1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Phys 452 Test 2 Today Mar 2: Review (Monday 5: end of Review) Wed Mar 7: New chapter Mon Mar 5 – Wed Mar 7 Next homework Friday Mar 9

3. Practice Test 2 Phys 452 1. Variational principle 2. Helium atom & variational principle 3. WKB approximation and tunneling 4. WKB approx for potential with a wall 5. WKB approx for potential with no wall

4. Techniques to solve for the allowed energies Phys 452 Hamiltonian Many particles Schrödinger Equation… … very hard to solve! ???

5. Techniques to solve for the allowed energies Phys 452 1. The perturbation theory (first, second order…) 2. The variational principle 2. The WKB approximation Test 2

6. Variational principle The trick: Phys 452 Ground state Expectation value on any normalized function 

7. Variational principle The method: Phys 452 Define your system, and the Hamiltonian H Pick a normalized wave function  Calculate You get an estimate of ground state energy Minimize

8. Variational principle the first excited state: Phys 452 First excited state Expectation value on a normalized function  normal to ground state

9. The ground state of Helium Phys 452 He atom 2 particles system Kinetic energy Interaction with proton Electron- electron interaction Zero-order Hamiltonian H 0 Perturbation Exact solution Ground state

10. The ground state of Helium Phys 452 He atom Second try: Use the variational principle to account for screening effect Same calculation except

11. The ground state of Helium Phys 452 Second try: Use the variational principle to account for screening effect He atom

12. The ground state of Helium Phys 452 Energy diagram He atom E 0 -109 eV -79 eV -75 eV First try -77.5 eV Second try

13. Hydrogen molecule ion H 2 + Phys 452 electron LCAO Technique (linear combination of atomic orbitals)

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

15. Phys 452 The WKB approximation The WKB approximation is based on the idea that for any given potential, the particle can be locally seen as a free particle with a sinusoidal wave function, but whose wavelength varies very slowly in space.

16. Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Non-classical region (E<V) Turning points

17. Phys 452 The WKB approximation Excluding the turning points:

18. Phys 452 Tunneling trough a barrier V(x) x V0V0 A B F -a+a Transmission coefficient

19. Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Patching – upward slope Linear approximation Patching region Overlap 1 Overlap 2 X=0

20. Phys 452 The WKB approximation General expression for the wave function Patching – upward slope

21. Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Patching – downward slope Linear approximation Patching region Overlap 1 Overlap 2 X=0

22. Phys 452 The WKB approximation Patching – downward slope General expression for the wave function

23. Phys 452 The WKB approximation Connection formulas Potential with no walls Potential with 2 walls Potential with 1 wall