PHYS264 SPRING 2008 collected notes page 1 Lectures Tuesday 15. January 2008 Wednesday 16. January 2008 Topics: Mean free path, Beams, Scattering Cross sections Comment: Preliminary version without text; Version with text coming soon

PHYS264 SPRING 2008 collected notes page 2 Tuesday 15. January 2008

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PHYS264 SPRING 2008 collected notes page 4 Wednesday 16. January 2008 Notes as recorded during the lecture: Exercise: Long range forces Coulomb 1/r energy 1/r^2 FORCE Dipole-Dipole 1/r^3 mean free path - one collision the probability reduces to 1/e Spheres with r and R collide. Cross section? Differential scattering for hard sphere Do the derivation - isotropic angular distribution The text will be edited

PHYS264 SPRING 2008 collected notes page 5 Wednesday 16. January 2008

PHYS264 SPRING 2008 collected notes page 6 Wednesday 16. January 2008 HOMEWORK: Differential scattering for hard sphere Do the derivation - isotropic angular distribution

PHYS264 SPRING 2008 collected notes page 7 Lecture Tuesday 22. January 2008 Topics: Scattering Cross Section Hard Sphere Scattering Comment: Preliminary version- Please, do the homework!

PHYS264 SPRING 2008 collected notes page 8 Notes Exercise

PHYS264 SPRING 2008 collected notes page 9 We need to know THETA as function of impact parameter b

PHYS264 SPRING 2008 collected notes page 10 From our notes:

PHYS264 SPRING 2008 collected notes page 11 Notes Exercise

PHYS264 SPRING 2008 collected notes page 12 Homeworks Consider scattering of two spheres on each other. How will the result change from the indicated one? Make a drawing illustrating the situation ___________________________________________ Find in literature the scattering of charged particles presentation. Consider the basics of Rutherford Cross Section Our text (Feuerstein) has Quantum results

PHYS264 SPRING 2008 collected notes page 13 Lecture Wednesday 23. January 2008 Topics: Scattering Cross Section WAVE PICTURE (QUANTAL) Comment: Preliminary version

PHYS264 SPRING 2008 collected notes page 14 Notes Exercise

PHYS264 SPRING 2008 collected notes page 15 We need to know THETA as function of impact parameter b

PHYS264 SPRING 2008 collected notes page 16 From our notes:

PHYS264 SPRING 2008 collected notes page 17 Notes Exercise

PHYS264 SPRING 2008 collected notes page 18 Homeworks Consider scattering of two spheres on each other. How will the result change from the indicated one? Make a drawing illustrating the situation ___________________________________________ Find in literature the scattering of charged particles presentation. Consider the basics of Rutherford Cross Section Our text (Feuerstein) has Quantum results

PHYS264 SPRING 2008 collected notes page 19 Radial currrent is characterized by its inverse proportionality to r^2 The expression is thus constant, j(r)*r 2 as any emission from point source in any conserved current density situation Big question: current density in plane geometry (surface waves energy density) What is constant there? Also J(  )*  2 ??? NO: IT IS J(  )*  in pl which is sonserved (Bessel functions and spherical Bessel)

PHYS264 SPRING 2008 collected notes page 20 Probability current density

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PHYS264 SPRING 2008 collected notes page 23 This was a bit wrong Corrected with patches 30% transparent

PHYS264 SPRING 2008 collected notes page 24 Remember the theta(b) is unique, b(theta) is not Exercise: find case with multivalued b(theta)

PHYS264 SPRING 2008 collected notes page 25 Lecture Tuesday 29. January 2008 Topics: Scattering Comment: Preliminary version

PHYS264 SPRING 2008 collected notes page 26 Remember the theta(b) is unique, b(theta) is not Exercise: find case with multivalued b(theta)

PHYS264 SPRING 2008 collected notes page 27 Cross Section for wave theory (quantal) Green's Function.... see next pages

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PHYS264 SPRING 2008 collected notes page 30 The discussion on the difference between QUANTUM AND OPTICAL WAVES has taken place on the next day

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PHYS264 SPRING 2008 collected notes page 35 Lectures Tuesday 29. January 2008 and Wednesday 30. January 2008 Topics: Scattering Kepler and Coulomb Problems Comment: Preliminary version

PHYS264 SPRING 2008 collected notes page 36 Remember the theta(b) is unique, b(theta) is not Exercise: find case with multivalued b(theta)

PHYS264 SPRING 2008 collected notes page 37 Cross Section for wave theory (quantal) Green's Function.... see next pages

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PHYS264 SPRING 2008 collected notes page 39

PHYS264 SPRING 2008 collected notes page 40 The discussion on the difference between QUANTUM AND OPTICAL WAVES has taken place on the next day

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PHYS264 SPRING 2008 collected notes page 45 Lecture Wednesday 30. January 2008 Topics: Scattering Kepler and Coulomb Problems Comment: Preliminary version

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PHYS264 SPRING 2008 collected notes page 51 Lecture Tuesday 5. February 2008 Topics: Scattering Comment: Preliminary version – to be edited

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PHYS264 SPRING 2008 collected notes page 61 Lecture Wednesday 20. February 2008 Quantum Scattering Cross sections Topics: Optical Theorem (the bright spot inside the shadow) Bessel functions and spherical Bessel functions The k f in the expression for scattering amplitude

PHYS264 SPRING 2008 collected notes page 62 The k f in the expression for scattering amplitude Optical Theorem (the bright spot inside the shadow) Bessel functions and spherical Bessel functions Partial waves – The angular part of Schrödinger equation (This part not yet done in final form)

PHYS264 SPRING 2008 collected notes page 63 The k f in the expression for scattering amplitude

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PHYS264 SPRING 2008 collected notes page 66 Bessel J n (x) Spherical Bessel j n (x)

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PHYS264 SPRING 2008 collected notes page 68 Lectures Tuesday 26 th and 27. February 2008 Topics: Born Approximation Partial Wave Methods Comment:matlab files and extra handwritten notes are to be found at our webpages

PHYS264 SPRING 2008 collected notes page 69 Topics:

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PHYS264 SPRING 2008 collected notes page 71 SPHERICAL HARMONICS Geodesy... whenever you believe that Earth is not a pancake, you MUST use spherical harmonics See wikipedia - Geodesy

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PHYS264 SPRING 2008 collected notes page 82 For sufficiently large l-values. l(l+1)/r^2 term will completely mask away any potential close to r=0; (This is not true for 1/r potential discuss the Infinite Range)

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PHYS264 SPRING 2008 collected notes page 84 The matlab routines schrod3.m, ypschr.m and pot.m are needed