1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Homework Phys 452 Wednesday Feb 29: assignment # 13 8.5, 8.6, 8.9, 8.13

3. Homework Phys 452 Friday Feb 25: assignment # 12 WKB &Tunneling: Pb 8.3 Pb 8.4 Pb 8.16 WKB & turning points: Connection formulae Pb 8.5 Pb 8.6 Pb 8.9 Pb 8.13 Tuesday Mar 1: assignment # 12

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

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

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

7. Quiz 19a Phys 452 What does physically happen to the wave function at the turning points? A. The amplitude goes to infinity B. The wave function is undefined C. The wave function collapses D. The wave function is finite and can be retrieved by continuity between the WKB approximated wave functions A. The wave function is finite and can be retrieved by patching the WKB approximated wave functions

8. Patching region Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V)

9. Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Patching region Linear approximation X=0 Pb 8.8

10. Phys 452 The WKB approximation Solving the Schrödinger equation in the patching region Solutions: Airy functionsand

11. Phys 452 The WKB approximation Solving the Schrödinger equation in the patching region Solutions: Airy functions

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

13. The WKB approximation Patching Phys 452 Continuity on overlap 1 ()() ()() Continuity on overlap 2

14. Phys 452 The WKB approximation Patching Upward slope left side (scattering state) right side (bound state)

15. 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 Pb 8.9

16. Phys 452 The WKB approximation Patching – downward slopePb 8.9 General expression for the wave function right side (scattering state) left side (bound state)

17. Phys 452 The WKB approximation Connection formula Different scenarios Potential with 2 walls Potential with 1 wall Potential with no walls

18. Quiz 19b Phys 452 Which one of these cases does NOT provide enough information to quantize the energy using the WKB approximation? A. A potential with two walls B. A potential with no walls but two turning points C. A potential with no walls but one turning point D. A potential with one wall and one turning point E. A potential with two walls and one turning point

19. Phys 452 The WKB approximation Connection formulas Potential with 2 walls

20. Phys 452 The WKB approximation Connection formulas Potential with 1 wall and 1 turning point Pb 8.5 Pb 8.6 Pb 8.13 Use the expression

21. Phys 452 The WKB approximation Connection formulas Potential with no walls but two turning points V(x) E Turning points Connect the two expressions Same phase modulo 

22. Homework Phys 452 Friday Feb 26: assignment # 13 WKB & turning points: Pb 8.9 (downward slope) Pb 8.5 Pb 8.6 Example of gravity potential Pb 8.13: logarithmic radial potential