Physics 452

1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

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

3. Phys 452 Practice Test 2 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. Phys 452 ??? Schrödinger Equation… … very hard to solve! Techniques to solvefor the allowed energies Hamiltonian Many particles

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

6. Phys 452 Ground state Expectation value on any normalized function y Variational principleThe trick:

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

8. Phys 452 First excited state Expectation value on a normalized function y normal to ground state Variational principlethe first excited state:

9. Phys 452 Kinetic energy Interaction with proton Electron- electron interaction Zero-order Hamiltonian H0 Perturbation Exact solution Ground state The ground state of Helium 2 particles system He atom

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

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

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

13. Phys 452 Hydrogen molecule ion H2+ electron LCAO Technique (linear combination of atomic orbitals)

14. Hydrogen molecule ion H2+ Phys 452 Equilibrium separation distance: Presence of a minimum: Evidence of bonding Step 4: Minimization

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

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

18. Phys 452 Tunneling trough a barrier V(x) V0 A F B x -a +a Transmission coefficient

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

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

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

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

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