LY THUYET HE NHIEU HAT

NGUYN VN TRUNG : 0915192169 Lí THUYT H NHIU HT Chng 1: Tớnh cht chung ca h nhiu ht 0- Khỏi nim v h nhiu ht 0.1- Nhiu : N : Vn k thut : s bin ; tng tỏc ; thay i v cht 0.2- Nhiu (N >>1) : khụng lm thay i cht 0.3- Nhiu (N >> 1) : lm thay i cht 0.4- H nhiu ht T=0K 0.5- H kớn 0.6- H T 0K Quan h gia C hc v Vt lý thng kờ (bao gm c c in v lng t) 1- H ht ng nht: 1.1- Nguyờn lý khụng phõn bit cỏc ht ng nht c hc lng t 1.2- Hm súng ca h cỏc ht ng nht 1.2.1- Tớnh i xng ca hm súng (1.1) Pij (q1, , qi , , qj , , qN) = (q1, , qj , , qi , , qN) + (q1, , qi , , qj , , qN) = + (q1, , qj , , qi , , qN) - (q1, , qi , , qj , , qN) = - - (q1, , qj , , qi , , qN) 1.2.2- Đặc điểm tính đối xứng hàm sóng 1.2.2.1- Tính đối xứng nh tất cặp biến : 1.2.2.2- Tính đối xứng hàm sóng phụ thuộc vào spin : Spin nguyên (0 ; ; ; .) Spin bán nguyên (1/2 ; 3/2 ; 5/2 ; .) 1.2.2.3- Tính đối xứng hàm sóng vĩnh cửu : 1.2.3- Dạng hàm sóng hệ hạt đồng không tơng tác pi (qi ) = ni (ri ) ( si ) ; q i = ( ri , s i ) ; pi = (ni , ) * pi (qi ) pk (qi ).dqi = (1.2) (1.3) (1.4) * * d r ( r ) ( s ) ( r i ni i i nk i ) ( si ) = ni , nk = pi pk Si (1.5) dri = dxi dy i dz i + (q1 , q2 , , q N ) = c p (q1 ) p (q2 ) . p (q N ) (q) N (1.6a) p1 (q1 ) p1 (q ) p1 (q N ) (q1 , q , , q N ) = p2 (q1 ) p2 (q ) p2 (q N ) N ! pN (q1 ) pN (q ) pN (q N ) (1.7a) Định thức Slater chứa đựng Nguyên lý loại trừ Pauli 2- Các đại lợng bảo toàn hệ nhiều hạt 2.1-Hamiltonian hệ nhiều hạt N H = ( / 2) ( i / mi ) + V (r1 , r2 , , rN ) (2.1a) i =1 H = ( / 2) ri r + r sin sin i + r sin + V (r , , ) i =1 ri ri i i i i i i i ( r (r1 , r2 , , rN ) ; (1 , , , N ) ; (1 , , , N ) ) N 2.2- Bảo toàn động lợng hệ nhiều hạt (2.1b) N P = i k (2.2) k =1 N N r& r r r r r r P = ( PH HP) = ( PV VP ) = ( kV V k ) = k V = Fk = Fint + Fext = Fext ih ih k =1 k =1 k =1 N Do đó: Fint = Fi = Fij = Nừu: F = ext i i j 2.3- Bảo toàn mô men động lợng hệ nhiều hạt N N L = k ; L z = i kz ; thay kz = i / k , k =1 k =1 N L z = i k =1 k N N L = ( L H HL ) = V = M z z z kz i k =1 k k =1 N M k =1 kz (2.4) (2.5a) = M z , int + M z , ext = M z (2.5b) CM : M z , int = Với Lz L2 bảo toàn 3- Biểu diễn tơng tác S (t ) = H S (t ) t S (t ) = [exp(iHt / )] H F (t ) = e iH t / F e iH t / Biểu diễn Shrodinger : Biểu diễn Heisenberg : i (3.1) H (3.2) (3.3) (3.4) (3.5) S H = [exp(iHt / )] S (t ) H = H + V Biểu diễn tơng tác : Fi (t ) = e iH t / FS e iH t / (3.6) i (t ) = [exp(iH t / )] S (t ) i (t ) i = Vi (t ) i (t ) t V (t ) = e iH t / V e iH t / (3.7) i (3.8) (3.9) S t i (t ) = i (t ) (i / ) Vi (t ' ) i (t ' )dt ' (3.10) i (t ) = i( ) (t ) + i(1) (t ) + i( ) (t ) + i (t ) = S (t , t ) i (t ) (3.11) (3.16) t0 t t1 t S (t , t ) =1 (i / ) Vi (t1 ) dt1 (1 / ) Vi (t1 ) dt1 Vi (t ) dt + + t0 t0 t0 t t1 t n t0 t0 t0 + (i / ) Vi (t1 ) dt1 Vi (t ) dt Vi (t n ) dt n + n Coi : t S (t , t ) = T exp(i / ) Vi (t ' ) dt ' t0 S (t , t1 ) S (t1 , t ) = S (t , t ) ; t > t1 > t Ký hiệu : (3.20) V (tV ) = S (t ) = S (t , tV ) S (t2 , t1 ) = S (t2 ) S (t1 ) (3.21) (3.17) (3.18) (3.19) (2.3) Trong ú : i (t ) = S (t ) i (tV ) (3.22) ; thay t = tV ==> i (tV ) = H Từ (3.2) : S (t ) = [exp(iHt / )] H i (t ) = [exp(iH t / )] [exp( iHt / )] H (t ) = S (t ) i (3.23) H Fi (t ) = S (t ) FH (t ) S (t ) M = 0H* T [ A H (t ) B H (t ' ) C H (t ' ' ) ] 0H Giả thiết (3.24) (3.25) t > t > t > M = S () T [ A i (t ) B i (t ' ) C i (t ' ' ) S ()] 0H 0* H M= Cuối : Chng : (3.26) S () 0H = e i 0H 0* T [ A (t ) B (t ' ) C (t ' ' ) S ()] H i i i (3.27) H (3.28) 0H* S () 0H Mt s phng phỏp gii bi toỏn h nhiu ht 4- Phơng pháp tách chuyển động khối tâm hệ : 4.1- Đặc điểm tơng tác: N H = ( / 2) ( i / mi ) + V (r1 , r2 , , rN ) (4.1a) V (r1 , r2 , , rN ) = V (r1 r2 , r1 r3 , , rN rN ) (4.1b) i =1 Sự phụ thuộc dẫn đến kết Décartes r ( x, y, z ) Jacobi ( , , ) : = [ (m1 x1 + m2 x2 ) : (m1 + m2 )] x3 ; = (m1 x1 : m1 ) x ; k k k = m j x j : m j xk +1 , với k = , , , N j =1 j =1 N = mi xi : mi N i =1 N i =1 (4.2a) (4.2b) Tơng tự cho toạ độ i i : : r , i = xi2 N N i =1 i =1 ( r, i / mi ) = ( , i / ài ) Có thể chứng minh đợc : + yi2 + zi2 ; , i = k ( àk ) =( m j ) + ( mk +1 ) j =1 i2 + i2 (4.3a) + k = , , , N N N = mi Khi (4.3b) i2 (4.3c) (4.3d) i =1 N H (r1 , r2 , , rN ) = ( / 2) ( r, i / mi ) + V (r1 , r2 , , rN ) i =1 N = H ' ( , , , N ) = ( / 2) ( , i / i ) + V ' ( , , , N ) i =1 (4.4) r ==> r + a ; H (r ) = H (r + a ) , ==> V (r ) =V (r + a ) : V (r1 , r2 , , rN ) = V ( x1 + a x , y1 + a y , z1 + a z , x + a x , y + a y , z + a z , , x N + a x , y N + a y , z N + a z ) = V ' ( , , , , , , , N + a x , N + a y , N + a z ) V '( , , , , , , , N + ax , N + a y , N + az ) = V ' ( , , , , , , , N , N , N ) , N H ' ( , , , N ) = ( / 2) ( , i / ài ) + V ' ( , , , N ) Kết (4.5) i =1 4.2- Phơng trình Shrodinger cho hệ đãtách chuyển động khối tâm: ( , , , N ) = ( , , , N ) G ( N ) (4.6) N [ ( / 2) ( , i / i ) + V ' ( , , , N )] ( , , , N )G ( N ) = E ( , , , N )G ( N ) i =1 N [ ( / 2) ( , i / i ) + V ' ( , , , N )] [ , N / N ] G = E 2G i =1 [ ( / 2) N ( , i / i ) + i =1 V ' ( , , , N )] ( , , , N ) = E1 ( , , , N ) [ , N / N ] G ( N ) = E2 G ( N ) i =1 (4.8c) (5.1) H i (ri ) = i + ui (ri ) 2mi i, j (4.8a) (4.8b) Với E1 + E2 = E Ví dụ : Xét hệ gồm hạt (N =2): Bai 5- Phơng pháp trờng trung bình 5.1- ý tởng phơng pháp trờng trung bình H = E N H = H i ( ri ) + (1 / 2) Vi j ( ri , r j ) (4.7) (5.2) H 'i (ri ) = i + ui (ri ) + Vef (ri ) 2mi i =1 N = (r1 , r2 , , rN ) = pi ( ri ) N r H = H 'i (ri ) với (5.3) (5.4) i =1 [ N i + ui ( ri ) + Vef ( ri )]pi ( ri ) =ipi ( ri ) 2mi i = E (5.5) Q[ ] = [ H E ] dq (5.6) i =1 * dq = dqi , q i = ( ri , s i ) ; N i =1 dq = N dr (5.7) i i =1 Si Q[ ] = *[ H E ] dq = 5.2- Thế hiệu dụng Vef hệ hạt boson (5.8) N * * ( q ) ( q ) [ pi i pk k H i (ri ) + (1 / 2) Vi j (ri , rj ) E ] dq i =1 i k i, j N + * [ H i ( ri ) + (1 / 2)Vi j ( ri , rj ) E ] pi ( qi ) p k (qk ) dq = , i =1 dq k (5.9) i k i, j N *p (q k ) *p (q i ) [ H i (ri ) + (1 / 2)Vi j (ri , r j ) E ] dq i k i k i i =1 i k i, j N + dqk p k (qk ) pi (qi ) [ H i* (ri ) + (1 / 2) Vij* (ri , rj ) E ] * dqi = i =1 ik N r r r p*i (qi )[ H i (ri ) + (1/ 2) Vi j (ri , rj ) E ] dqi = ik i =1 i, j (5.10) ik i, j ik (5.11) ' ' ' Vi j (ri , r j ) = V ik + Vij c1 = i k * pi (5.12) ik j k i i, j (qi ) [ H i (ri )] pi (q i ) dqi i k i k (5.13a) i k c = *pi (qi ) [(1 / 2) ' Vij (ri , r j )] pi (qi ) dqi i k j k i k i k (5.13b) i k Vef (rk ) = *pi (qi ) [ ' Vik ( ri , rk )] pi ( qi ) dqi ik ik ik (5.14) i k * pi (qi ) [ H i (ri )] dqi = [c1 + H k (rk )] pk (qk ) (5.15a) * pi (qi ) [(1 / 2) ' Vij (ri )] dqi = [c + Vef (rk )] pk (q k ) (5.15b) ik i k i ik i ik i, j (qi ) E dqi = E p k (qk ) * pi (5.15c) i k [ H k (rk ) + V ef (rk )] pk (q k ) = k pk (q k ) (5.17a) k = E c1 c2 5.3- Thế hiệu dụng hệ hạt fermion (q1 , q ) = (5.17b) [ (q1 ) (q ) (q ) (q1 )] (1.7b) [ (q ) (q ) (q ) (q )] ( H + H + V E ) [ (q ) (q ) (q ) (q )] dq dq (5.18) + [ (q ) (q ) (q ) (q )] ( H + H + V E ) [ (q ) (q ) (q ) (q )] dq dq = (q ) (q ) ( H + H + V E ) (q ) (q ) dq dq = = (q ) (q ) ( H + H + V E ) (q ) (q ) dq dq * * (q ) (q1 ) ( H + H + V12 E ) (q ) (q1 ) dq1dq = (q ) (q ) ( H + H + V E ) (q ) * * (q1 ) (q2 )V12 (q2 ) (q1 ) dq1dq2 = (q ) (q )V (q ) (q ) dq dq (q ) (q )V (q ) (q ) dq dq = (q ) (q ) V (q ) (q ) dq dq * * 1 * 2 * * 1 * * 2 * 2 12 1 2 2 1 2 12 1 2 1 2 * 1 * 12 1 2 * 1 2 12 1 2 * 2 * 1 * 2 12 * 1 2 2 12 1 (q2 ) dq1dq2 * 2 * * 1 * 1 * 12 1 2 * 2 12 1 2 ( q1 ) *2 (q ) ( H + H E ) (q ) ( q1 )dq1 dq = (q ) (q ) ( H + H E ) (q ) (q ) dq dq = * * (q1 ) (q2 ) ( H + H E ) (q2 ) (q1 )dq1dq2 = * * 2 1 2 2 (q ) (q1 ) ( H + H E ) (q1 ) (q ) dq1 dq = * * (q ) (q ) ( H + H + V E ) (q ) (q ) dq dq (q1 ) (q2 )V12 (q2 ) (q1 ) dq1dq2 + + * (q1 ) * (q ) ( H + H + V12 E ) (q1 ) (q ) dq1 dq (q ) (q ) V (q ) (q )]dq dq = dq1 * (q1 ){[ * (q ) ( H + H + V12 E ) (q ) dq ] (q1 ) (q ) V (q ) (q ) dq } + + dq1 (q1 ){[ (q ) ( H 1* + H 2* + V12* E ) * (q ) dq ] * (q1 ) (q ) V12* * (q ) * (q1 ) dq } = * * 1 2 12 1 2 * * * 1 2 * 2 12 1 2 * 1 2 2 1 r [ H1 +Vef (r1 )]1 (q1 ) = 11 (q1 ) = E 02 , H ( q ) = 02 ( q ) 2 2 12 2 (5.19a) (q1 ) * Vef ( r1 ) = *2 ( q2 )V12 (r1 , r2 ) (q ) dq2 (q2 )V12 (r1 , r2 ) (q ) dq2 (q1 ) [ H + Vef (r2 )] (q2 ) = 2 (q2 ) = E 01 H 11 ( q1 ) = 011 (q1 ) ; (q ) * * Vef (r2 ) = (q1 ) V12 (r1 , r2 ) (q1 ) dq1 (q1 ) V12 (r1 , r2 ) (q1 ) dq1 (q ) j (qi ) * Vefi (ri ) = ' *pi (q j ) Vij (ri , r j ) p j (q j ) dq j ' p j (q j ) Vij (ri , r j ) pi (q j ) dq j i (qi ) j j 6- Phơng pháp lợng tử hoá lần thứ hai 6.1- ý tởng phơng pháp (q1 , q2 , , q N ) = c p (q1 ) p (q2 ) . p (q N ) (q) N 6.2- Toán tử sinh hạt, toán tử huỷ hạt toán tử số hạt cho hệ hạt boson: a i ., N = N i ., N i a i+ ., N a i+ a i , N i = N i a i+ , N i N i + ., N = = N i N i , N i = N i , N i N i , N đợc : i = N i , N i (6.6) a i a k+ a k+ a i = ik a i a k a k = ai+ a k+ a k+ a i+ = 6.3- Toán tử sinh hạt, toán tử huỷ hạt toán tử số hạt cho hệ hạt fermion: Ni = : a i , N = 0, = ; a i , N =1, = , N = 0, i i (6.7) (6.8) i a i+ , N i = 0, =0 ; (6.1) (6.5) Do : a i+ , N i =1, (5.20c) (6.4) N i = ai+ a i Ký hiệu (5.20b) (6.3) +1 i (5.19b) (6.2) i i (5.20a) = , N i =1, a i , Ni , = N i , Ni 1, ; a i , N i + 1, = N i , Ni , (6.9) a i+ , Ni , = N i , Ni +1, ; ai+ , Ni 1, = N i , Ni , (6.10) N i , N i , = a i+ a i , N i , = N i a i+ , N i 1, = N i , N N i = a i+ a k a i+ + a i+ a k = ik a i a k + a k a i = a i+ a k+ + a k+ a i+ = 6.4- Hamilton phơng pháp lợng tử hoá lần thứ hai H = H a + Va , b + Va , b, c + a a,b a , b, c i , (6.11a) (6.11b) (6.12) (6.13) (6.14) a + u (ra ) ma < H a > = i Ni (6.15) i = < i H a i > = i* (qa ) H a i (qa )dqa (6.17) Ha = a i (6.16) H a = i N i = i ai+ a i (6.18) i Vik = i* (q a ) V ( q a , q b ) k (qb )dq a dqb Vik => VikNk => VikNkNi => < Va , b > = a,b Vik N i N k i, k H = a i + i a i + i H = (H ) a i+ a k + a i ,k i ,k (6.19) Vik N i N k i ,k Va ,b = a ,b 1 Vik N i N k = Vik ai+ ak+ ak i,k i,k Vik a i+ a i a k+ a k + i ,k (Va ,b ) ik ,m a i+ a a k+ a m i ,k ,,m (6.22) (6.23a) ( H a ) i ,k = i ,k = i* (q a ) H a k (q a )dq a Trong đó: Vik ,m = i* (q a ) k* (qb ) V (q a , q b ) (q a ) m (q b )dq a dq b (q a ) = i ( q a )a i i + (q a ) = i* (q a )a i+ i ( q a ) + ( qb ' ) + (q a ' ) (qb ) = (q a qb ' ) ab (q a ) (q a ' ) (q a ' ) (q a ) = + (q a ) + (q a ' ) + (q a ' ) + (q a ) = F (1) = f (q a ) a F (1) = + ( qa ) f (qa ) (qa )dqa H = + (q a ) H a ( q a )dq a + + (q a ) + (qb )V (q a , qb ) ( qb ) (q a )dq a dqb (6.21) (6.23b) (6.23c) (6.24a) (6.24b) (6.25a) (6.25b) (6.25c) (6.26) (6.27) (6.28) Chng 3: Hamiltonian v phng trỡnh Shrodinger cho mt s h nhiu ht 7- Phng trỡnh Shrodinger cho h cỏc electron v cỏc ion tinh th 7.1- Phng trỡnh Shrodinger tng quỏtcho h cỏc electron v cỏc ion H (r , R ) = E ( r , R) H = i ri RJ + V (r , R) 2m J 2M J (7.1) V (r , R ) = V1 (r , R ) + V2 ( R ) V1 ( r , R) = Ve e (r ) + Ve I (r , R ) V2 ( R ) = V I I ( R ) (7.4) (7.5) (7.6) 7.2- Gn ỳng on nhit v cỏc phng trỡnh Shrodinger cho h cỏc electron v cho h cỏc ion ( r , R ) = ( r , R ) ( R ) (7.7) r r r h2 r r r r RrJ V2 ( R )]1 (r , R) ( R ) = E1 (r , R ) ( R ) [ ri + V1 (r , R )] ( r , R ) ( R ) [ J 2M J i 2m [ ri + V1 (r , R)] (r , R) + RJ + V2 ( R)] ( R) = E (r , R) , [ (r , R) ( R) J 2M J X J i 2m [ ] [ i ri + V1 (r )] (r ) = (r ) 2m RJ J 2M J W = E Ee [ V1 (r ) = V1 (r , R ) + V2 ( R) + Vef ( R )] ( R) = W ( R) Vi : (7.13) 8- Trng thỏi v nng lng ca electron mng tinh th i 2m ri + V1ef (ri ) (r ) = (r ) [ ri + V1ef (ri )] ni (ri ) = i ni ( ri ) 2m ( = i ) ; V (r ) = V (r ) + V (r , R ) + V (r , R ) 1ef i ef e i iI i I i J i J (8.1a) (8.1b) (8.2) i Vi I (ri , R I ) = Vi J (ri , R ) = zI e2 ri R I J I zJ e2 ri R J 8.1- Phng trỡnh Shrodinger cho electron trng hp liờn kt mnh Nguyờn t cụ lp Tinh th f =3 7N d =2 5N 4s 3p =1 p 3N a) b) s Hỡnh 1N =08.1 : Cỏc mc nng lng ca electron Hỡnh 8.2 : Hin tng chng a) nguyờn t cụ lp b) tinh th 8.2- Phng trỡnh Shrodinger cho electron trng hp liờn kt yu (r ) * ' * ' nj i Vef e (ri ) = nj (r j ) Vij (ri , r j ) nj (r j ) dr j (8.5) nj (r j ) Vij (ri , r j ) ni (r j ) dr j j j ni ( ri ) k (r + a ) = k (r ) k (r ) = k (r ) exp (ik r ) (8.6) ; (8.7) (8.1b) [ r + Vef e (ri )] ni (ri ) = i ni ( ri ) 2m i Vi Mụ hỡnh Kronig-Penney : Vef e ( X J + a ) = Vi ( X J ) (8.8) Vef e ( X J ) = ( X J na ) V0 b n = lim (cV0 ) = const c V0 c x O a Hỡnh 8.3 S th nng ca mụ hỡnh Kronig-Penney 9- Dao ng mng tinh th 9.1- Phng trỡnh Shrodinger cho cỏcdao ng mng tinh th biu din to V J ( R ) = V2 ( R ) + Vef ( R) (9.1) u n = (u n , x , u n , y , u n , z ) = lch ca nguyờn t v trớ cõn bng nỳt mng th n V J ( R) = V J (u ) = V J (0) + (V J / u n , ) u n , + + (1 / 2) ( V n , J / u n , u n ', ) u n , u n ', + ( V + (1 / 6) n , n ', , J / u n , u n ', u n '', ) u n, u n ', u n '', + n , n ', n '', , , (V J / u n , ) = 9.2- Phng trỡnh Shrodinger cho cỏc phonon biu din lng t hoỏ ln th hai VJ ( R ) = An ,n ' xn xn ' (9.4) n,n ' [ V J ( R ) = An x n2 n ] (9.5) H ph = p /(2 M n ) + M n x / = H n n ú n n n (9.6) (9.7) (9.8a) (9.8b) (9.9) (9.10) (9.11) (9.13) (9.16) n H n = p /(2 M n ) + M n x / 2 n n n A = M / x i (1 / M ) p A = M / x + i (1 / M ) p = A A + A + A + H = H n = ( / ) + A + A H E = E E => HA E = ( E )A E ; HA + E = ( E + )A + E A = ==> E0 = / T (9.13) Cn ==> E n = ( + n) ; C n n = ( A + ) n ==> n = , 1, , , (9.22) = < ( A + ) n ( A + ) n > = < A n ( A + ) n > A n ( A + ) n = A n n ( A + ) n + A n ( A + ) n A 2 C = n < A n ( A + ) n > = n C = n! n C n (9.21) n (9.23) 2 C0 = ; ú C n = n! n v C n = n! n n Cui cựng : A = => A + n = n! [ M / x + (1 / M )( / x ) ] ( x) = 0 ( x) = C exp[m x / ( ) ] m ( x) dx = ==> C = 1/ (9.24) (9.25) (9.26) m / exp[ m x / ( ) ] , ú ( x) = (9.27) 10- Hamiltonian cho h cỏc spin 10.1- Trng hp h cỏc electron linh ng M = g àB N N (10.1) V N + N = N H = H1 + H N H1 = i =1 r 2m i H = H2 = V 1ef (10.2) (10.3a) (10.3b) (10.3c) (ri ) i ( r ) nj i * nj (r j ) Vij (ri , r j ) ni (r j ) dr j i , j =1 ni (ri ) N (10.3d) i j E =< H > (10.4) p1 (q1 ) p1 (q ) p1 (q N ) p2 (q1 ) p2 (q ) p2 (q N ) N ! p N (q1 ) p N (q ) pN (q N ) p j (q j ) = k (r j ). ( s j ) = exp (ik j r j ). ( s j ) j V N E d = < H > = * (q1 , q , , q N ) ( r ) (q1 , q , , q N ) dq1 , dq , , dq N 2m i =1 = (q1 , q , , q N ) = (10.5) (10.6) (10.7) i dq dr = i (10.8a) i Si dri = dxi dy i dz i Ed = < H1 > = N j =1 k j 2m 2k k < k F 2m =2 V V Ed = 8m (10.9) dk = dk x dk y dk z = dk k (10.8b) V k d k = 2m ki < k F kF k dk = V k F5 10m (10.10) kF V Vk F3 V N = = dk = k dk = 4 k k 3F + k 3F = k F3 (10.16) H2 = 2V N exp [ i(k i , j =1 i j j k i )(r j ri )]Vij (ri , r j ) dr j e2 ri r j N H2 = exp [ i ( k j k i ) r ]Vij ( r ) dr = 2V i , j =1 2V (10.17) Vi , j (ri , r j ) = i j V (k j k i ) = Et = H = e2 V Et = V k < k F k '< k F e2 k j ki (10.18) N i , j =1 i j V (k j k i ) (10.20) e2 V k ,k '< kF k k ' (10.21) dk ' dk e k V = k F F ( ) dk kF k < k F k k' x2 1+ x F ( x) =1 + ln 2x x Et = N (10.24) me 4 Ry = a0 = (10.25) ; (4 ) me 3 Et = N k F a0 R y N k a0 R y 2 F 3 E d = N ( k F a ) R y + N (k F a ) R y 5 3 3 E = E d + Et = N [ (k F a ) (k F a )]RY + N [ (k F a ) (k a )]RY 5 F 3 N = N = N / , k F = k F = k F : E N = E dN + E tN = N [ (k F a ) (k F a )]RY N = N v N = , k F = 21 / k F ; k F = EM < EN (10.22) (10.23) k F a0 R y 3 1/ E M = E dM + EtM = N [ 2 / (k F a ) ( k F a )]RY 5 = 0,352125 (10.32) k F a < 1/ 2 + (10.19) (10.26) (10.27) (10.28) (10.29) (10.30) (10.31) Et > Ed (10.33) Y nghia ca iu kin (10.32) H= ,k k F ci+ c+ cm c j ( k E ) ak + k ak'Vkk' =V0 = *V dr kk ' k k' k '> k V0 < F < k ; k ' F< F + D V kk' = , ngoa `i khoa ? ng V0 tre ^ n k a = k'0 ak = ( k E )0akkhi V0 a k' ( k E ) k '>kF k ' k '> k F 1 = V0 k >kF ( E k ) 1 = V0 k >kF ( E k ) (11.11) k > F , E > Trng thỏi liờn kt C F lk = F C E = C < F (11.12) 1 = V0 k >kF ( k EC ) g ( ) g ( F ) g ( F )V0 = ln kTC lk / V0 F + D g ( ) d = V0 ( EC ) F lk + D lk F + D E = g ( F )V0 F + D EC + D d = g ( F ) = g ( F ) ln F V0 ( EC ) F EC F TC ( D / k ) exp(1 / ) lk = F C lk = D exp(1 / ) 11.2- Toỏn t hai ht v trng thỏi chõn khụng ca h siờu dn ck+ = a k+ a +k ; ck = a ka k H = ( k / 2) (a k+ a k + a +k a k ) + (1 / 2) V a + a + a a kk ' k k k ' k ' k k ,k ' nk n k = = ; nk nk =1 = nk =1 n k = n k = nk n k ck+ ck = a k+ a +ka ka k = a k+ a ka +ka k = n kn k = n k = n k H = k ck+ ck + (1 / 2) V c + c kk ' k k ' k k ,k ' (0) = u (0) + v c (0) = (u k + vkck+ ) k (0) k k + k k k k k k u +v =1 (0) = (u k + vkck+ ) k (0) k v : ck (0) = ck (0) = ck [uk' + vk'ck+' ] nk =0 = a k ak [uk' + vk'ak+' a+k' ] n = = k = [u k' k k ' k k' k' + + k ' k ' k ' + v a a k' + + k k k ] nk =0 a k a k [u k + v a a ] nk =0 [u k + vka +k a k+ ] nk =0 a k a ku k nk =0 v ka +k a k+ nk =0 = u kvka k a k nk =0 [ a k+ a +k nk =0 ] = = u kv ka k a k [a k+ a +k nk =0 ] nk =0 = u kvka k a k a k+ a +k nk =0 nk =0 u kvka k a k a k+ a +k nk =0 nk =0 a kakvkak+ a+k k (0)v ka+kak+ k (0) ~ vka ka k+ k (0)a k+ vka ka +ka +k k (0) = Phơng pháp hàm Green lợng tử Chng 4: ý tởng phơng pháp 12- Phơng pháp hàm Green lợng tử nhiệt độ T=0K 12.1- Định nghĩa hm Green lợng tử nhiệt độ T= 0K G ( x, x' ) = i T[ H ( x) H+ ( x' )] < T[ i ( x) i+ ( x' ) S ( )] > G ( x, x' ) = i < S ( ) > (12.1a) (12.1b) r i lim G ( x, x ')dr = a i+ a i = N t ' t + r r r ' r Vì : i G ( x, x' ) N = n(r )dr => n( x) = i tlim 't+ r ' r FH(1) (t ) = i [ lim f ( x)G ( x, x ' )]dr , t't+ r ' r 12.2- Hm Green cho hệ hạt fermion iG ( x, x' ) = T [e i H t / S (q )e i H t / e i H t '/ S+ (q ' )e i H t '/ ] = = T [e i E0 ( t t ') / i (r ) k* (r ' ) a i e i H ( t t ') / a k+ ] i k iG ( x, x' ) = T [e i E0 ( t t ') / i (r ) i* (r ' ) a i e i H ( t t ') / a i+ ] i (12.2) (12.3) d4p i [ p ( r r ') t ] G ( x x' ) = G ( p , ) e (2 ) H (r , t ) = V a pe a +p e i i e ( p '') a +p ''a p '' t / ( p '') a +p ''a p '' t / p '' H (r , t ) = V G (0) p p '' i pr / =e =e (12.4) e i H t / a p e i H t / i i ; d p = dpd ( p '') a +p ''a p '' + ( p ) t / p '' ( p '') a +p ''a p '' ( p ) t / p '' a p a +p e i[ p r ( p ) t ] / a p p H ( x) H+ ( x' ) , t > t ' ( x, x' ) = i T[ H ( x) ( x' )] = i + H ( x' ) H ( x ) , t ' > t + H N p , t > t ' i i[ p ( r r ') ( p ) ( t t ' )] / e V p N p , t ' > t , p < p + N p = a p a p = , p > p N p , t > i G ( ) ( x ) = e i [ p r ( p ) t ] / V p N p , t > r rr N pr , t > i [ ( p )/ h] t (0) r (0) i [( pr / h) t ] r G ( p , ) = G ( x )e dr dt = i dt e N pr , t < G ( ) ( x, x ' ) = = i ( p p0 ) dt e i [ ( p ) / ] t , z > ) , z < + i ( p0 p) dt e i [ ( p ) / ] t ( Vi: ( z ) = ds F (s) F (s) = ds i F (0) +0 + s + i s + i s 0 ( p p0 ) ( p0 p) G ( ) ( p, ) = + = r h ( p ) + i sign ( p p0 ) ( p ) + i ( p ) i 12.3- Hm Green phonon e ist dt = lim e ist t dt = i lim D( x, x' ) = i < T[H ( x)H ( x' )] > k i[ k r0 ( k) t ] k ( r , t )e k+ (r , t )e i[ k r ( k ) t ] u (r , t ) = u + u k k [u i (r , t ), u j (r ' , t )] = i (r r ' ) ij bk = (k ) / u k bk+ = ( k ) / u k+ { = } K = [u (r , t )] dr E = < H >= < K > = = kk ' i [ k r ( k ) t ] i [ k ' r ( k ') t ] + i [ k r ( k ) t ] i [ k ' r ( k ') t ] + ( k ) ( k ' ) bk bk ' e e + b k bk ' e e k , k ' kk ' { b + + b + b >] = (k ) [ N + (1 / 2)] E = ( / 2) ( k ) [ < b 0 k k' k k' k k H ( x) = V D ( ) ( x) = r D (0) (k , ) = D (0) ( x) e k { (k ) / bk e i[ k r ( k ) t ] + bk+ e i[ k r ( k ) t ] k i 2V rr i [ k r t ] k H = ( k ) [ N k + (1 / 2)] } { i[ k r ( k ) t ] i[ k r ( k ) t ] ( k ) ( t ) e + ( t ) e k } r r ei[ ( k )]t , t > i0 (k ) r dr dt = dt ei[ +0 ( kr )]t , t < i ( k ) i [ ( k ) ] t i [ +0 ( k ) ] t = + dt e dt e 0 r (k ) 02 (k ) 1 r D (k , ) = = (k ) + i + (k ) i 02 (k ) + i (0) 12.4- Định lý Wick (H0)* T [ A (q1 , t1 ) B (q2 , t2 ) C (q3 , t3 ) X (qm1 , tm1 )Y (qm , tm )] (H0 ) ( q n , t ) + (q n , t n ) ==> < T[ (q n , t ) + (q n , t n )] > < T [ A (q1 , t1 ) B (q , t ) C (q3 , t ) D (q , t ) X (q m , t m )Y (q m , t m )] > = = < T[ A (q1 , t1 ) B ( q2 , t2 )] >< T[C (q3 , t3 ) D (q4 , t4 )] > < T [ X (qm , tm )Y (qm , tm )] > < T[ A (q , t )C (q , t )] >< T[ B (q , t ) D (q , t )] > < T [ X ( q , t )Y (q , t )] > 1 3 2 m m m m < T[ ( x) + ( x' ) S ()] > G ( x, x ' ) = i < S () > t1 S () =1 (i / ) V (t1 ) dt1 (1 / ) V (t1 ) dt1 V (t ) dt + + t1 t n + (i / ) n V (t1 ) dt1 V (t ) dt V (t n ) dt n + V (t ) = g + ( x) ( x)( x ) dr g2 = G (1) ( x, x ' ) = 2 p0 m (g / ) dx1 < T[ ( x) + ( x ' ) + ( x1 ) ( x1 )( x1 )] > < S ( ) > < T[ ( x ) + ( x' )] >< T[ + ( x1 ) ( x1 )] >< [ ( x1 )] > < [ ( x1 )] > = < [ ( x1 )] > = G (1) ( x, x ' ) = Dễ dàng chứng tỏ tất bậc lẻ gia số hàm Green G ( n+1) ( x, x' ) không } G ( 2) i( g / ) ( x, x ' ) = dx1 dx < T[ ( x ) + ( x' ) + ( x1 ) ( x1 ) ( x1 ) + ( x ) ( x ) ( x )] > < S ( ) > < T[ ( x ) + ( x' ) S ()] > = < T[ ( x) + ( x' ) S ()] > k < S () > G ( x, x' ) = i < T[ ( x) + ( x' ) S ()] > (12.1c) => Hàm Green G ( x, x' ) biểu thị qua hàm Green hệ hạt không tơng tác G ( ) ( x, x' ) 13- Phơng pháp hàm Green lợng tử nhiệt độ T 0K Khụng hc vỡ mụn Phng phỏp hm Green cú mt chng vờ hm Green nhit T 0K 14- Gin Feynman 14.1- Gin Feynman trng hp T=0K 14.1.1- Nhng quy tc ch yu ca ky thut gin < T[ ( x) + ( x' ) S ()] > k G ( x, x ' ) = i < S () > S () = S (,) =1 (i / ) V (t1 ) dt1 V (t1 ) dt1 V (t ) dt + + (i ) n + n ! n V (t1 ) dt1 V (t ) dt V (t n ) dt n + n t i ( i ) dt1 dt n < T [ ( x) + ( x ') V (t1 ) V (t n )] > n < S () > n = n ! h VS = + ( r1 ) + (r2 ) U S (r1 r2 ) (r2 ) (r1 ) dr1 dr2 ( x1 x ) = U (r1 r2 ) (t1 t ) Ky hiu: + + 4 V (t1 ) dt1 = ( x1 ) ( x2 ) ( x1 x2 ) ( x2 ) ( x1 ) d x1 d x2 Khi : n=1 G (1) = d x1 d x < T [ ( x) + ( x' ) + ( x1 ) + ( x ) ( x ) ( x1 )] > ( x1 x ) < S ( ) > G ( x, x ') = < T[ ( x) + ( x' ) + ( x1 ) + ( x ) ( x ) ( x1 )] > = < T[ ( x ) + ( x1 )] > < T [+ ( x2 ) ( x2 )] > < T [ ( x1 ) + ( x ')] > < T[ ( x) + ( x1 )] > < T [ + ( x ) ( x1 )] > < T [ ( x ) + ( x ' )] > + + < T[ ( x ) + ( x )] > < T [ + ( x1 ) ( x1 )] > < T [ ( x ) + ( x' )] > < T[ ( x) + ( x )] > < T [ + ( x ) ( x )] > < T [ ( x ) + ( x ' )] > + + < T[ ( x) + ( x ' )] > < T [ + ( x1 ) ( x1 )] > < T [ + ( x ) ( x )] > + < T[ ( x) + ( x ' )] > < T [ + ( x ) ( x )] > < T [ + ( x ) ( x )] > Thay G (0) : (0) (0) i G(0) ( x, x1 ) G(0) ( x2 , x2 ) G(0) ( x1 , x ') i G( ) ( x, x1 ) G( ) ( x1 , x ) G( 0) ( x , x' ) + i G ( x, x ) G( 0) ( x1 , x1 ) G ( x , x' ) (0) ( 0) i G ( x, x ) G( ) ( x , x1 ) G( ) ( x1 , x' ) i G ( x, x ' ) G( ) ( x1 , x1 ) G( ) ( x , x ) + (0) + i G ( x, x' ) G( ) ( x , x1 ) G( ) ( x1 , x ) Gin Feynman: G (1) phự hp vi gin trờn hỡnh 14.1 x2 a) a) b) x x x1 x x1 x2 x1 x b) x x1 x c) x x2 x1 d) x x x2 x x x1 x2 x Hỡnh 14.1 Gin liờn kt: x x x a) x b) Hỡnh 14.2 n ( i ) < T [ ( x) ( x ') S ( )] > = n n = n ! + dt dt n < T [ ( x) + ( x' ) V (t1 ) V (t n )] > = n ( i ) = A( n, m) dt1 dt m < T [ ( x) + ( x' ) V (t1 ) V (t m )] > k n n = m = n ! dt m +1 dt n < T [ V (t m+1 ) V (t n )] > n! A(n, m) = m ! ( n m) ! n n ( i ) m < T [ ( x) ( x' ) S ()] > = = m n = m = m ! + dt dt m < T [ ( x) + ( x' ) V (t1 ) V (t m )] > k ( i ) n m dt m +1 dt n < T [ V (t m +1 ) V (t n )] > nm ( n m) ! (i ) m m m = m ! = + dt1 dt m < T [ ( x) ( x' )V (t1 ) V (t m )] > k ( i ) k k k = k ! Vỡ dt m +1 dt m + k < T [ V (t m +1 ) V (t m + k )] > dt m+1 dt m+ k < T [ V (t m+1 ) V (t m+ k )] > = ( i ) k k ! dt k =0 k m +1 (14.7) dt1 dt k < T [V (t1 ) V (t k )] > dt m + k < T [ V (t m +1 ) V (t m + k )] > =< S () > (14.8a) x ( i ) m dt1 dt m < T [ ( x) + ( x' ) V (t1 ) V (t m )] > k = < T [ ( x) + ( x' ) S ( )] > k m m ! m= < T [ ( x) + ( x' ) S ( )] > = < T [ ( x) + ( x' ) S ( )] > k < S ( ) > G ( x, x' ) = i < T [ ( x) + ( x' ) S ()] > k (14.8b) (14.9) (14.10) 14.1.2- Ky thut gin khụng gian ta Xột Tng tỏc hai ht: b) a) d) c) i) h) e) ) k) g) Hỡnh14.3 a) b) c) d x1d x d x3 d x G(0 1) ( x x1 )G(10)2 ( x1 x )G(20) ( x2 x' )G(30)3 (0)G(40)4 (0)V ( x1 x3 )V ( x2 x ) d x1d x2 d x3 d x4 G(0 1) ( x x1 )G(10)2 ( x1 x2 )G(30)3 ( x2 x3 )G(40)4 ( x2 x4 )G(40) ( x4 x' )V ( x1 x2 )V ( x3 x4 ) d x1 d x d x3 d x G( 1) ( x x1 )G(10)2 ( x1 x )G(30)3 ( x x3 )G(30) ( x3 x' )G(04) (0)V ( x1 x )V ( x x3 ) ) d x d x d x d x G ( x x )G ( x x' )G ( x e) d x d x d x d x G ( x x )G ( x x' )G g) d x d x d x d x G ( x x )G ( x x )G d) d x1 d x d x d x G ( x x1 )G ( x1 x )G 3 ( x x )G ( x x' )G 4 (0)V ( x1 x )V ( x x ) 4 4 4 4 4 ( 0) 4 ( 0) 4 4 4 ( 0) (0) 1 (0) (0) (0) (0) 1 (0) 1 (0) (0) (0) x3 )G(30)4 ( x3 x )G( 04) ( x x )V ( x1 x )V ( x3 x ) (0) ( x x )G(30)2 ( x x )G( 04) (0)V ( x1 x )V ( x x ) (0) 3 ( x x )G(30) ( x3 x' )G( 04) (0)V ( x1 x3 )V ( x x ) 4 4 (0) (0) (0) (0) ( 0) h) d x1 d x d x3 d x G ( x x1 )G ( x1 x )G 3 ( x x3 )G 4 ( x3 x )G ( x x' )V ( x1 x )V ( x x3 ) i) d x1 d x d x d x G ( x x1 )G ( x1 x )G 3 ( x x )G 4 ( x x )G ( x x' )V ( x1 x )V ( x x ) 4 4 (0) (0) (0) ( 0) (0) k) d x1 d x d x3 d x G ( x x1 )G ( x1 x )G ( x x' )G ( x3 x )G ( x x3 )V ( x1 x3 )V ( x x ) 4 4 (0) (0) (0) (0) ( 0) (t ) dt = + ( x ) + ( x ) ( x x ) ( x ) ( x ) dx dx = V 1 2 1 +1 ( x1 ) +2 ( x ) ( x1 x ) ( x1 x3 ) ( x x ) ( x ) ( x3 ) dx1 dx dx3 dx = = +1 ( x1 ) +2 ( x ) ( x1 x ) ( x1 x ) ( x x3 ) ( x ) ( x3 )dx1 dx dx3 dx = = +1 ( x1 ) +2 ( x ) (10)2 ( x1 x , x3 x ) ( x ) ( x3 ) dx1 dx dx3 dx 4 = (10)3 d x1 d x (10)2 ( x1 x , x x ) < T [ ( x) + ( x' ) +1 ( x1 ) +2 ( x ) ( x ) ( x3 )] > = G (1) = (10)3 ( x1 x , x x ) = ( x1 x ) [ ( x1 x ) ( x x ) ( x1 x ) ( x x3 ) ] = i d x1 d x (10)2 ( x1 x , x3 x4 )G ( 0) ( x x1 )G ( 0) ( x3 x )G ( 0) ( x4 x' ) ( x1 x , x3 x ) = mt hỡnh vuụng Hinh 14.4 (14.13) = gin 14.4 cỏc gin 14.3 a , b , c , d ==> 14.5 a ; cỏc gin 14.3 , e , g , h ==> 14.5 b ; cỏc gin 14.3 i , k ==> 14.5 c c) b) a) Hinh14.5 a) d x1 d x8 (10)2 ( x1 x , x3 x ) (50)6 ( x5 x6 , x7 x8 ) .G ( 0) ( x x1 )G ( 0) ( x3 x5 )G (0) ( x7 x' )G (0) ( x x )G ( 0) ( x8 x6 ) b) d x1 d x8 (10)2 ( x1 x , x3 x )(50)6 ( x5 x , x x8 ) .G ( ) ( x x1 )G ( 0) ( x3 x' )G ( 0) ( x x5 )G ( 0) ( x7 x )G ( 0) ( x8 x ) c) d x1 d x8 (10)2 ( x1 x , x3 x )(50)6 ( x5 x6 , x7 x8 ) G ( 0) ( x x1 )G (0) ( x3 x5 )G (0) ( x7 x )G (0) ( x x6 )G ( 0) ( x8 x' ) Quy tc tớnh phn b sung bc n ca hm Green : 1) Ve tt c cỏc gin liờn kt cú cú cu trỳc tụ pụ khụng tng ng nhau; (0) 2) Mi ng cho tng ng vi mt hm Green G ( xi x j ) ; (0) 3) Mi t giỏc cho tng ng vi mt hm ( x1 x , x3 x ) ; 4) Ly tớch phõn theo cỏc ta ca tt c cỏc nh v ly tng theo cỏc bin spin ; 5) Biu thc nhn c nhõn vi h s (i)n(1/2)n-(m/2), ú m l s gin biu din khụng i xng tng ng vi gin i xng ang xột Du ca gin cng tng ng vi du trng hp khụng i xng Gin bc ba ve trờn hỡnh 14.6 Hỡnh 14.6 (i) d x1 d x12 (10)2 ( x1 x2 , x3 x4 )(50)6 ( x5 x6 , x7 x8 )(90)10 11 12 ( x9 x10 , x11 x12 ) G ( 0) ( x x1 )G ( 0) ( x3 x5 )G (0) ( x7 x9 )G ( 0) ( x4 x10 )G (0) ( x12 x6 )G ( 0) ( x8 x2 )G ( 0) ( x11 x' ) 14.1.3- Ky thut gin khụng gian xung lng Xột Tng tỏc hai ht (1b ) (0) ( 0) ( 0) G ( x, x ' ) = i G ( x x1 ) G ( x1 x ) G ( x x' ) ( x1 x )d x1 d x d4p ( 0) G ( x1 x2 ) = G ( p ) eip ( x1 x ) / ( ) ( 0) d 4q ( x1 x2 ) = ( q ) eiq ( x1 x ) / (2 ) (0) (0) G ( p ) = G ( x) e ipx / d x p = ( p, ) , q = ( q , ) , p ( x1 x ) = p (r1 r2 ) (t1 t ) (1b ) G ( x, x' ) = i (2 ) 16 d p1d p d p3 d qd x1d x e i[ p1 ( x x1 )+ p1 ( x1 x2 )+ p1 ( x2 x ') + q ( x1 x2 )] / (0) (0) (0) G ( p1 ) G ( p ) G ( p ) ( q ) = = i (2 ) d p1d p2 d p3d q ei[ p1 x p2 x '] / ( p1 p2 q) ( p2 p3 + q) .G( ) ( p1 ) G( 0) ( p ) G( ) ( p ) (q ) = i (2 ) d p1d p d p3 d q e i[ p1x p2 x '] / ( p1 p q) ( p p3 + q) (0) (0) (0) G ( p1 ) G ( p ) G ( p ) (q ) (1b ) G ( p, p ' ) = i d p d q e i[ p1x p2 x '] / ( p p q ) ( p p '+ q ) ( 0) ( 0) ( 0) G ( p ) G ( p ) G ( p ' )( q ) (1b ) G ( p, p ' ) = G (1b ) ( p ) ( p p ' ) (2 ) G G (1a ) (1b ) d 4q ( p) = i G ( p) G ( ) ( p q )(q )G ( ) ( p ) (2 ) (0) d p1 ( 0) ( p) = 2i G ( p) (0) G ( p1 ) lim ei t G ( ) ( p) t +0 (2 ) (0) a) p p1 p b) q p p-q Hỡnh14.10 p p1+q1+q 2+q p1+q 2+q q1 p1+q q2 p p-q1 p1 q1+q2+q q3 p-q1-q p-q4 p-q1-q 2-q q4 p-q1-q 2-q 3-q Hinh 14.11 i (2 ) 20 p [G ( ) ( p )] d q1 d q (q1 )(q )(q3 )(q )(q1 + q + q3 ) .G ( 0) ( p q1 )G ( 0) ( p q1 q )G ( 0) ( p q1 q q )G ( 0) ( p q1 q q q )G ( 0) ( p q ) . d p1 G ( 0) ( p1 )G ( 0) ( p1 + q3 )G ( ) ( p1 + q + q )G ( ) ( p1 + q1 + q + q3 ) 14.1.4- Phng trỡnh Dyson c) b) a) Hinh14 G = G ( ) + G (1) + G ( ) + = + + + + + + + + + + + { = [ + + + + + + + + + + + + + + + + + + G G + + [ (0) = + [ + + + Hỡnh 14.16 + ] ] [ + } + + ] = + + = + + + (1) = ( p) = i ==> ] + } + + [ ] + + + ] + [ ] + [ = + + + [ ] + [ ] + + + + G =G [ + + [ ( 0) ] d p1 ( ) , ( p, p1; p1 , p)G ( p1 ) (2 ) + ] + : Hỡnh 14.17 => Hỡnh 7.5 c Hỡnh 7.6 Hỡnh 7.17 ( 2) ( p ) = = gin ve trờn hỡnh 14.18 (0) , ( p, p1 ; p , p + p1 p ) , ( p , p + p1 p ; p1 , p ) d p1d p2 (2 )8 Hỡnh 7.6 ( 2) ( p) = Gà ( p2 )G ( p1 )G ( p + p1 p2 ) ==> Hỡnh 14.18 (0) ): Thay = (1) + ( ) vo phng trỡnh (14.19) (chỳ y G = E0 ( p ) d p1 ( 0) [ E0 ( p )]G ( p ) i , ( p, p1 ; p1 , p )G ( p1 )G ( p ) + (2 ) + ( ), ( p, p1 ; p , p + p1 p )à , ( p , p + p1 p ; p1 , p ) d p1 d p Gà ( p )G ( p1 )G ( p + p1 p ) G ( p ) = (2 ) [...]... cỏc gin liờn kt cú cú cu trỳc tụ pụ khụng tng ng nhau; (0) 2) Mi ng cho tng ng vi mt hm Green G ( xi x j ) ; (0) 3) Mi t giỏc cho tng ng vi mt hm ( x1 x 2 , x3 x 4 ) ; 4) Ly tớch phõn theo cỏc ta ca tt c cỏc nh v ly tng theo cỏc bin spin ; 5) Biu thc nhn c nhõn vi h s (i)n(1/2)n-(m/2), trong ú m l s gin trong biu 1 3 2 4 din khụng i xng tng ng vi gin i xng ang xột Du ca gin cng tng ng vi du... 0,352125 (10.32) k F a 0 < 1/ 3 2 2 + 1 (10.19) (10.26) (10.27) (10.28) (10.29) (10.30) (10.31) Et > Ed (10.33) Y nghia ca iu kin (10.32) H= ,k