Quantum mechanics II Winter 2011 Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

Homework Phys 452 Today Mar 30: assignment # 20 11.1, 11.2, 11.4 Friday Apr 1: assignment #21 11.5, 11.6, 11.7 Sign up for the QM & Research presentations Next week, W April 6 or F April 8

Scattering Phys 452 Quantum treatment Spherical wave Plane wave Relationship with cross-section

Scattering Phys 452 Partial wave analysis Develop the solution in terms of spherical harmonics, Solution to Coulomb potential Radiation zone intermediate zone Scattering zone

in the scattering region Phys 452 Scattering Partial wave analysis General solution where V~0: Wave function in spherical coordinates Rayleigh’s formula Scattered waves To be determined by solving the Schrödinger equation in the scattering region + boundary conditions Total cross-section for large r

Scattering Phys 452 Partial wave analysis (Pb 11.3) Example: Hard-sphere scattering Boundary conditions Exploiting Total cross-section

Scattering- Partial wave analysis Phys 452 Scattering- Partial wave analysis Pb 11.4 Spherical delta function shell Assumption (low energy) Outside: Inside: Boundary conditions at r = a Continuity of Discontinuity of In spherical coordinates Find a relationship between a0 , a and a...

Scattering Phys 452 Phase - shifts Physical representation in 1D wall Physical representation in 3D

Scattering Phys 452 Phase – shifts and interference effects Physical representation in 1D wall interference

Quiz 32 Phys 452 In scattering and interference processes, the phase shift depends on the wavelength True False

Scattering Phys 452 Phase - shifts If If Asymptotic behavior at Outgoing spherical wave Incoming spherical wave If

Scattering Phys 452 Phase - shifts Partial wave amplitude Phase shift Connecting the asymptotic behavior at Partial wave amplitude Phase shift

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

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

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

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