [ V ( 2mπkB T ) ] N 3/2 (7.24) (2πħ) 3 F = k B T ln Z [ 3 = Nk B T 2 ln(2πmk BT ) + ln V ] N + 1 3 ln(2πħ) (7.25) 7.7 (7.25) 7.7 (4.1) n P n S = k B P n ln P n (7.26) : W n n P n = 1 W S = k B ln W = k B W 1 W ln 1 W W = k B P n ln P n n=1 (7.26) P n = 1 Z e βen 54

( ) F S = = ( kb T ln Z ) T V T ( = k B ln Z + ) E n e βen k n B T Z ( = k B ln Z + ( ln(pn Z) ) ) P n n = k B P n ln P n (7.26) n (7.26) H = n P n log 2 P n (7.27) (7.27) n P n 55

8 8.1 x p h. (8.1) h x a (8.1) p h/a (8.2) T k B T p mk B T 1 (8.2) p h a mk B T p (8.3) (8.3) k B T h2 ma 2 (8.4) 2 1 (order estimate) 2 m (8.4) 56

8.1 (8.3) 8.2 N m i d 2 r i dt 2 = F i(r 1, r 2, ); i = 1, 2,, N (8.5) 2 2 6N (r 1, p 1 ), (r 2, p 2 ),, (r N, p N ) N 6N 6N (phase space) (representative point) 8.2.1 1 { ṗ = kx mẍ = kx (8.6) ẋ = p/m { x = A cos(ωt + θ) k p = mωa sin(ωt + θ), ω m (8.7) p m ω A A x 8.1: 57

A θ E E = p2 2m + 1 2 kx2 = 1 2 ka2 (8.8) A 2 x p S S = πa 2 mω (8.9) 3 S E = k 2πmω S = ω 2π S (8.10) ( E = n + 1 ) ħω; n = 0, 1, 2, (8.11) 2 (8.10) n S ( S = n + 1 ) h (8.12) 2 n h f 2f h f 4 E Ω cl (E) E h f Ω cl (E) = 1 h f dx 1 dx f dp 1 dp f (8.13) H(x,p)<E 5 N N! E E E W (E) W (E) = Ω cl (E) Ω cl (E E) (8.14) 3 S 4 f 2f 5 h (8.1) (8.13) 58

8.2.2 (8.12) S = pdq = nh (8.15) (1/2) (8.15) 8.3 (Ergodic hypothesis) 2f H(x, p) = E 2f 1 8.4 1/h f Z Z = n e En/k BT (8.16) 1 h f e H(q,p)/k BT f dq i dp i (8.17) i=1 1/N! 59

q p P (q, p) P (q, p) = 1 Z e H(q,p)/k BT (8.18) 8.4.1 1 N 1 H = N i=1 ( p 2 i 2m + 1 ) 2 mω2 x 2 i (8.19) Z = 1 [ dx h N 1 dx N dp 1 dp N exp 1 N ( p 2 i k B T 2m + 1 ) ] 2 mω2 x 2 i i=1 [ [ 1 = dx 1 dp 1 exp 1 ( p 2 1 h k B T 2m + 1 )]] N 2 mω2 x 2 1 [ 1 = 2πmkB T ] N [ ] N kb T 2πk B T/mω h 2 = (8.20) ħω ( ) kb T F = Nk B T ln ħω ( ) F S = T (8.21) (8.22) E = d dβ ln Z = Nk BT (8.23) 8.2 (8.18) 1 2 k BT 8.3 L V = L 3 N Z P S E 60

9 A B A 9.1 A E A N A B E B N B E T N T E A + E B = E T, N A + N B = N T (9.1) W (E T, N T ) W A (E A, N A ) W A (E B, N B ) W (E T, N T ) = W A (E A, N A ) W B (E B, N B ) N A +N B =N T, E A +E B =E T = [ 1 ( exp S A (E A, N A ) + S B (E B, N B )) ] (9.2) k B N A E A E B = E T E A, N B = N T N A S A S B A B A E A N A P (E A, N A ) [ ( )] exp 1 S A (E A, N A ) + S B (E B, N B ) kb P (E A, N A ) = (9.3) W (E T, N T ) E A N A E A N A (9.3) d ( ) S A (E A, N A ) + S B (E B, N B ) = 0 (9.4) de A d ( ) S A (E A, N A ) + S B (E B, N B ) = 0 (9.5) dn A 61

(9.4) ds A de A = ds B de B 1 T A = 1 T B (9.6) (9.5) 1 ( ) S(E, N, V ) = µ N T E,V (9.7) ds A dn A = ds B dn B µ A T A = µ B T B (9.8) µ E A N A T A (E A, N A ) = T B (E B, N B ), µ A (E A, N A ) = µ B (E B, N B ) A 9.1 µ A > µ B T A = T B A B 9.2 (reservoir) N n P n,n P n,n A n B n E n P n,n P n,n W B (E T E n, N T N) [ ] 1 = exp S B (E T E n, N T N) k B (9.9) E T E n, N T N S B (E T E n, N T N) = S B (E T, N T ) S B E E n S B ET,N T N N + ET,N T const. 1 T E n + µ T N. 1 de = T ds P dv + µdn ds = 1 T de + P T dv µ dn T 62

T µ S B E = 1 ET,N T T, S B N = µ ET,N T T. P n,n [ P n,n exp 1 ( En µn )] (9.10) k B T T µ N n P n,n E n P n,n = 1 [ Ξ exp 1 ( En µn )] (9.11) Ξ(T, µ) = k B T exp N=0 n [ 1 ( En µn )] k B T e µn/kbt Z(T, N) (9.12) N=0 Ξ(T, µ) (Grand Partition Function) 2 (9.11) T µ (Grand Canonical Ensemble) 9.2 A B Ξ Ξ = Ξ A Ξ B n A B (n A, n B ) E n E n = E A n A + E B n B N = N A + N B 2 N N T (9.12) N N T n N 63

9.2.1 N P (N) N P (N) = n N=0 P n,n = n N=0 [ 1 Ξ(T, µ) exp 1 k B T N [ N N = NP (N) = Ξ(T, µ) exp 1 k B T N=0 n n ( En (N) µn )]. (9.12) µ Ξ µ = ( ) [ N exp 1 ( En µn )] k B T k B T N = k B T 1 Ξ ( En (N) µn )] Ξ µ = k BT ln Ξ (9.13) µ 9.2.2 (9.13) µ (7.18) ( ) N (N ) 2 k B T = N (9.14) µ T N (N N )2 1 N N 9.3 (9.14) 9.3 Ω(T, V, µ) Ω(T, V, µ) k B T ln Ξ(T, V, µ) (9.15) ( ) Ω = N, µ T,V ( ) Ω = P (9.16) V T,µ 64

(9.13) Ω (7.22) Ω V E n ( ) Ω V T,µ = ( ) k B T ln Ξ(T, V, µ) V = E n 1 V Ξ e β(en µn) N n = ( 1 P n (V )) Ξ e β(en µn) = P (T, µ) N n P n (V ) = E n / V n T, V, µ P V (9.16) Ω(T, V, µ) V Ω(T, V, µ) V (9.16) P Ω(T, V, µ) = P (T, µ)v (9.17) (7.20) Ξ(T, V, µ) = e µn/kbt Z(T, V, N) N = [ exp 1 ( ) ] F (T, V, N) µn k B T N [ exp 1 ( ) ] F (T, V, N0 ) µn 0 k B T N 0 T, V, µ N 0 (T, V, µ) 7.3 N 0 Ω(T, V, µ) = F (T, V, N) µn (9.18) N 0 N 0 Ω 9.4 (9.18) (9.17) G = F + P V = Nµ 65

9.4 Z(T, V, N) = 1 [ V ( 2πmkB T ) ] N 3/2 (9.19) N! (2πħ) 3 Ξ(T, V, µ) = e µn/kbt Z(T, V, N) = N=0 1 N! N=0 = exp [ e µ/k BT [ e µ/k BT Ω(T, V, µ) = k B T e µ/k BT N = ( ) Ω µ T,V (9.17) V ( 2πmkB T ) ] N 3/2 (2πħ) 3 V (2πħ) 3 ( 2πmkB T ) 3/2 ]. V (2πħ) 3 ( 2πmkB T ) 3/2 = e µ/k BT V (2πħ) 3 ( 2πmkB T ) 3/2 P V = Ω = Nk B T 9.5 F F (T, V, N) = k B T ln Z(T, V, N) ( ) F (T, V, N) µ = N n E E 7 Z(T, V, N) = [ exp E ] n(v, N) k n B T [ W (E, V, N) exp E ] k B T = E [ exp 1 k B T = E [ CN a exp 1 k B T 66 T,V ( E S(E, V, N)T ) ] ( ) ] E 0 S(E 0, V, N)T

C a E 0 ( ) ( ) S E S(E, V, N)T = 0, = E E E=E0 1 T E 0 T, V, N E 0 (T, V, N) ( ) k B T ln Z(T, V, N) E 0 (T, V, N) S E 0 (T, V, N), V, N T F (T, V, N) N = ( ( ) ) E 0 (T, V, N) S E 0 (T, V, N), V, N T N ( ) (( ) ) S S(E0 ) E0 = T T 1 N E 0,V E 0 V,N N ( = T µ ) ( ) 1 T T T 1 E0 N = µ 67

10 T P T P T P T P 10.1 10.1 V P P V E E B E + P V + E B = 7 V n P n (V ) [ P n (V ) exp 1 ( En (V ) + P V )] (10.1) k B T T V n E n (V ) 10.1: T 68

10.1.1 V P (V ) P (V ) = P n (V ) Z(V ) e P V/k BT n [ = exp 1 ( ) ] F (V ) + P V (10.2) k B T Z(V ) (7.4) F (V ) (7.20) V 0 F (V ) + P V ( ) F (V ) + P V = 0 (10.3) V ( ) F V T = P P F V T N (10.3) V 0 T, P, N V 0 (T, P, N) 10.1 (10.2) 10.1.2 (10.2) V V 0 F (V ) + P V = F (V 0 ) + P V 0 + 1 ( ) 2 F 2 V 2 (V V 0 ) 2 + T V =V0 = F (V 0 ) + P V 0 + 1 (V V 0 ) 2 (10.4) 2κ T V 0 κ T 1 ( ) V V P T ( ) 2 F V 2 T ( ) P = V T = 1 κ T V 69

P (V ) exp [ (V V ] 0) 2 2k B T κ T V 0 (10.5) (V V0 ) 2 = k B T κ T V 0 (10.6) 10.2 T P T P (T P ensemble) T P V n E n P n [ 1 P n (V ) = Y (T, P, N) exp 1 ( En (V ) + P V )] (10.7) k B T Y (T, P, N) T P Y (T, P, N) dv [ exp 1 ( En (V ) + P V )] (10.8) 0 k n B T = Y (T, P, N) = = = 0 0 0 0 dv Z(T, V, N) e P V/k BT (10.4) T P dv Z(T, V, N) e P V/k BT [ dv exp 1 ( ) ] F (T, V, N) + P V k B T [ dv exp 1 k B T = 2πk B T κ T V 0 exp ( F (T, V 0, N) + P V 0 + 1 [ 1 k B T ( F (T, V0, N) + P V 0 ) ] k B T k B T ln Y (T, P, N) = F (T, V 0, N) + P V 0 1 2 k BT ln ( 2πk B T κ T V 0 ) G(T, P, N) = F (T, V 0, N) + P V 0 )] (V V 0 ) 2 + 2κ T V 0 N ln N T P G(T, P, N) = k B T ln Y (T, P, N). (10.9) 70

10.3 V (V V ) 2 T P T P (10.8) Y P = 1 dv V e (En+P V )/k BT (10.10) k B T n 2 Y P = 1 dv V 2 e (En+P V )/k BT (10.11) 2 (k B T ) 2 n 1 Y Y P = 1 V k B T (10.12) 1 2 Y Y P = 1 V 2 2 (k B T ) 2 (10.13) k B T P (k B T ) 2 2 P 2 ln Y = ln Y = V (10.14) ( ) 2 V V (10.15) T P (10.9) G(T, P, N) V = (10.16) P (10.12) P V P = 1 ( V ) 2 V 2 = 1 k B T k B T ( V V ) 2 ( ) 2 V V = k B T V ( P = k BT V 1 V ) V P = k B T V κ T (10.17) κ T (10.6) 71

10.4 T P Z(T, V, N) (7.24) Y (T, P, N) = = 0 0 = 1 N! dv Z(T, V, N) e P V/k BT [ ] (2πmkB T ) 1/2 3N V N e P V/k BT dv 1 N! 2πħ [ (2πmkB T ) 1/2 2πħ I N (a) 0 dx x N e ax = ( 1) N dn da N ] 3N ( kb T P N! [ ] (2πmkB T ) 1/2 3N ( kb T Y (T, P, N) = 2πħ P 0 ) N+1 dx x N e x 0 dx e ax = ( 1) N dn da N 1 a = N! 1 a N+1 ) N+1 G(T, P, N) = k B T ln Y = Nk B T ln [ k B T P ( ) 3 ] 2πmkB T 2πħ (10.18) N G = µn µ [ ( ) 3 ] k B T 2πmkB T µ(t, P ) = k B T ln (10.19) P 2πħ T P 10.5 T P 10.1 T P (10.1) W (E, V ) E V A B E T V T E T = E A + E B, V T = V A + V B. A n E n V A n P n B W B (E T 72

E n, V T V ) P n W B (E T E n, V T V ) [ ] 1 = exp S B (E T E n, V T V ). (10.20) k B A E T E n, V T V S B S B (E T E n, V T V ) S B (E T, V T ) + S B ( ) S B ( ) En + V E B V B EB =E T V B =V T = S B (E T, V T ) 1 T E n P T V T P 1 T = S B P E B, EB =E T T = S B V B V B =V T (10.1) EB =E T V B =V T EB =E T V B =V T 73

11 11.1 1. 2. 3. 1 11.1.1 1 74

ψ(x 2, x 1 ) = ψ(x 1, x 2 ) (11.1) ψ(x 2, x 1 ) = ψ(x 1, x 2 ) (11.2) (Boson) (Fermion) N N 11.4 (Pauli exclusion principle) 11.1.2 S ħ (S = 0, 1, ) (S = 1/2, 3/2, ) S = 1/2 S = 1 3 He S = 1/2 4 He S = 0 11.2 75

i n i n i n i { n i = 0, 1 (Fermion) (n 0, n 1, n 2, n 3, ), n i = 0, 1, 2, 3, (Boson) (occupation-number representation) i e i N E N = i n i, E = i e i n i 11.3 11.3.1 N N n ( n e n ne n P n (9.11) [ ] n(e µ) P n exp (11.3) k B T T µ n { 0, 1 Fermion n = (11.4) 0, 1, 2, 3, Boson 76

11.3.2 n Ξ = 1 + e β(e µ) e β(e µ) n = 0 P 0 + 1 P 1 = 1 + e β(e µ) 1 = e β(e µ) + 1 f F (e) (11.5) (Fermi distribution function) Ξ = e βn(e µ) = n=0 1 1 e β(e µ) (9.13) n = n=0 n P n = k B T µ ln Ξ = 1 e β(e µ) 1 f B(e) (11.6) (Bose distribution function) 11.3.3 N µ N T i f F 1 f B 5 4 0.5 3 2 1 0 µ e 0 e 11.1: f F (e)( ) f B (e)( ). µ µ 77

ϵ i n i n i = i i 1 e β(ϵ i µ) ± 1 = N (11.7) T µ ϵ i µ T 1 e = µ n = 1/2 e < µ n 1 e > µ n 0 e < µ e f B (e) < 0 µ 11.3.4 T = 0 µ i (11.7) i n i 1 (11.8) µ ϵ i e β(ϵ i µ) 1 n e β(e µ) (11.9) (Boltzmann distribution function) 2 11.4 1 G = Nµ G/ T P,N = S < 0 2 5 1/N! 78

11.4.1 2 Ĥ 0 (x) = ħ2 + V (x) (11.10) 2m x2 x ϕ(x) x Eϕ(x) = Ĥ0(x)ϕ(x) (11.11) E ϕ(x) (E n, ϕ n (x)) : E n ϕ n (x) = Ĥ0(x)ϕ n (x), (n = 1, 2, 3, ). (11.12) Ĥ0(x) ϕ n (x) (n = 1, 2, 3, ) ϕ(x) ϕ(x) = a n ϕ n (x) (11.13) n=1 11.4.2 Ĥ(x 1, x 2 ) = ħ2 2m 2 x 2 1 + V (x 1 ) ħ2 2m 2 x 2 2 + V (x 2 ) + U(x 1 x 2 ) = Ĥ0(x 1 ) + Ĥ0(x 2 ) + U(x 1 x 2 ) (11.14) x 1 x 2 U(x 1 x 2 ) ϕ(x 1, x 2 ) Eϕ(x 1, x 2 ) = Ĥ(x 1, x 2 )ϕ(x 1, x 2 ) (11.15) ϕ(x 1, x 2 ) x 2 x 1 ϕ n (x) ϕ(x 1, x 2 ) = a n (x 2 )ϕ n (x 1 ) (11.16) n=1 a n (x 2 ) x 2 a n (x 2 ) a n (x 2 ) = b n,m ϕ m (x 2 ) (11.17) m=1 79

ϕ(x 1, x 2 ) = b n,m ϕ n (x 1 )ϕ m (x 2 ) (11.18) n=1 m=1 ϕ n (x 1 )ϕ m (x 2 ) (n, m = 1, 2, 3, ) (11.19) (11.19) (11.14) U(x 1 x 2 ) = 0 Ĥ(x 1, x 2 ) = Ĥ0(x 1 ) + Ĥ0(x 2 ) (11.20) (11.19) E n + E m ) ) (E n + E m ϕ n (x 1 )ϕ m (x 2 ) = (Ĥ0 (x 1 ) + Ĥ0(x 2 ) ϕ n (x 1 )ϕ m (x 2 ). (11.21) 11.1 (11.21) (11.12) 11.4.3 ϕ(x 1, x 2 ) ϕ(x 2, x 1 ) ϕ(x 1, x 2 ) ϕ(x 1, x 2 ) = +ϕ(x 2, x 1 ) (11.22) ϕ(x 1, x 2 ) = ϕ(x 2, x 1 ) (11.23) (Boson) (Fermion) (11.22) (11.18) b n,m ϕ n (x 1 )ϕ m (x 2 ) = b n,m ϕ n (x 2 )ϕ m (x 1 ) n=1 m=1 = n=1 m=1 n=1 m=1 b m,n ϕ m (x 2 )ϕ n (x 1 ) (11.24) n m b n,m = b m,n (11.25) 80

(11.18) ϕ(x 1, x 2 ) = b n,n ϕ n (x 1 )ϕ n (x 2 ) n=1 + n=1 m=n+1 ) b n,m (ϕ n (x 1 )ϕ m (x 2 ) + ϕ n (x 2 )ϕ m (x 1 ) ( ) ϕ n (x 1 )ϕ n (x 2 ), ϕ n (x 1 )ϕ m (x 2 ) + ϕ n (x 2 )ϕ m (x 1 ) (11.26) (11.27) n n m (11.18) (11.23) b n,m ϕ n (x 1 )ϕ m (x 2 ) = b n,m ϕ n (x 2 )ϕ m (x 1 ) n=1 m=1 = n=1 m=1 n=1 m=1 b m,n ϕ m (x 2 )ϕ n (x 1 ) (11.28) b n,n = 0, b n,m = b m,n ; (n m) (11.29) (11.18) ϕ(x 1, x 2 ) = n=1 m=n+1 ) b n,m (ϕ n (x 1 )ϕ m (x 2 ) ϕ n (x 2 )ϕ m (x 1 ) (11.30) ( ) ϕ n (x 1 )ϕ m (x 2 ) ϕ n (x 2 )ϕ m (x 1 ) ; (n m) (11.31) 81

12 12.1 N q i (i = 1, 2,, N) m i d 2 q i dt 2 = F i = V (q 1, q 2,, q N ) q i 12.1.1 q i p i dq i dt = H = {q i, H} p i (12.1) dp i dt = H = {p i, H} q i (12.2) H N p 2 i H = + V (q 1, q 2,, q N ), 2m i i=1 {A, B} {A, B} N i=1 ( A B A B ) q i p i p i q i (12.3) 82

(12.1) (12.2) 12.1.2 N (q i, p i ), (i = 1, 2,, N) 6N 6N N f(q, p, t) 1 A = A(q, p) q p A t A t = A(q, p)f(q, p, t)dqdp da dt = A q dq dt + A p dp dt = A q H p + A p H q = {A, H} (12.4) 12.1.3 Liouville Liouville Liouville v ( q, ṗ) v = ( qi + ṗ ) i = q i i p i i ( H ) H = 0 q i p i p i q i 1 f N f(q 1, p 1,, q N, p N, t) N (q 1, p 1,, q N, p N ) (q, p) 83

Lagrange df dt = 0 (12.5) Lagrange df dt = f t + f q dq dt + f p dp dt = f t + f q H p f p H q (12.5) (12.3) f t = {f, H} (12.6) (12.4) 12.1.4 f can = 1 Z e βh ; Z = e βh dqdp f can t = {f can, H} = 1 Z {e βh, H} = 0 (12.7) 12.1 (12.7) 12.2 j fv f t + j = 0 (12.8) v (12.6) 84

12.2 ϕ(x, t)schrödinger ϕ(x, t) iħ = Ĥϕ(x, t) (12.9) t A ϕ Â A ϕ A ϕ = ϕ (x, t)âϕ(x, t)dx (12.10) 12.2.1 ϕ(x) ϕ (x) ϕ(x) ϕ (x) ϕ(x 1 ) ϕ(x 2 ).. ( ϕ (x 1 ), ϕ (x 2 ), ϕ (x 3 ), ϕ, (12.11) ) ϕ (12.12) Dirac ϕ ϕ 2 ψ ϕ ψ (x)ϕ(x)dx (12.13) (12.9) Schrödinger iħ d ϕ(t) = Ĥ ϕ(t) (12.14) dt ( ) iħ d ϕ(t) = ϕ(t) Ĥ (12.15) dt A ϕ = ϕ Â ϕ (12.16) 85

{u i (x)} u i (x)u i (y) = δ(x y) (12.17) i u i u i = 1 (12.18) i 12.2.2 (pure state) (mixed state) c 1 ψ 1 + c 2 ψ 2 ψ ψ 1 ψ 2 ψ 12.2.3 ϕ i w i (density operator) ˆρ i ϕ i w i ϕ i (12.19) 2 w i w i [0, 1], w i = 1 (12.20) i A 2 2 A = i w i A ϕi. 2 86

(12.16) A = w i A ϕi = w i ϕ i  ϕ i i i = ( ) w i ϕ i  u j u j ϕ i i j = w i ϕ i  u j u ϕi j = uj ϕi wi ϕ i  u j i j j i = ( ) u j ϕ i w i ϕ i  u j. j i A = j [ ] u j ˆρ u j = Tr ˆρ (12.21) Tr ( ) (12.14) (12.15) iħ d dt ˆρ(t) = ( iħ d ) dt ϕ i w i ϕ i + ϕ i w i (iħ d ) dt ϕ i i i = i Ĥ ϕ i w i ϕ i i ϕ i w i ϕ i Ĥ = Ĥˆρ ˆρĤ iħ d dt ˆρ(t) = [ ˆρ(t), Ĥ ] (12.22) Heisenberg O (Heisenberg ) iħ d dtô = [ Ô, Ĥ ] (12.23) w 1 = 1 w i = 0 (i 1) ϕ 1 (12.19) ˆρ = ϕ 1 ϕ 1 ˆρˆρ = ϕ 1 ϕ 1 ϕ 1 ϕ 1 = ϕ 1 ϕ 1 = ˆρ ˆP ˆP = ˆP (12.24) 87

(12.24) 3 12.3 (12.24) w i 1 12.2.4 E E ( ) e βe /Z ˆρ can ˆρ can = i e βe i Z E i E i ; Z = i e βe i (12.25) Ĥ [ ] ˆρ can = e βĥ Z ; Z = Tr e βĥ (12.26) ˆρ can (12.22) iħ d dt ˆρ can = [ ˆρ can, Ĥ ] = 0 (12.27) 12.4 (12.25) (12.26) Â f(â) f(x) f(x) = n=0 1 n! f (n) (0)x n f(â) = n=0 1 (n) f (0)Ân n! 12.5 (12.27) 3 ˆP = e 1 e 1 + e 2 e 2 (12.24) Tr ˆP = 2 (12.20)

( ) T P ( ) ( ) (E, V, N) T, (V, N) T, (V ), µ T, P, (N) Pn = 1 W (E, V, N) e En(V,N)/k BT Z(T, V, N) e ( En(V,N) Nµ ) /k BT Ξ(T, V, µ) e ( En(V,N)+P V ) /k BT Y (T, P, N) T P W (E, V, N) Z(T, V, N) = n e En(V,N)/k BT Ξ(T, V, µ) = = N=0 N=0 ( ) e En(V,N) Nµ /k BT n e Nµ/k BT Z(T, V, N) = Y (T, P, N) = 0 0 dv ( ) e En(V,N)+P V /k BT n dv e P V/k BT Z(T, V, N) Ω(T, V, µ) = kbt ln Ξ S(E, V, N) = kb ln W F (T, V, N) = kbt ln Z = V P (T, µ) G(T, P, N) = kbt ln Y = Nµ(T, P ) (*)