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Time independent H o  o = E o  o PowerPoint Presentation
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Time independent H o  o = E o  o

Time independent H o  o = E o  o

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Time independent H o  o = E o  o

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  1. Time dependent Schrödinger [Ho + V(t)] = iħ/t Time independent Hoo= Eoo Time dependent [Ho + V(t)] = iħ/t Harry Kroto 2004

  2. Atoms Molecules Basically only electronic transitions >10000 cm-1 Harry Kroto 2004

  3. C We have to solve the Time independent problem Hoo= Eoo Harry Kroto 2004

  4. Atoms Molecules Basically only electronic transitions electronic transitions E > 10000 cm-1 Vibrational transitions E = 100-10000 cm-1 Rotational transitions E = 0.1 – 100 cm-1 >10000 cm-1 Harry Kroto 2004

  5. The Born-Oppenheimer Separation H = E H = Hel + Hvib + Hrot+ … Harry Kroto 2004

  6. The Born-Oppenheimer Separation H = E H = Hel + Hvib + Hrot+ …  = el vib rot …  =i i Harry Kroto 2004

  7. The Born-Oppenheimer Separation H = E H = Hel + Hvib + Hrot+ …  = el vib rot …  =i i E = Eel + Evib + Erot +… E= i Ei Harry Kroto 2004

  8. The Born-Oppenheimer Separation H = E H = Hel + Hvib + Hrot+ …  = el vib rot …  =i i E = Eel + Evib + Erot +… E= i Ei We shall often use Dirac notation mm and m* n Harry Kroto 2004

  9. Time independent Hoo= Eoo Harry Kroto 2004

  10. Time independent Hoo= Eoo Stationary States mo m Harry Kroto 2004

  11. Time independent Hoo= Eoo Stationary States mo m m  o Harry Kroto 2004

  12. D Selection Rules Need to solve the Time Dependent Problem Harry Kroto 2004

  13. Time independent Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t m  o Harry Kroto 2004

  14. Time independent Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e m  o Harry Kroto 2004

  15. Time independent Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e Ee (t) = Eeocos2t Ee(t) Radiation field e Electric dipole moment m  o Harry Kroto 2004

  16. Time independent Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e Ee (t) = Eeocos2t Ee(t) Radiation field e Electric dipole moment = mam(t)m m  o Harry Kroto 2004

  17. Fermi’s Golden Rule x Io I l Harry Kroto 2004

  18. Fermi’s Golden Rule x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004

  19. Fermi’s Golden Rule x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004

  20. Fermi’s Golden Rule x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004

  21. Fermi’s Golden Rule x Io I l Beer Lambert law I= Io e-l Harry Kroto 2004

  22. Fermi’s Golden Rule x Io I l Beer Lambert law I= Io e-l  is the absorption coefficient  = (83/3hc)n em2(Nm-Nn)(o-) Harry Kroto 2004

  23.  = (4/3ħc) nem2 (Nm-Nn) (o-) Harry Kroto 2004

  24.  = (4/3ħc) nem2 (Nm-Nn) (o-) • ① • Square of the transition moment nem2 Harry Kroto 2004

  25.  = (4/3ħc) nem2 (Nm-Nn) (o-) • ① ② • Square of the transition moment nem2 • Frequency of the light  Harry Kroto 2004

  26.  = (4/3ħc) nem2 (Nm-Nn) (o-) • ① ② ③ • Square of the transition moment nem2 • Frequency of the light  • Population difference (Nm- Nn) Harry Kroto 2004

  27.  = (4/3ħc) nem2 (Nm-Nn) (o-) • ① ② ③ ④ • Square of the transition moment nem2 • Frequency of the light  • Population difference (Nm- Nn) • Resonance factor - Dirac delta function (0) = 1 Harry Kroto 2004

  28. C Solution > Energy Levels For the H atom we shall just use the Bohr result E(n) = - R/n2 D Selection Rules n no restriction l = ±1 E Transition Frequencies E = - R[ 1/n22 – 1/n12] Harry Kroto 2004

  29. Harry Kroto 2004

  30. Harry Kroto 2004

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  35. Harry Kroto 2004

  36. Harry Kroto 2004

  37. Hot gas cloud –the famous Orion Nebulae At the centre is the Trapezium Cluster of very hot new stars Harry Kroto 2004

  38. Collisions in the Interstellat Medium ISM In space the pressures are low Very low If n = number of molecules per cc (mainly H) then 2b = 103/n yrs per collision 3b = 1023/n2 yrs per collision Number densities are anything from n = 1-1000 Harry Kroto 2004

  39. Einstein Coefficients n Bn<-m m Harry Kroto 2004

  40. Einstein Coefficients n Bn<-m Bn->m m Harry Kroto 2004

  41. Einstein Coefficients n Bn<-m Bn->m An->m m An->m/ Bn->m = 8h3/c 3 Harry Kroto 2004

  42. Einstein Coefficients n Bn<-m Bn->m An->m m A = 1.2 x 10-37 3n em2 transitions per sec Spontaneous emission lifetime   (sec) = 1/A = 1037/3 sec Harry Kroto 2004

  43.  (sec) = 1037/3   (cm-1)  (Hz) 3 (Hz3)  (sec) H (1420 MHz) 21cm 0.05 1.5x109 3x1027 1010 * H2CO rotations 1cm 1 3 x 1010 3x1031 106 CO2 vibrations 10 103 3 x 1013 3 x 1040 10-3 Na D electronic 500nm 2x104 1.5 x 1014 6 x 1044 10-7 H Lyman  100nm 105 3 x 1015 3 x 1046 10-9 Calculations assume e = 1Debye 1yr = 3 x 107 sec * magnetic dipole Harry Kroto 2004

  44. Harry Kroto 2004

  45. Bohr radius an = aon2 ao = 0.05 nm Harry Kroto 2004