Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Friday Mar 30: assignment # , 11.2, 11.4 Tuesday Apr 3: assignment # , 11.6, 11.7 Sign up for the QM & Research presentations On Friday April 6 or Monday April 9

Homework Phys 452 Fri April 6 & Mon April 9 assignment # 24 Research &QM presentations Briefly describe your research project and how Quantum Mechanics can help you or can be connected to your research field If no direct connection between your research and QM, mention one topic of QM that could potentially be useful or that you particularly liked 2-3 minutes / student (suggested 2-3 transparencies)

Phys 452 Scattering A classical geometrical view b Impact parameter Scattering angle Scattering solid angle Incident cross-section Differential cross-section

Phys 452 Scattering A classical geometrical view b Impact parameter Scattering angle Incident cross-section Scattering solid angle To be determined in specific situations Example: Hard-sphere scattering

Phys 452 Scattering Rutherford scattering 1.Conservation of energy 2.Conservation of angular momentum 3. Change of variable to express r(  ) 4. Integration: to find  max in terms of b etc 5 Relationship between  max and  6 Final relationship Pb 11.1 b   r q1q1 q2q2

Phys 452 Scattering Quantum treatment Plane wave Spherical wave Scattering amplitude

Phys 452 Quiz 33 Which one of these statements describes best the quantum treatment of scattering? A.The incident wave is described by the initial state of the wave function B.The scattered wave is described by the final state of the wave function C.Only the scattered wave is the solution to the Schrödinger equation D.The global solution for the wave function includes the sum of the incident and scattered waves E. The quantum theory of scattering has no physical analogy with the classical theory of scattering

Phys 452 Scattering Quantum treatment Plane wave Spherical wave In 3D In 2D In 1D Pb 11.2 Elastic process

Phys 452 Scattering Quantum treatment Plane wave Spherical wave Differential cross-section

Phys 452 X-ray Resonant Magnetic Scattering Electric dipolar transition E1 ( L 2,3 : 2p 3d) edges) chargemagnetism Interaction photon spin / magnetic moment S photon M ’’ e’ z x k q   ’’ e k’ q=k’-k

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, 