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Physics 452

Physics 452. Quantum mechanics II Winter 2011. Karine Chesnel. Phys 452. Sign up for the QM & Research presentations Next week, W April 6 or F April 8. Homework #24 20 pts. Homework. Today Apr 1 : assignment #21 11.5, 11.6, 11.7. Wed Apr 6 : assignment #22 11.8, 11.10, 11.11, 11.13

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Physics 452

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  1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

  2. Phys 452 Sign up for the QM & Research presentations Next week, W April 6 or F April 8 Homework #24 20 pts Homework Today Apr 1: assignment #21 11.5, 11.6, 11.7 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

  3. Phys 452 Today April 1st: Born approximation Monday 4th: Born approx., Compton effect Wednesday 6th: research & QM presentations I Friday 8th :research & QM presentations II Class- schedule 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 Relationship with cross-section Quantum treatment Spherical wave Plane wave

  7. Phys 452 Scattering To be determined by solving the Schrödinger equation in the scattering region + boundary conditions Total cross-section Partial wave analysis Connecting all three regions and expressing the Global wave function in spherical coordinates Rayleigh’s formula Scattered waves

  8. Phys 452 Scattering Scattering amplitude Scattering Cross-section Phase - shifts

  9. Phys 452 Scattering – phase shift • In region 1 • In region 2 2) Continuity at boundary: 3) Identify the phase shift Pb 11.5 Reflection against a wall 1) Solve the Schrödinger equation wall Region 1 Region 2

  10. Phys 452 Scattering – Phase shift Boundary conditions We found Express the phase shift: using Express in terms of functions and Pb 11.6 Hard sphere scattering (Pb 11.3)

  11. Phys 452 Inside: Continuity of Boundary conditions Discontinuity of Express in terms of and Scattering- phase shifts Pb 11.7 Spherical delta function shell (Pb 11.4) Do NOT do the assumption Outside:

  12. 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

  13. Phys 452 Quiz 32 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 • D. To express the scattering factor in terms of scattering vector • E. To find the scattering factor in case of low energy

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

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

  16. Phys 452 Born formalism Green’s function Pb 11.8 Check that it satisfies the Helmholtz equation Integral form of the Schrödinger equation Using Fourier Transform of Helmholtz equation and contour integral with Cauchy’s formula, one gets:

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

  18. Phys 452 Born approximation • (First order) Born approximation Scattering vector

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

  20. 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

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

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

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