2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n F (k) k=0 2 V (X) = (X X ) 2 = n (x i X ) 2 P i = X 2 X 2 n P i Z n P i P i Z P i P i f(x) = n f(x i ) P i = Z n f(x i )P i

.2. 3.2 X [x min, x max ] X x [x, x + x] P (x) x P (x) xmax P (x)dx = x min f(x) = X n = Z = f(x) = Z xmax x min xmax x min xmax x min xmax x min f(x)p (x)dx x n P (x)dx P (x)dx f(x)p (x)dx n P (x) = δ(x x i )P i x P (x) x x+ x n x δ(x x i )P i Lesbergue (3)

4 x 2 x P (x) x dp (x) x P (x) dx.3 S I 3 S I n P i log P i log P S I 0 P i 0 S I = 0 S I P i S I P i Lagrange λ λ I S I + λ( P i ) = P i (log P i + λ) + λ di dp i = log P i λ = 0 P i i e λ x i λ P i = n 4 S max I = n log n = log n X n 2 3 I 4 λ λ P i P i = e λ λ

.4. Gauß 5 5 S I = log P = xmax x min P (x) log P (x)dx Lagrange I S I + λ( P (x)dx) = P (x)(log P (x) + λ) + λ P (x) δi = [log P (x) + λ + ]δp (x)dx = 0 δp (x) P (x) = e λ P (x) x.4 Gauß X (, ) Gauß Z = Gauß P (x) e (x x 0 ) 2 2σ 2 P (x)dx = P (x) = e (x x 0 ) 2 2σ 2 2πσ e (x x 0 ) 2 2σ 2 dx = 2πσ X X = 2πσ xe (x x 0 ) 2 2σ 2 = x 0 V (X) = 2πσ x 2 e (x x 0 ) 2 2σ 2 x 2 0 = σ 2 5 Lesbergue P (x) x

7 2 2. ( ) 0 2 3 H = H i + H int H int = i,j U ij + i,j,k V i,j,k + Ψ n Schrödinger H Ψ n = n Ψ n n(= 0,, 2...) H n n n 2 3 (solid-state physics) (condensedmatter physics) 2 0 3

8 2 n [, + ] ( 4 W (, ; ) W (, ; ) = + δ( m )d W (, ; ) m W (, ; ) Ω(; ) 5 Ω(; ) Ω 0 (; ) W (, ; ) = Ω 0 ( + ; ) Ω 0 (; ) = Ω 0(; ) Ω(; ) = Ω 0(; ) W (, ; ) W (, ; ) ε 0 /ε 0 /ε 0 ε 0 O() ε 0 4 5

2.2. 9 O( 0 ) W (, ; ) W (, ; ) W (, ; ) O( 0 ) / Ω W (, ; ) = Ω( ; ) ε 0 + O(( )2 ) Ω(; ) = Ω( ε 0 ; ) Ω Ω(; ) = ε 0 Ω( ε 0 ; ) H = H i Schrödinger H i ψ (i) l i = ε (i) l i ψ (i) l i Ψ n = n = 6 n (l, l 2,..., l ) ψ (i) l i ε (i) l i 2.2 i ε ( i) = 0, ε 0 6

0 2 /2 0 ɛ 0 = 0 ɛ 0 ε 0 ε 0 ω() ω() = C m = ε (i)! m!( m)! (m ε 0 ) ω() ω() ω() /ε 0 = /2 ε 0 ω() ε 0 [, + ] W (, ; ) = + δ( mε 0 )ω(mε 0 )d = m=0 + δ( mε 0 )ω( )d ω() 7 ω( ) = ω() + ω ( ) + W (, ; ) = ω() ε 0 Ω(; ) = ω() ε 0 = ε + O(( )2 ) m=0 Γ( + ) Γ( ε 0 + )Γ( + ε 0 ) ε 0 Ω( ε 0 ; ) 8 7 ω() 8 Stirling

2.3. Stirling Γ(n + ) ( n ) n 2πn (n ) e Ω(; ) = [ ( ) ε ( 0 2 ) ] ε 0 2 2πε0 ε 0 ε 0 ε 0 Ω( ε 0 ; ) O( 0 ) 2 023 O( 0 ) ε 0 9 0 Ω(; ) = [ ( ) ε 0 ( ) ] ε 0 2πε0 ε 0 ε 0 2.3 ω m Ĥ = 2m ˆp2 + mω2 2 ˆq2 ε l = hω(l + )(l = 0,, 2,...) 2 hω Ĥ = n = 2 hω + L n hω, L n = 2m ˆp2 i + mω2 2 ˆq2 i l i = 0,, 2,... 9 0 Ω(; )

2 2 n l i 2 hω hω hω ω() L = Ē hω 2 ω() = L+ C = (L + )! l!( )! hω [, + ] W (, ; ) = 3 + δ( L hω)ω( )d L=0 W (, ; ) = ω() hω Ω(; ) = ω() hω = hω Γ( Ē hω + ) Γ( Ē hω + )Γ() Stirling Ω(; ) = 2π hω ( + ( hω hω ) hω + 2 ) + hω 2 ( ) 2 O( 0 ) O( 0 ) O(/) 4 Ω(; ) = ( + 2π hω hω ( hω ) + hω ) hω 2 L L L + L+ C 3 ω() 4

2.4. 3 2.4 2.4. L m H = 2m (ˆp2 x + ˆp 2 y + ˆp 2 z) 5 Schrödinger h2 2m Ψ n(x, y, z) = n Ψ n (x, y, z) Ψ n (x, y, z) = L 3 exp[ ī h (pxx + p yy + p z z)] 6 p x = h L n x, p z = h L n z, p z = h L n z(n x, n y, n z = 0, ±, ±2...) n = 2m ((p2 x + p 2 y + p 2 z) n (n x, n y, n z ) (p x, p y, p z ) h L 7 h L p x, p y, p z h L [p x, p x + p x ], [p y, p y + p y ], [p z, p z + p z ] ( ) 3 L p x p y p z h 8 p 2 x + p 2 y + p 2 z = 2m Ω 0 () 2m ( L h ) 3 Ω 0 () = 4π 3 (2m)3/2 ( L h 5 0 6 7 8 ) 3

4 2 [, + ] W (, ) = Ω 0 ( + ) Ω 0 () = Ω 0() = 2π g() W (, ) = g() g() = 2π ( ) 3 L (2m) 3/2 h ( ) 3 L (2m) 3/2 h /2 g g() = ɛ 0 g( ɛ 0 ) g() g() /2 ɛ 3/2 0 g() 9 ɛ 0 = m ( ) 2 h L g() = 25/2 π ɛ 3/2 0 ɛ0 2 20 ɛ 0 ( 9 ɛ 0 = h ) 2 2m L ɛ 0 ( 20 h L δ frac2m h 2 L) δ 2 h m L δ = h 6m L δ h m L ɛ 0

2.4. 5 2.4.2 H = 2m n = 2m (ˆp 2 ix + ˆp 2 iy + ˆp 2 iz) ((p 2 ix + p 2 iy + p 2 iz) p ix = h L n ix,..., (n ix,... = 0, ±, ±2...) n 3 (n x, n y, n z,..., n z ) 3 2 ( h 3 V ) Π p ix p iy p iz 22 ( ) V h 3 Π p ix p iy p iz p 2 ix + p 2 iy + p 2 iz = 2m 3 Ω 0 (; ) 2m 3 ( ) 3 ( Ω 0 (; ) = π3/2 L Γ( 3 2 + = π3/2 2 )(2m)3/2 h Γ( 3 2 + ) Stirling ( 4πe Ω 0 (; ) = 3π 3 ) 3/2 ɛ 0 ) 3/2 2 22 p p

6 2 W ( ; ) W ( 3/2 ) 23 2 s 2 t t 2 s 2 Ω 0 (; ) 24 2 25! Ω 0 (; ) Ω 0 (; ) = ( ) 3/2 ( 4πe 4πe 5/3 =! 3π 3 6π 3 ɛ ) 3/2 ɛ = 2/3 ɛ 0 = h2 m ( ) 2/3 V 26 ɛ V ɛ 0 ɛ! 27 Ω(; ) = ( 4πe 5/3 6π 3 ɛ ) 3/2 28! 23 ɛ 0 24 25 26 ɛ 27 28

2.5. 7 29 ɛf 0 g(ɛ)dɛ = ɛ F 30 3 ɛ F ɛ ɛf 0 ɛg(ɛ)dɛ = ɛ5/2 F ɛ 3/2 0 ɛ5/2 = ɛ ɛ 3/2 0 32 ɛ 33 ɛ 2.5 3 r i p i 6 6 6 6 ( ) 34 29! 30 ɛ F ( 3 4π ) 3/2 3 32 2 33 epsilon epsilon 34 ( )

8 2 35 36 Π i r i p i A Π i r i p i 37 A [ ] 3 A 2.5. L 2m H = p 2 i p 2 i = 2m L ( ) Π dx i dy i dz i θ 2m Π dp xi dp yi dp zi 0 (θ(x) ) V 2m 3 A Ω cl 0 (; ) = A V π3/2 Γ( 3 2 + )(2m)3/2 38 A =!h 3!h 3 39 35 36 Liouville 37 A A 38 cl 39 A p 2 i

2.5. 9 ɛ ɛ 40 2.5.2 H = p 2 i 2m + mω2 q 2 2 p 2 i + (mωx) 2 = 2m L ( ) Π dx i dy i dz i Π dp xi dp yi dp zi θ 2m p 2 i + (mωx) 2 0 2 0 () = π A (mω) Γ( + ) ( 2m) 2 = ( ) 2π = (eh) ( A Γ( + ) ω A 2π hω Ω cl Ω cl () = Ωcl 0 () = A 2π (eh) Ω() = hω [ = hω 2π Γ( Ē hω +) Γ( Ē hω +)Γ() hω (+ hω ) + hω 2 ( hω ) hω + 2 ( ) 2 hω { [ = hω 2π + hω ( )] + hω ( ) 2 ( ) hω 2 Stirling 4 ] } ) 40 4 ( )

20 2 hω hω hω hω 42 [ lim + hω hω ( ] ) hω = e Ω(; ) = [ ( ) ( e ) + ] 2 hω 2π hω ( ) + 2 e 2 / 2 Ω(; ) = ( ) hω 2π e hω A = h 43 44 f ( 2f)A h f 45! 46 42 43 ɛ 0 44 = 45 [ ]/h f 46!!

2 3 3. 2 3 0 23 ( ) [, + ] 4 2 3 4

22 3 3.2 5 A = lim T A(t)dt T T A(t) A t A(t) 0 23 6 0 T i S i (T ) Si(T ) T T i 0 S i (T ) P i lim T T A P i A = i P i A i A i A i P i 7 8 P i [, + ] 5 m/s m (/ ) 6 0 6 7 P i P i 8

3.2. 23 P i 9 W (, ; ) P i = W (, ; ) [, + ] W (, ;) A = W (, ; ) 0 V = V = V W = = W H = W U i i U = 2 U(S, V, ) 3 S = log P = k B P i log P i 9 0, V 2 trivial 3 U (S, V, ) T U(T, V, ) i i i i A i

24 3 k B 4 S = k B i W log W = k B log W (, ; ) 56 (U, V, ) 7 T = P T = ( ) S U ( ) S V 2 V, U, S +2 = S + S 2 W +2 = W W 2 3.3 ( 0 23 ) ( 0 ɛ 0 ). [, + ] 4 5 6 k B S = log W T S k B T SI 7

3.3. 25 8 2. 3. 9 S = k B log Ω(; ) = k B log Γ ɛ [ ( ) 0 U = k B log U U log ΓΓ ( ) U + 2 log 2π + O ( ) ] + log ɛ 0 0 23 log 00 20 O(log ) log O( 0 ) O ( ) 0 20 [] U O() 2 log ɛ 0 ɛ 0 22 O() ɛ 0 log ɛ 0 O(log ) 23 S(U, ) = k B [ U log U + ( U ) ( log U )] ( ) S T = = k [ ( )] B ɛ0 log U ɛ 0 U 8 2 3 2 3 2 3 9 20 2 V V O() 22 < ɛ 0 23

26 3 U = exp(βɛ 0 ) + β k BT c = ( ) U = k B (βɛ 0 ) 2 exp(βɛ 0 ) T [exp(βɛ 0 ) + ] 2 U βɛ 0 = ɛ0 k BT ɛ 0 ɛ 0 24 kbt k B T ɛ 0 25 ɛ0 k BT U = 2 ( βɛ 0 2 ) ɛ 0 ɛ0 k BT c = k B 4 (βɛ 0) 2 T U 2 c 0 26 ɛ0 k BT exp( βɛ 0 U = exp( βɛ 0 ) c = k B (βɛ 0 ) 2 exp( βɛ 0 ) T 0 U 0 27 ɛ 0 ɛ 0 exp( βɛ 0 ) k B T ɛ 0 log k B T 28 24 u = U t = k BT ɛ 0 25 ɛ 0 26 T U U > 2 0 < U < 2 27 28

3.4. 27 3.4 ( 0 23 ) ( ω). [, + ] 2. 3. Ω(; ) = hω Γ( Ē hω + ) Γ( Ē hω + )Γ() = hω Γ(( hω + )) Γ(( hω + ))Γ() hω O( 0 )( ) 29 u Ω(; ) = hω hω Γ((u + )) Γ((u + ))Γ() = 2π hω (u + )(u+ x (u+ Stirling 2 ) 2 ) ( )( log Ω(; ) = [ (u + 2 ) log(u + ) (u + 2 ) log u ( 2 ) log( ) + 2 log 2π + log hω log( + a) a + O() O( 0 ) 30 [ log Ω(; ) (u + ) log(u + ) u log u + 2 log + ] log hω + O( 0 ) 3 U U S(U, ) = k B [( + ) log( hω hω + ) ] U hω log U ] hω 29 O() 30 log 3 log hω hω log 2 )

28 3 T = ( ) S = k ( B U hω log + hω ) U U hω = exp(β hω) 32 c = ( ) U = k B (β hω) 2 exp(β hω) T [exp(β hω) ] 2 hω k BT ( ( U hω = k BT hω ) ) hω 2 + O k B T c = k B ( + O ( ( ) )) 2 hω k B T T U k B T c k B U = k B T c = k B = = 33 hω k BT exp( β hω) U = exp( β hω) c = k B (β hω) 2 exp( β hω) T 0 U 0 34 k B T hω log 3.5 T 0 0 a (a O( 0 ) ) 0 32 U β hω = hω hω k B T hω ω 33 = = 34

3.6. 29 3.6 [, + ] [( 3 S = k B log Ω(; ) = k B 2 ) ( ) 4πe 5/3 log + 3 ɛ log 3 2 6π + ] log ɛ ɛ = h2 m ( ) 2/3 V S(U, V, ) = 3 ( ) 4πe 5/3 2 k U B log 3 ɛ k B T = 2U 3 c v = 3 2 kb pv = 2U 3 u 0, v 0, s 0 [ S(U, V, ) = s 0 + 3 2 log U + log V ] u 0 v 0 0 U 3 = ɛ 4πe 5/3 u 0 v 2/3 3 m 0 = 4πe 5/3 h 2 u 0, v 0 U ɛ

30 3 35 k B T ɛ 36 T 0 3738 Ω(; ) = ɛ 0 Ω( ; ) S = k B log Ω(; ) = k B log Ω( ; ) + k B log ɛ 0 O(log ) Ω S = k B log Ω( ; ) S = k B log Ω(; )ɛ 0 Ω ω Ω( ; ) ω( )) S = k B log ω( U ) 3.7 35 U 36 ɛ 37 T 0 38 u 0, v 0

3.7. 3 3.7. 2 [, + ] 2 [ 2, 2 + 2 ] Ω, Ω 2 Ω ( ; )Ω 2 ( 2 ; 2 ) 2 [, + ]( ) W (, ) = + 2 + d d 2 Ω ( ; )Ω 2 ( 2 ; 2 ) = + d 0 d d 2 Ω ( ; )Ω 2 ( 2 ; 2 )δ( + 2 ) = + d 0 d Ω ( ; )Ω 2 ( ; 2 ) = 0 d Ω ( ; )Ω 2 ( ; 2 ) Ω() = 0 d Ω ( ; )Ω 2 ( ; 2 ),2 Ω ( ; )ɛ () 0 = k B S (, ) Ω 2 ( 2 ; 2 )ɛ (2) 0 = k B S 2 ( 2, 2 ) ɛ () 0, ɛ(2) 0 39 [ ] Ω() = 0 d exp k B (S (, ) + S 2 (, 2 )) ɛ () 0 ɛ(2) 0 (S (, ) + S 2 (, 2 )) = 0 Ω() [ ] [ ] exp k B (S () + S 2 ( )) 0 d exp 2 2k B (S 2 (, ) + S 2 (, 2 ) = )( ) 2 [ ] = exp k B (S () + S 2 ( )) 39 log

32 3 (, ) 40 [ 2 ] d exp 2k B 2 (S (, ) + S 2 (, 2 ) = )( ) 2 πk B = S + S 2 (S, S 2 S ) 4 2 2 S() = k B log Ω() = S ( ) + S 2 ( ) 2 42 (S (, ) + S 2 (, 2 )) = = 0 S = S 2 2 2 40 4 S + S 2 < 0 42