Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Thu Apr 5: assignment # , 11.10, 11.11, Tuesday April 10: assignment # , 11.18, Sign up for the QM & Research presentations Fri April 6 or Mon April 9 Homework #24 20 pts

Class- schedule Phys 452 Today April 4: Born approximation, Compton effect Friday April 6 : research & QM presentations I Mon. April 9 : research & QM presentations II Wed. April 11: FINAL REVIEW 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 Easy formula to calculate f(  )? q or f(q)?

Phys 452 Born formalism Max Born ( ) German physicist Nobel prize in 1954 For interpretation of probability of density  Worked together with Albert Einstein (Nobel Prize 1921 Photoelectric effect ) Werner Heisenberg (Nobel Prize 1932 Creation of QM)

Phys 452 Quiz 35a What is the main idea of the Born approximation? A.To develop a formalism where we express the wave function in terms of Green’s functions B. To use Helmholtz equation instead of Schrödinger equation C. To find an approximate expression for  when far away from the scattering center for a given potential V D. To express the scattering factor in terms of scattering vector E. To find the scattering factor in case of low energy

Phys 452 Born formalism Max Born ( ) German physicist Nobel prize in 1954 For interpretation of probability of density  Born approximation: The main impact of the interaction is that an incoming wave of direction is just deflected in a direction but keeps same amplitude and same wavelength. One can express the scattering factor In terms of wave vectors

Phys 452 Born formalism Solution Schrödinger equation Helmholtz equation Helmholtz Green’s function George Green British Mathematician

Phys 452 Born formalism Green’s function Integral form of the Schrödinger equation Using Fourier Transform of Helmholtz equation and contour integral with Cauchy’s formula, one gets: Pb 11.8

Phys 452 Born approximation First Born approximation

Phys 452 Quiz 35b When expressing the scattering factor as following A.The potential is spherically symmetrical B. The wavelength of the light is very small C. This scattering factor is evaluated at a location relatively close to the scattering center D. The incoming wave plane is not strongly altered by the scattering E. The scattering process is elastic What approximation is done?

Phys 452 Born approximation Scattering vector

Phys 452 Born approximation Case of spherical potential Low energy approximation Examples: Soft-sphere Yukawa potential Rutherford scattering

Phys 452 Born approximation Soft sphere potentialPb Scattering amplitude Approximation at low E Case of spherical potential Develop and to third order

Phys 452 Scattering – Phase shift Pb Yukawa potential Expand

Phys 452 Scattering- phase shifts Spherical delta function shell (Pb 11.4) Pb Low energy case For any energy Compare results with pb 11.4

Phys 452 Scattering – Born approximation Pb Gaussian potential Integration by parts f has also a Gaussian shape in respect to q Total cross- section Differential cross- section don’t forget that

Phys 452 Born approximation Impulse approximation  p momentum I impulse Deflection Step 2. Evaluate the impulse I Step 3. Evaluate the deflection  Pb 11.14: Rutherford scattering b   r q1q1 q2q2 Step 1. Evaluate the transverse force Step 4. deduct relationship between b and 

Phys 452 Born approximation Impulse and Born series Unperturbed wave (zero order) Deflected wave (first order) Extending at higher orders Zero order First order Second orderThird order propagator See pb 11.15

Phys 452 Born approximation Pb 11.18: build a reflection coefficient Delta function well: Finite square well -aa Pb Pb Back scattering (in 1D) See pb 11.17

Phys 452 Quiz 35 Compton scattering essentially describes: A. The scattering of electrons by matter B. The scattering of high energy photon by light atoms C. The scattering of low energy photons by heavy atoms D. The scattering of lo energy neutrons by electrons E. The scattering of high energy electrons by matter

Phys 452 Compton scattering January 13, 1936 Arthur Compton ( , Berkeley) American physicist Nobel prize in 1927 For demonstrating the “particle” concept of an electromagnetic radiation

Phys 452 Compton scattering Phys rev. 21, 483 (1923)

Phys 452 Compton scattering Electromagnetic wave Particle: photon Classical treatment: Collision between particles Conservation of energy Conservation of momentum

Phys 452 Compton scattering Homework Compton problem (a): Derive this formula from the conservation laws Compton experiments Final wavelength vs. angle

Phys 452 Compton scattering Quantum theory Photons and electrons treated as waves Goal: Express the scattering cross-section Constraint 1: we are not in an elastic scattering situation So the Born approximation does not apply… Constraint 2: the energy of the photon and recoiled electron are high So we need a relativistic quantum theory We need to evaluate the Hamiltonian for this interaction and solve the Schrodinger equation

Phys 452 Compton scattering Quantum theory Klein – Gordon equation: relativistic electrons in an electromagnetic field Vector potential Interaction Hamiltonian (perturbation theory) Vector potential momentum Energy at rest

Phys 452 Compton scattering Quantum theory

Phys 452 Compton scattering Quantum theory Electron in a scattering state with First order perturbation theory to evaluate the coefficients: Homework Compton problem (b): Show that

Phys 452 Compton scattering Quantum theory We retrieve the conservation laws: Furthermore we can evaluate the cross-section: (d): Compare to Rutherford scattering cross-section Homework Compton problem (c): Evaluate  in case of (Thomson scattering)