1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Phys 452 Wednesday Feb 22: assignment # 11 8.1, 8.2, 8.7, 8.14 extended to Thursday Feb 23 Friday Feb 24: assignment # 12 8.3, 8.4, 8.16 Homework

3. Phys 452 Techniques to find approximate solutions to the Schrodinger equation • The perturbation theory 2. The variational principle 3. The WKB approximation

4. Phys 452 Gregor Wentzel Leon Brillouin Hendrik Kramers German 1898- 1978 Dutch 1894- 1952 French 1889- 1969 The WKB approximationWentzel- Kramers - Brillouin

5. 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.

6. Phys 452 Finite box Infinite space The free particle

7. Phys 452 Flat potential Scattering state Bound state E V

8. Phys 452 Varying potentialThe WKB approximation Turning points E Classical region (E>V) V(x)

9. Phys 452 Locally constant or varying very slowly In respect to wavelength E Classical region (E>V) The WKB approximation V(x)

10. Phys 452 with The WKB approximationClassical region

11. Phys 452 solution real part imaginary part The WKB approximationClassical region

12. Phys 452 and assumption The WKB approximationClassical region solution

13. Phys 452 where Incidentally The WKB approximationClassical region solution

14. Phys 452 Quiz 17a In the WKB approximation, what can we say about the solution for the wave function ? • The amplitude and the wavelength are fixed • The amplitude is fixed but the wavelength varies • The wavelength varies but the amplitude is fixed • Both the wavelength and the amplitude vary • There are multiple wavelengths for a given position

15. Phys 452 Phase is a function of x The WKB approximationClassical region solution

16. Phys 452 Develop the function as power of The WKB approximationClassical region Pb 8.2 Another way to write the solution: where f(x) is a complex function

17. Phys 452 How to use this Formula? When the phase is known at specific points: Gives information on the allowed energies The WKB approximationClassical region

18. Phys 452 In which situation can we apply the formula ? Quiz 17b • For any type of potential and any energy value • Only when • Only when • Only when the potential exhibits 1 turning point • Only when the potential exhibits 2 turning points

19. Phys 452 The WKB approximationClassical region Example Infinite Square well

20. Phys 452 The WKB approximationClassical region Pb 8.1

21. Phys 452 Turning points E Classical region (E>V) The WKB approximationat turning points V(x)

22. Phys 452 Connection formula (eq 8.51) E Classical region (E>V) The WKB approximationat turning points V(x)

23. Phys 452 Pb 8.7 Pb 8.14 Hydrogen atom Harmonic Oscillator The WKB approximationat turning points