Physics 452 Quantum mechanics II Winter 2012 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 Tuesday Apr 3: assignment #21 11.5, 11.6, 11.7 Thursday Apr 5: assignment #22 11.8, 11.10, 11.11, 11.13 Tuesday April 10: assignment #23 11.14, 11.18, 11.20
Phys 452 Today April 2: Partial waves, phase shifts Wed April 4 : Born approx., Compton effect Friday April 6 : research & QM presentations I Monday 9 :research & QM presentations II Class- schedule Treats and vote for best presentation In each session
Phys 452 Research and QM presentation Template My cool research! …or doing simulations or theory As an experimentalist In the lab …
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
Phys 452 Scattering Relationship with cross-section Quantum treatment Spherical wave Plane wave
Phys 452 Scattering Develop the solution in terms of spherical harmonics, solution to a spherically symmetrical potential Radiation zone intermediate zone Scattering zone Partial wave analysis
Scattering Phys 452 Radiation zone Intermediate zone Physical Solution Hankel functions General Solution Scattering zone Geometrical considerations Solve the Schrödinger equation with potential V Partial wave amplitude Partial wave analysis
Phys 452 Scattering when with Differential cross-section Total cross-section Partial wave analysis Connecting intermediate and radiation zone Orthogonality of Legendre polynomials
Phys 452 Scattering Rayleigh’s formula 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 JlBessel functions
Phys 452 Scattering Bessel function Legendre polynomial Hankel function Partial wave analysis
Phys 452 Scattering Boundary conditions Exploiting Total cross-section Partial wave analysis (Pb 11.3) Example: Hard-sphere scattering
Phys 452 Assumption (low energy scattering) Outside: Inside: Continuity of Boundary conditions Discontinuity of Find a relationship between a0and (a,a...) Scattering- Partial wave analysis Pb 11.4 Spherical delta function shell
Phys 452 Scattering Physical representation in 1D Physical representation in 3D wall Phase - shifts
Phys 452 Scattering wall interference Phase – shifts and interference effects Physical representation in 1D
Phys 452 Quiz 34 In scattering and interference processes, the phase shift generally depends on the wavelength • True • False
Phys 452 Scattering Outgoing spherical wave Incoming spherical wave If Phase - shifts Asymptotic behavior at If
Phys 452 Scattering Connecting the asymptotic behavior at Partial wave amplitude Phase shift Phase - shifts
Phys 452 Scattering Scattering amplitude Scattering Cross-section Phase - shifts
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
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)
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:
