1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Homework Phys 452 Wed Feb 16: assignment # 10 Pb 7.8, 7.9, 7.13, 7.17 Friday Feb 18: assignment # 11 8.1, 8.2, 8.7, 8.14 Announcements Monday: Holiday Next class: Tuesday at 1pm

3. H atom Hydrogen molecule ion H 2 + Phys 452 H atom electron

4. Hydrogen molecule ion H 2 + Phys 452 electron Step 1: Hamiltonian Step 2: trial wave function Overlap integral: Normalization:

5. Hydrogen molecule ion H 2 + Phys 452 Use the trial wave function: Hydrogen Hamiltonian with first nucleus Hydrogen Hamiltonian with second nucleus Cross terms Step 3: expectation value of H

6. Hydrogen molecule ion H 2 + Phys 452 Direct integral Dexchange integral X Pb. 7.8 Eigenstates of Individual hydrogen atoms

7. Hydrogen molecule ion H 2 + Phys 452 where direct integral exchange integral Finally…!

8. Hydrogen molecule ion H 2 + Phys 452 Step 4: Minimization First include the proton-proton interaction ! where x = R/a

9. Hydrogen molecule ion H 2 + Phys 452 Step 4: Minimization Presence of a minimum: Evidence of bonding Equilibrium separation distance:

10. Quiz 15 Phys 452 The binding energy for the hydrogen molecule ion H 2 + is experimentally found to be 2.8eV. What can we predict about the binding energy E estimated with the variational principle? A. E > 2.8 eV B. E < 2.8 eV C. E = 2.8 eV D. Can be any value E. Can not tell

11. Hydrogen molecule ion H 2 + Phys 452 Pb 7.8 Calculation of Direct integralExchange integral

12. Hydrogen molecule ion H 2 + Phys 452 Pb 7.9 Presence of a minimum: Evidence of bonding For symmetrical state What about antisymmetrical state?

13. Hydrogen atom H Phys 452 Pb 7.13 Another trial wave function Hamiltonian Calculate …and minimize it use spherical coordinates

14. Helium - like system Phys 452 Pb 7.17 “Rubber – band” model for He a) Change of variable b) Exact solution (harmonic oscillators) c) Evaluatewith ground state of 3D HO