Physics 452

1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

2. Phys 452 Homework #24 30 pts : research overview 10pts connection with QM 10pts Presenting 10pts Homework Wed Apr 6: assignment #22 11.8, 11.10, 11.11, 11.13 Friday April 8: assignment #23 11.14, 11.18, 11.20 + Compton question QM & Research presentations Next week, W April 6 or F April 8

3. Phys 452 Class- schedule Today 4th: Born approx., Compton effect Wednesday 6th: research & QM presentations I Friday 8th :research & QM presentations II Treats and vote for best presentation In each session Next week April 11 & 13: Final Review

4. Phys 452 Research and QM presentation Make a connection with a topic covered in Quantum Mechanics: A principle An equation An application Template Focus on one physical principle or phenomenon involved in your research

5. Phys 452 Scattering q Easy formula to calculate f(q,f)? Quantum treatment Spherical wave Plane wave or f(q)?

6. Phys 452 Born formalism Born approximation: The main impact of the interaction is that an incoming wave of direction is just deflected in a direction but keeps same amplitude and same wavelength. One can express the scattering factor In terms of wave vectors Max Born (1882-1970) German physicist Nobel prize in 1954 For interpretation of probability of density y

7. Phys 452 Born formalism Green’s function Integral form of the Schrödinger equation Using Fourier Transform of Helmholtz equation and contour integral with Cauchy’s formula, one gets:

8. Phys 452 Born approximation • Spatial approximation • (First order) Born approximation The incoming wave is scattered in a wave of same amplitude, just different direction

9. Phys 452 Quiz 34 When expressing the scattering factor as following What approximation is done? • The potential is spherically symmetrical • B. The wavelength of the light is very small • C. This scattering factor is evaluated at a location relatively • close to the scattering center • D. The incoming wave plane is not strongly altered by the scattering • E. The scattering process is elastic

10. Phys 452 Born approximation Scattering vector

11. Phys 452 Born approximation • Low energy approximation • Case of spherical potential • Examples: • Soft-sphere • Yukawapotential • Rutherford • scattering

12. Phys 452 Born approximation Case of spherical potential Develop and to third order Pb 11.10 Soft sphere potential • Scattering amplitude • Approximation at low E

13. Phys 452 Scattering – Phase shift Expand Pb 11.11 Yukawa potential

14. Phys 452 • More generalspherical potential Scattering- phase shifts Pb 11.13 Spherical delta function shell (Pb 11.4) • Low energy case

15. Phys 452 Scattering – Born approximation Differential cross- section don’t forget Total cross- section Pb 11.20 Gaussian potential Integration by parts f has also a Gaussian shape in respect to q

16. Phys 452 Born approximation impulse momentum I p Deflection q1 Pb 11.14: Rutherford scattering q Step 1. Evaluate the transverse force r b f q2 Impulse approximation q Step 2. Evaluate the impulse I Step 3. Evaluate the deflection q Step 4. deduct relationship between b and q

17. Phys 452 Born approximation Unperturbed wave (zero order) Deflected wave (first order) Extending at higher orders propagator Zero order First order Second order Third order Impulse and Born series See pb 11.15

18. Phys 452 Born approximation Back scattering (in 1D) See pb 11.17 • Delta function well: • Finite square well -a a Pb 11.16 Pb 11.17 Pb 11.18: build a reflection coefficient

19. Phys 452 Quiz 35 Compton scattering essentially describes: A. The scattering of electrons by matter B. The scattering of high energy photon by light atoms C. The scattering of low energy photons by heavy atoms D. The scattering of lo energy neutrons by electrons E. The scattering of high energy electrons by matter

20. Phys 452 Compton scattering Arthur Compton (1892-1962, Berkeley) American physicist Nobel prize in 1927 For demonstrating the “particle” concept of an electromagnetic radiation January 13, 1936

21. Phys 452 Compton scattering Phys rev. 21, 483 (1923)

22. Phys 452 Compton scattering Electromagnetic wave Particle: photon Classical treatment: Collision between particles • Conservation of energy • Conservation of momentum

23. Phys 452 Compton scattering Compton experiments Final wavelength vs. angle Homework Compton problem (a): Derive this formula from the conservation laws

24. Phys 452 Compton scattering Goal: Express the scattering cross-section We need to evaluate the Hamiltonian for this interaction and solve the Schrodinger equation Quantum theory Photons and electrons treated as waves Constraint 1: we are not in an elastic scattering situation So the Born approximation does not apply… Constraint 2: the energy of the photon and recoiled electron are high So we need a relativistic quantum theory

25. Phys 452 Compton scattering • Klein – Gordon equation: relativistic electrons in an electromagnetic field Energy at rest Vector potential momentum • Vector potential • Interaction Hamiltonian (perturbation theory) Quantum theory

26. Phys 452 Compton scattering Quantum theory

27. Phys 452 Compton scattering with First order perturbation theory to evaluate the coefficients: Homework Compton problem (b): Show that Quantum theory Electron in a scattering state

28. Phys 452 Compton scattering Furthermore we can evaluate the cross-section: (Thomson scattering) Homework Compton problem (c): Evaluate s in case of Quantum theory We retrieve the conservation laws: (d): Compare to Rutherford scattering cross-section