1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

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

3. Phys 452 Homework Friday Feb 25: assignment # 12 Tuesday Mar 1: 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

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

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

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

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

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

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

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

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

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

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

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

15. The WKB approximation Phys 452 Patching region Overlap 2 E Patching – downward slope V(x) Pb 8.9 Linear approximation Overlap 1 X=0 Non-classical region (E<V) Classical region (E>V)

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

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

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

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

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

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

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