Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Sign up for the QM & Research presentations Next week, W April 6 or F April 8 Homework #24 20 pts Tuesday Apr 3: assignment # , 11.6, 11.7 Thursday Apr 5: assignment # , 11.10, 11.11, Tuesday April 10: assignment # , 11.18, 11.20

Class- schedule 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 Treats and vote for best presentation In each session

Research and QM presentation Phys 452 Template As an experimentalist In the lab … …or doing simulations or theory

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

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

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

Phys 452 Scattering Partial wave analysis Intermediate zone Scattering zone Solve the Schrödinger equation with potential V Physical Solution Hankel functions General Solution Partial wave amplitude Geometrical considerations Radiation zone

Phys 452 Scattering Partial wave analysis Connecting intermediate and radiation zone Differential cross-section Total cross-section when with Orthogonality of Legendre polynomials

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

Phys 452 Scattering Partial wave analysis Legendre polynomialBessel function Hankel function

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

Phys 452 Scattering- Partial wave analysis Spherical delta function shell Pb 11.4 Assumption(low energy scattering) Outside: Inside: Boundary conditions Continuity of Discontinuity of Find a relationship between a 0 and (a, 

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

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

Phys 452 Quiz 34 In scattering and interference processes, the phase shift generally depends on the wavelength A.True B.False

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

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

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

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

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

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