1. 1/18 [1]1-1 Schrdinger +Ze 1 Schrdinger 1.1 (1-1) 2 1 2 1 L2 1 Ze 2 2r 2 2 = E (1.1) 2 me r r r r 4 0 r L 1 1 2 L2 = 2 sin + (1.2) sin sin 2 2 1.1 Schrdinger nlm (r , , ) = Rnl (r )Ylm ( , ) ll (1.3) Rnl (r )Ylm ( , )(1-3)l Schrdinger [4] 4.1 )1-2 Schrdinger (1.3) nlm (r , , ) n, l, ml 3lA. n E (1.4) me e 4 Z 2 E=(1.4) 2n 2 2 n (principle number n 1, 2, 3, 4.K, L, M, N.2B. l n n nlm ( r , , ) L 2. 2/18 L = l (l + 1) (1-5) (azimuthal quantum number) l n1 l0 l 0, 1, 2, 3.s, p, d, f. s, p, d, f C. ml (Ampre ) m (magnetic quantum number, l ml l) n l ml Zeeman D. n = 1, l = 0, ml=0 1s n = 2, l = 0, ml=0 2s n = 2, l = 1, ml=1, 0, +1 2p(x,y,z) n = 3, l = 0, ml=0 3s n = 3, l = 1, ml=1, 0, +1 3p(x,y,z) n = 3, l = 2, ml=2, 1, 0, +1, +2 3d(z2, x2-y2, xy, xz, yz) n = 4, l = 0, ml=0 4s E. Schrdinger ( ) (n, l, ml) 12 S Z SZ 1 S = s( s + 1) s=(1.6) 2 3. 3/18 1 S Z = m S mS = (1.7) 2ms +1/2 (up spin), 1/2 (down spin) 4 (n, l, ml, mS) 4. 4/18F.(a)F-1 (Paulis principle) 12p 2p2s2s2 4 1s(n, l, ml, mS) 1s+3e +eF-2. F-1 (b) 2p 2pA~E 2s2s1s 1s 1 1 +3e +e n l, ml 1.4 1.5 1s22s1 1s22p1 2s 2p 1 1.5 2s 2p l 1.5(a)2s 1s 2s (a) (+32)e = e 1.5(b) 2p 1.6 1.5 1.5(a) 1s 2s 1.8 1s 2s 2s 1s +Ze (12 1 12 2 12 < U ( R ) >= (2.3)3 4 0 R 6 kT 1, 2 2 T [K] E1 3-4 2= 0l 1(ii) ( ) (H2) (O2) 1=1l1 +2 E1 2 E 2= E1 1=1l1 2.7 ( ) 10. 10/18 ( 2.7 ) E=(2.4)(i) 3.5 (4.40)( 1)(i)[4](2.4) E U(R) 2 1 1 1 2 U ( R) = 2 4 0 R6 (2.5) (2.5) 1 (i) (iii)( ) London O2, H2, N2 2 He, Ar E (ii) U(R) 2.7 3 1 I1 I 2 1 2U ( R) = (2.6) 2 4 0 I 1 + I 2 R6i, i 1 I (: (2.6) , , )2-5.XHY XH X = O,N, Cl Y H X 2.7 eV 101 [eV] 2.8 DNA 11. 11/18DNA DNA 2.8 2.9 4 2.9 DNA 4 DNA 2.10 4 2.11 2.12 DNA 2.13 2.10 DNA DNA 2.11 2.13 DNA 2.12 12. 12/18[3] 3 (amorphous) 3.1 a, b, c 3 3 3 (unit cell) 3.1 (space lattice) (lattice 3.1 point) 3.2 (crystal axis) a, b, c , , 6 (lattice constant) 3.1 3.3 14 (Bravais lattice) (7 )(body-centered)3 (faced-centered)(2 )(base-centered)(2 ) 3.2 3.1 3.3 13. 13/18[4]4-1 Schrdinger ? v x y = A sin 2 t (4.1) v(4.2) x x y = A sin 2 t + A sin 2 + t v vx = 2 A sin 2 cos 2 t (4.2)v x t (4.3) x ( x) = 2 A sin 2(4.3) v y = ( x ) cos 2 t (4.4) x 2 2 t (4.5) 2 ( x) 4 2 2+ ( x) = 0(4.5) x 2 v2 () 1 3 2 ( x, y, z ) 2 ( x, y, z ) 2 ( x, y, z ) 4 2 2 +++ ( x, y , z ) = 0 (4.6)x 2 y 2 z 2 v2 E E T V E = T +V(4.7) 1 m T= mv 2 2p = mv (4.7)p2 E=+V(4.8)2m p = 2 m( E V )(4.9) 14. 14/18 (4.10) de Broglie () h h ==(h: Planck ) (4.10) p mv(4.9)h = 2 m( E V )(4.11)v = (4.11)(4.6) H ( x, y, z ) = E ( x, y, z ) (4.12)2 2 22 H = 2 + 2 + 2 + V ( x, y , z ) :2m x yz (h, = 2 (4.12) Schrdinger 4-2 1 Schrdinger (4.12)( x = r sin cos , y = r sin sin , z = r cos ) 2 1 2 1 L2 H = r + V (r , , ) (4.13)2m r 2 r r r 2 2 L 1 1 2 L2 = 2 sin +(4.14) sin sin 2 2 1 V +Ze (4.15) 1 Ze 2 V =(4.15)4 0 r(4.13) 2 1 2 1 L2 1 Ze 2 r 2 2 = E (4.16) 2me r 2 r r r 4 0 r e (4.16) Schrdinger m ( Quantum Mechanics, L. I. Schiff , , ) nlm (r , , ) = Rnl (r )Ylm ( , ) l l (4.17) 15. 15/18(4.17) Rnl (r ) Ylm ( , ) l 2l4(n l 1)! Z 3 2 n 2l +1R nl (r ) = 4n [ (n + 1)!] a 0 3 n e Ln + l ( 2 / n ) (4.18a) n l 1 [(n + l )!] 2 s k L 2 l +1 n+l( s) = k =0 (1) k + 2l +1 ( n l 1 k )!(2l + 1 + k )!k!(Laguerre) (4.18b) 4 0 2 = (Z / a0 )r , a0 =me e 2 :=0.5292 [](2l + 1)(l | ml |)! | m |Ylm ( , ) = (1) ( m +|m |) / 2l l Pl (cos )eim l l(4.19a) l4 (l + | ml |)! 1 |ml |d |m |l Pl ( w) = (1 w ) |ml | 2 2 Pl ( w) dw|m | l(Legendre) (4.19b)l 1 d Pl ( w) =( w 2 1) : Legendre (4.19c)2 l l! dw l nlm (r , , )lHnlm (r , , ) = Ennlm (r , , )ll(4.20)mee 4 Z 2e2 Z 2E == n (4 0 ) 2 2n 2 22a0 n 2 (n = 1, 2, 3, .)L2nlm (r , , ) = 2l (l + 1)nlm (r , , )ll(l = 0, 1, 2,, n1)(4.21)LZ nlm (r , , ) = mnlm (r , , )l l (ml = l, (l1),1, 0, 1, (l1), l) (4.22) L Z = i LZ Z nlm (r , , )2(4.20)~(4.22) l H, L , LZ (L Z ) n, l (l + 1) ,E ml r~r+dr,~ +d, ~ +d 16. 16/18 | nlm (r , , ) |2 r 2 sin dr d d = r 2 | Rnl (r ) |2 | Ylm ( , ) |2 sin d dll | Ylm ( , ) |2l ( 1.3 1.4 | Ylm ( , ) |2 l)3.3 Zeeman I S = IS(4.23) Ampre a e v I (4.23)e ev I = =(4.24)T 2a S S = a L L = meav 2 (4.25) eL = (4.25)2 me +Z B U eHLZL U = B = Z B = = B B Z (4.26) 2 meeB = 2me : Bohr (Bohr magneton)(4.26) Z LZ (4.22)(4.26) U U = B Bml (4.27) ml 3-4 2 ? Heitler Lomdon 2 A( a 1 ) 3.1() 17. 17/18B( b 1 ) 3.1 (4.28) 22 e2 111 111 e2 H =1 2 ++ + + (4.28)2m 22me4 0 ra1 ra 2 rb1 rb 2 r12 4 0 R (4.28) 1, 2 2 (1, 2) ][2 (1,2) 2 (a, b) (1,2)(a, b) (4.26) 4.29 E H = E(4.29)(4.29) 1,2 2 1s a(1)b(2) 1, 2 a, b (4.30) 1, 2 (1,2) = C [ a (1) b (2) + a (2) b (1)] (4.30)C | (1,2) | 1,2 d1 d22 1 (1,2) (1,2)d 1 d 2 =1 * (4.31) C (4.32)1C=2(1 + S 2 )(4.32)S = a (1) b (1)d 1 = a (2) b (2)d 2(4.33) S 2 (4.29)(4.31) (1,2) (4.33)1 = [ a (1) b (2) + a (2) b (1)] (4.34)2(1 + S 2 ) E (4.29)(4.31)E = * (1,2)H (1,2)d 1 d 2(4.35)(4.34)(4.35) E (4.36) 18. 18/18J+K E = 2E H + (4.36)1+ S 2EH 1s 21 1 EH = a (1) 2 me 1 + a (1)d 14 0 ra1 (4.37): (4.37)a b 1 2 (4.37) K J, (4.38), (4.39) e2 1 1 1 1 J= a (1) b (2) + a (1) b (2) d 1 d 2 (4.38)4 0 r12 rb1 ra 2 R e2 1 1 1 1 K= a (1) b (2) + a (2) b (1)d 1 d 2(4.39)4 0 r12 rb1 ra 2 R (4.36) 1 2EH 2 2 J K 2 2 2 2 J K (4.38) K (4.39)(4.38) 1s 2 3-5 ( ) (CO) 2 q[C] 2 l [m] = ql [Cm] 1 2 R 1 2 E1 (4.40) 19. 19/181 1 3( 1 R1R )E1 = R3 (4.40) 4 0 R 5 E1 2 U(R)U (R ) = 2 E 1 (4.41) (3.40)(3.41)1 1 2 3( 1 R 2 RU )()U ( R) = R3 (4.42) 4 0 R 5 2 2 < U ( R ) >= U ( R; ) P ( )(4.43)P() P() U(R,)Maxwell-Boltzmann e U(R, )/kT (T [K]) U ( R , ) U ( R , )e kT sin d< U >=U ( R , ) (4.44)e kT sin d(4.45) 2 1 12 2 1 2< U >= (4.45) 3 4 0 R 6 kT3-6 ( )(H2)(O2) E E (4.46) = 1U = E 2 (4.47) 2 (i)(4.40) E E 20. 20/18 (4.47) (4.40)(4.47) E 2 2 1 1 3( RRR ) U ( R) = 3 2 4 0 R R5 21 1 2 = 6 + .....2 4 0 R (4.48)[ ] 1 21. 21/18 2.1, 2.7 ISBN 4-339-01191-6 1.1, 1.3, 1.4, 3.1 () ISBN 4-7598-0097-2 () ISBN 4-7598-0098-0 1.6~ 1.8 ()James E . Huheey ISBN 4-8079-0236-9 2.2(a), (b), 2.3, 2.4, 3.1, 3.1~ 3.3 ISBN 4-563-03159-3 2.8~ 2.13 ( ) Lubert Stryrer ISBN4-8079-0130-3