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Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel Phys 452 Homework Wed Apr 6: assignment #22 11.8, 11.10, 11.11, 11.13 Friday April 8: assignment #23 11.14, 11.18, 11.20 + Compton question QM & Research presentations Next week, W April 6 or F April 8 Homework #24 30 pts : research overview connection with QM Presenting 10pts 10pts 10pts Phys 452 Class- schedule Today 4th : Born approx., Compton effect Wednesday 6th : research & QM presentations I Friday 8th :research & QM presentations II Treats and vote for best presentation In each session Next week April 11 & 13: Final Review Phys 452 Research and QM presentation 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 q Plane wave Spherical wave ikz eikr A e f , r Easy formula to calculate f(,)? or f(q)? Phys 452 Born formalism Born approximation: The main impact of the interaction is that an incoming wave of direction k is just deflected in a direction k ' but keeps same amplitude and same wavelength. Max Born (1882-1970) One can express the scattering factor f , German physicist In terms of wave vectors Nobel prize in 1954 For interpretation of probability of density k,k ' Phys 452 Born formalism Using Fourier Transform of Helmholtz equation and contour integral with Cauchy’s formula, one gets: eikr G r 4 r Green’s function r 0 r ik r r m 2 2 e 0 3 V r r d r r0 0 0 r0 Integral form of the Schrödinger equation Phys 452 Born approximation • Spatial approximation r eikr ikr0 G r r0 e 4 r r0 • (First order) Born approximation The incoming wave is scattered in a wave of same amplitude, just different direction r Aeik ' r f , 0 r Aeikr m 2 2 e i k 'k r V r d 3r Phys 452 Quiz 34 When expressing the scattering factor as following f , m 2 2 iq .r 3 e V r d r What approximation is done? 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 Phys 452 Born approximation k' Scattering vector q q 2k sin / 2 k f , m 2 2 e iq .r V r d r 3 4 sin / 2 Phys 452 Born approximation • Low energy approximation q.r 1 Examples: f , m 2 V r d r 3 2 • Soft-sphere • Case of spherical potential 2m f , 2 rV r sin qr dr q0 • Yukawa potential • Rutherford scattering Phys 452 Born approximation Pb 11.10 Soft sphere potential • Scattering amplitude Case of spherical potential f , rV0 sin qr dr V0 0 • Approximation at low E qa Develop sin qa and 1 cos qa to third order Phys 452 Scattering – Phase shift Pb 11.11 Yukawa potential e r V r r f , e r sin qr dr 0 Expand 1 iqr iqr sin qr e e 2i f 1 2 q2 Phys 452 Scattering- phase shifts Pb 11.13 V 0 Spherical delta function shell (Pb 11.4) V (r ) r a • Low energy case V 0 f , r a d 3r • More general spherical potential f , r r a sin qr dr f 2 m a 2 2 Phys 452 Scattering – Born approximation Pb 11.20 Gaussian potential V r Ae f , re r 2 r2 sin qr dr 0 Integration by parts f e Differential cross- section d f d f d 2 Total cross- section q2 /4 f has also a Gaussian shape in respect to q 2 don’t forget q 2k sin / 2 Phys 452 Born approximation Impulse approximation I F dt impulse momentum p I Deflection Pb 11.14: tan I p Rutherford scattering q1 Step 1. Evaluate the transverse force F Step 2. Evaluate the impulse I Step 3. Evaluate the deflection b Step 4. deduct relationship between b and r q2 Phys 452 Born approximation Impulse and Born series r 0 r G r r 0 V r0 r0 d 3r0 Unperturbed wave (zero order) Deflected wave (first order) Extending at higher orders propagator r 0 r GV GVGV GVGVGV ... Zero order First order Second order Third order See pb 11.15 Phys 452 Born approximation Pb 11.16 Pb 11.17 Pb 11.18: build a reflection coefficient Back scattering (in 1D) R m 2 k • Delta function well: V x x 2 2 ikx e V x dx R 2ikx e x dx • Finite square well a -a a R 2ikx e V0dx a 2 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 Arthur Compton (1892-1962, Berkeley) American physicist Nobel prize in 1927 For demonstrating the “particle” concept of an electromagnetic radiation January 13, 1936 Phys 452 Compton scattering Phys rev. 21, 483 (1923) Phys 452 Compton scattering Classical treatment: Collision between particles • Conservation of energy • Conservation of momentum Electromagnetic wave Particle: photon Phys 452 Compton scattering Compton experiments Final wavelength vs. angle Homework Compton problem (a): Derive this formula from the conservation laws 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… We need to evaluate the Hamiltonian for this interaction and solve the Schrodinger equation Constraint 2: the energy of the photon and recoiled electron are high So we need a relativistic quantum theory Phys 452 Compton scattering Quantum theory • Klein – Gordon equation: relativistic electrons in an electromagnetic field 2 2 2 2 2 4 c i qA m c 2 t momentum Vector potential A A0 As • Vector potential • Interaction Hamiltonian (perturbation theory) 2 1 H i qA mc 2 m q2 H ' 2 As . A0† m Energy at rest Phys 452 Compton scattering Quantum theory q2 H ' 2 As . A0† m i k '.r 't As As e i k .r t A0 A0e ' q 2 As A0 i k ' k r ' t H'2 e '. m Phys 452 Compton scattering Quantum theory Electron in a scattering state r , t cp pd p 3 with p r ,t mc3 2 3 ei p.r Et / E First order perturbation theory to evaluate the coefficients: mc 2 0 3 c p ' i dt d p H ' c p p' p 2 1 Homework Compton problem (b): Show that 1 cp' E E ' ' p p ' k k ' 0 i 2 4 3 q mc A0 As . ' d p cp 2 E.E ' Phys 452 Compton scattering Quantum theory We retrieve the conservation laws: p ' k ' p k E ' ' E Furthermore we can evaluate the cross-section: 2 q k' d 2 . ' d k ' 4 0 mc 2 k 2 2 Homework Compton problem (c): Evaluate in case of k' k (d): Compare to Rutherford scattering cross-section (Thomson scattering)