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Time dependent Schr ö dinger [H o + V(t)] = i ħ /t. Time independent H o o = E o o. Time dependent [H o + V(t)] = i ħ /t. Harry Kroto 2004. Atoms Molecules. Basically only electronic transitions . >10000 cm -1. Harry Kroto 2004.

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## Time independent H o o = E o o

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**Time dependent Schrödinger**[Ho + V(t)] = iħ/t Time independent Hoo= Eoo Time dependent [Ho + V(t)] = iħ/t Harry Kroto 2004**Atoms Molecules**Basically only electronic transitions >10000 cm-1 Harry Kroto 2004**C We have to solve the Time independent problem**Hoo= Eoo Harry Kroto 2004**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**The Born-Oppenheimer Separation**H = E H = Hel + Hvib + Hrot+ … Harry Kroto 2004**The Born-Oppenheimer Separation**H = E H = Hel + Hvib + Hrot+ … = el vib rot … =i i Harry Kroto 2004**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**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 mm and m* n Harry Kroto 2004**Time independent**Hoo= Eoo Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m m o Harry Kroto 2004**D Selection Rules**Need to solve the Time Dependent Problem Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t m o Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e m o Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e Ee (t) = Eeocos2t Ee(t) Radiation field e Electric dipole moment m o Harry Kroto 2004**Time independent**Hoo= Eoo Stationary States mo m Time dependent [Ho + V(t)] = iħ/t V(t) = -Ee(t)e Ee (t) = Eeocos2t Ee(t) Radiation field e Electric dipole moment = mam(t)m m o Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Beer Lambert Law I= Io e-l Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Beer Lambert law I= Io e-l Harry Kroto 2004**Fermi’s Golden Rule**x Io I l Beer Lambert law I= Io e-l is the absorption coefficient = (83/3hc)n em2(Nm-Nn)(o-) Harry Kroto 2004** = (4/3ħc) nem2 (Nm-Nn)**(o-) Harry Kroto 2004** = (4/3ħc) nem2 (Nm-Nn)**(o-) • ① • Square of the transition moment nem2 Harry Kroto 2004** = (4/3ħc) nem2 (Nm-Nn)**(o-) • ① ② • Square of the transition moment nem2 • Frequency of the light Harry Kroto 2004** = (4/3ħc) nem2 (Nm-Nn)**(o-) • ① ② ③ • Square of the transition moment nem2 • Frequency of the light • Population difference (Nm- Nn) Harry Kroto 2004** = (4/3ħc) nem2 (Nm-Nn)**(o-) • ① ② ③ ④ • Square of the transition moment nem2 • Frequency of the light • Population difference (Nm- Nn) • Resonance factor - Dirac delta function (0) = 1 Harry Kroto 2004**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**Hot gas cloud –the famous Orion Nebulae**At the centre is the Trapezium Cluster of very hot new stars Harry Kroto 2004**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**Einstein Coefficients**n Bn<-m m Harry Kroto 2004**Einstein Coefficients**n Bn<-m Bn->m m Harry Kroto 2004**Einstein Coefficients**n Bn<-m Bn->m An->m m An->m/ Bn->m = 8h3/c 3 Harry Kroto 2004**Einstein Coefficients**n Bn<-m Bn->m An->m m A = 1.2 x 10-37 3n em2 transitions per sec Spontaneous emission lifetime (sec) = 1/A = 1037/3 sec Harry Kroto 2004** (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**Bohr radius**an = aon2 ao = 0.05 nm Harry Kroto 2004

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