1 / 32

Born Approximation and Compton Scattering in Quantum Mechanics

This presentation covers the Born approximation, its application in scattering calculations, and Compton scattering in quantum mechanics.

scroggins
Download Presentation

Born Approximation and Compton Scattering in Quantum Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

  2. Phys 452 Homework Thu Apr 5: assignment #22 11.8, 11.10, 11.11, 11.13 Tuesday April 10: assignment #23 11.14, 11.18, 11.20 Sign up for the QM & Research presentations Fri April 6 or Mon April 9 Homework #24 20 pts

  3. Phys 452 Class- schedule Today April 4: Born approximation, Compton effect Friday April 6 : research & QM presentations I Mon. April 9 : research & QM presentations II Wed. April 11: FINAL REVIEW Treats and vote for best presentation In each session

  4. Phys 452 Research and QM presentation Template My cool research! …or doing simulations or theory As an experimentalist In the lab …

  5. 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

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

  7. Phys 452 Born formalism Worked together with Albert Einstein (Nobel Prize 1921 Photoelectric effect) Werner Heisenberg (Nobel Prize 1932 Creation of QM) Max Born (1882-1970) German physicist Nobel prize in 1954 For interpretation of probability of density y

  8. Phys 452 Quiz 35a What is the main idea of the Born approximation? • To develop a formalism where we express the wave function • in terms of Green’s functions • B. To use Helmholtz equation instead of Schrödinger equation • C. To find an approximate expression for y when far away from • the scattering center for a given potential V • D. To express the scattering factor in terms of scattering vector • E. To find the scattering factor in case of low energy

  9. 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

  10. Phys 452 Born formalism Helmholtz equation Solution Helmholtz 1821 - 1894 Green’s function George Green British Mathematician 1793 - 1841 Schrödinger equation

  11. 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: Pb 11.8

  12. Phys 452 Born approximation • First Born approximation

  13. Phys 452 Quiz 35b 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

  14. Phys 452 Born approximation Scattering vector

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

  16. 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

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

  18. Phys 452 • For any energy Scattering- phase shifts Pb 11.13 Spherical delta function shell (Pb 11.4) • Low energy case • Compare results with pb 11.4

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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

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

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

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

  28. 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

  29. 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

  30. Phys 452 Compton scattering Quantum theory

  31. 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

  32. 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

More Related