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2 8¹ 2/26 1 a f 2 a f Ì ëö8ú Ù

3 üf và 3/26 -fƒmãƒpš^ üfµfôn fxnµþf1æ - Ýü$½ƒpŠ^\r - X fµ fôn p pøp ÝíNµlfNÔn þfín BEC NµvàÔnÆ

4 Schrödinger Equation 4/26 Louis de Broglie > Åâƒ5 fgψ( r, t >Å gdâf >² Åψ( r, t e i(ωt k r ŠâÅ5Ÿ kω = E/, k = 2π/λ = p/ l ψ( r, t = e i(et p r/ i ψ(r, t = Eψ(r, t t (Ü i ψ = pψ, 2 2 ψ = p 2 ψ, Ú²;å Æ E = p 2 /2m k 2 (i + t 2m 2 ψ = (E p2 ψ = 0 2m Erwin Schrödinger ( Born in Vienna, Austia Nobel Prize in Physics (1933

5 fschrödinger Equation 5/26 'X E i p t 2 (i + t 2m 2 ψ = (E p2 ψ = 0 2m XJĽ UþE عkžm Eψ( r = Ĥ( rψ( r a f Ĥ = T + V = ] [ 2 2µ 2 Ze2 4πɛ 0r BŒÆs Å ½

6 a fschrödinger Equation 6/26 [ 1 r 2 Ĥ = T + V = ] [ 2 2µ 2 Ze2 4πɛ 0r Ù µ = mm m+m ü âfzÿþ" ƒpš^ %³ é5µ f! N Wigner-Eckart½n Fermi 7½K I(ëên 2 = 1 r 2 \Schrödinger r r2 r + 1 r 2 sin θ r r2 r + 1 r 2 sin θ θ ( sin θ θ + ( sin θ + θ θ 1 r 2 sin 2 θ 2 φ + 2µ 2 1 r 2 sin 2 θ 2 φ 2 ] (E V(r 2 ψ = 0

7 a fschrödinger Equation lcþ 7/26 [ ] r r2 r + 2µr2 (E V(r ψ 2 [ 1 = sin θ θ lcþψ(r, θ, φ = R(rY(θ, φ = 1 Y sin θ Ú\~êλ ü 1 r 2 ( sin θ θ + 1 sin 2 θ 1 d dr R dr r2 dr + 2µr2 (E V(r 2 ( sin θ 1 θ θ Y sin 2 θ d dr dr r2 dr + sin θ Y ] 2 ψ φ 2 2 φ 2 [ 2µ (E V(r λ ] R = 0 (1 2 r 2 ( sin θ Y + λ sin 2 θ = 1 2 Y θ θ Y φ 2

8 a fschrödinger Equation lcþ 8/26 qœuy lcþ Ú\~êm 1 sin θ d dθ ( sin θ dθ dθ + ( λ λ sin 2 θ = 0 (2 d 2 Φ dφ 2 + m2 Φ = 0 (3 [ 1 d dr 2µ r 2 dr r2 dr + (E V(r λ ] R = 0 (1 2 r 2

9 a fschrödinger Equation ΦÚΘ 9/26 (3 k84 {ü 3 \à ÙÏ/ª Φ(φ = A e imφ + B e imφ, 3Ônþ Φ(φ = Φ(φ + 2Nπ m ê = m = 0, ±1, ±2, Œ±ÙAΦ m(φ = Ae imφ " (2 x = cos θ l d dx [ (1 x 2 dθ dx ] + (λ m2 Θ = 0 1 x 2

10 a fschrödinger Equation Θ(θ 10/26 Θ(x = (1 x 2 m 2 v=0 a vx v = (1 x 2 m 2 v(x u k(1 x 2 v 2( m + 1xv + (λ m 2 m v = 0 = {(v + 1(v + 2a v+2 [v(v 1 + 2( m + 1v λ + m + m 2 ]a v}x v = 0 v=0 þª á ^ kxêñ 0 a v+2 = v = k =a k+2 = 0 v(v 1 + 2( m + 1v λ + m + m2 a v (v + 1(v + 2 λ = k(k 1 + 2( m + 1k + m + m 2 = (k + m (k + m + 1

11 a fschrödinger Equation Θ(θ 11/26 -l = k + m K λ = l(l + 1 dukú m Ñ škê Ïd l = 0, 1, 2, m l m = 0, ±1, ±2, ± l ë V4 [ d (1 x 2 dθ ] + (l(l + 1 m2 Θ = 0 dx dx 1 x 2?ê Úƒ Θ(θ = BP m l (cos θ Å Y l,m(θ, φ

12 a fschrödinger Equation R(r 12/26 [ 1 d dr 2µ r 2 dr r2 dr + (E V(r λ ] R = 0 2 r 2 R(r = u(r/r ²L?ê { µz 2 e 4 E n = (4πɛ n = µc2 α 2 Z 2 2 2n 2 n = 1, 2, 3,, l = 0, 1, 2,, n 1 m = 0, ±1, ±2,, ±l ψ(r, θ, φ = R n,l(ry m l (θ, φ α = e2 4πɛ 0 c, ab = 4πɛ 2 µe 2

13 Energy Levels E n (ev l= n s 1 3p 2 3d s 2p n = 1, 2,, ØÓÄþU? o ëyu? s 图 n 1 13/26

14 14/26 :5ïÄåP Ù 8 z^ ψ(r, θ, φ = R n,l (ry m l (θ, φ R n,l (r = N nl e ρ/2 ρl 2l+1 2Z n+1 (ρ, ρ = r na µ 0 drr 2 π 0 2π dθ sin θ dφ ψ(r, θ, φ 2 = 1 0

15 R ρ r nl (r 其中, 2Z μ 2Z 2 2 = = r, aμ = 4 πε0 / μe = a0m/ μ na m na μ 0 考察核质量无穷大时, 前几个径向波函数为, R ( r = 2( Z / a exp( Zr/ a 3/ R ( r = 2( Z / 2 a (1 Zr/ 2 a exp( Zr/ 2 a 3/ /2 R21( r = ( Z / 2 a0 ( Zr/ a0exp( Zr/ 2 a0 3 R ( r = 2( Z / 3 a (1 2 Zr/ 3a + 2 Z r / 27 a exp( Zr/ 3 a 3/ /2 R31( r = ( Z / 3 a0 (1 Zr/ 6 a0( Zr/ a0exp( Zr/ 3 a /2 2 R32( r = ( Z / 3 a0 ( Zr/ a0 exp( Zr/ 3 a 再来看看角向波函数 (angular function, 为球谐函数 1/2 15/26

16 Y m l (θ, φ 前几个球谐函数为 Y Y Y 1 = (4 π 0,0 1/2 1,0 1, ± 1 Y Y 2,0 2, ± 1 3 = 4π 1/2 3 = 8π cosθ 1/2 sinθ ± i e φ 1/2 2 (3cos 1 5 = θ 16π 15 = 8π 1/2 sinθcosθ 1/ ± 2i Y2, 2 sin θe φ ± = 32π ± i e φ 1/2 3 (5cos 3cos 7 Y3,0 = θ θ 16π Y Y 2,0 2, ± 1 5 = θ 16π 15 = 8π 2 (3cos 1 1/2 sinθcosθ 1/ ± 2i Y2, 2 sin θe φ ± = 32π ± i e φ 1/2 3 (5cos 3cos 7 Y3,0 = θ θ 16π Y 3, ± 1 1/2 2 ± i sin θ(5cos θ 1 e φ 21 = 64π 1/ ± 2i Y3, 2 sin θcosθe φ ± = 32π 1/ ± 3i Y3, 3 sin θe φ ± = 64π 在有些应用场合, 用 L 2 的另外一套本征函数即 Y (,,cos θ φ = NΘ l lm 16/26

17 AÇ Ý 17/26 AÇ Ýρ(r, θ, φ = ψ(r, θ, φ 2 3 r:?d r AÇ ρ(r, θ, φd r = ψ(r, θ, φ 2 r 2 sin θdrdθdφ = R 2 nl(rr 2 dr Yl m (θ, φ 2 dω zæa^ ; p z = p 0 p x = 1 2 (p 1 + p 1 p y = 1 i 2 (p 1 p 1

18 AÇ Ý 18/26 ÿþºkã^ º m Ym l (θ, φ

19 AÇ Ý» R nl (r 1.0 r 2 R nl 2 (r 0.4 1s R r/a r/a /26

20 AÇ Ý» 20/26 R nl (r R 20 R 21 r 2 R nl p 2s r/a r/a

21 AÇ Ý» 21/ d 3p 3s R nl (r R 30 R 31 R 32 r 2 R nl 2 (r r/a r/a 0

22 AÇ Ý» + 22/26

23 AÇ Ý 23/26 AÇ Ý P[ψ nlm (r, θ, φ] = ψ nlm (r, π θ, π + φ = R nl (ryl m (π θ, π + φ = ( 1 l ψ nlm (r, θ, φ 5 Úõ>fNÅ é5«oœ

24 24/26 1 Ÿþ µâf3 %å $Ä"Ù /ª µ V(r = α r s, α > 0. žy²3åp^ 0 < s < 2 2y²3E 0 NC 3Ãõ åpu?" 2 OŽ fä>fúføƒm ÕáÚ³U< V >Úü öƒmkúå³u< G >ƒmé< F > / < G >" 3 a fåψ nlm éu2s 2p>f O Ž< r k > k = 1, 2 "

25 Schrödinger s Grave Alpbach Ž ë z Alpbach C dm6 Œ -c/. / 25/26

26 Ì ëö8ú Ù I 26/26 MŽƒ?Í C ÔnÆ 12 I ÆEâŒÆÑ 2008 ISBN: B. H. Bransden and C. J. Joachain Physics of atoms and molecules (2nd Edition Pearson Education Limited, 2003, ISBN: X. Christopher J. Foot Atomic Physics Oxford University Press, 2005, ISBN: