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Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel Phys 452 Class- evaluation Please take some time to submit your class evaluation on line by April 13 http://studentratings.byu.edu Quiz on Wed April 11 5 points Click if you have submitted evaluation or commit to submit the evaluation by Friday Phys 452 Quiz scores Quiz 1 – 10: part I (January) Quiz 11 – 19: part II (February) Quiz 20: test 2 review Quiz 21- 30: part III (March-April) Review your quiz score, and if you see missing entries, let me know by Friday Phys 452 Class scores W 2012 Tentative grade scale Homework (40%) 92.0 Quizzes (10%) 89.9 Midterms (30%) 76.9 Final (20%) Class average 86.3 Complete all your homework by Friday!! Prepare well for the final!! A AB+ B BC+ C CD+ D D- 93 89 85 80 75 70 60 55 50 45 40 Phys 452 Final test Saturday, April 14 2:30pm – 5:30 pm Where: classroom SWKT 337 Time – limited: 3 hours Comprehensive Closed book, closed notes Bring a calculator Phys 452 Final test Choose 5 out of 6 problems Sample Test (2011) 1. Two particle & perturbation theory Ch 5 & 6 2. Hydrogen fine structure Ch 6 3. Variational principle Ch 7 4. WKB approximation Ch 8 5. Two level & magnetic resonance Ch 9 6. Scattering Ch 11 Phys 452 Final test Choose 5 out of 6 problems Planned test (2012) 1. Ch 6: perturbation theory 2. Ch 7: variational principle 3. Ch 8: WKB approximation 4. Ch 9: Time dependant pertubation 5. Ch 10: Adiabatic approx. 6. Ch 11: Scattering Phys 452 Techniques to solve for the allowed energies 1. The perturbation theory 2. The variational principle 3. The WKB approximation Quiz 32 a For which of these techniques you need to have a first estimate of the eigenstates? A. B. C. D. E. 1 2 3 1&2 All of them Phys 452 Perturbation theory H H H' 0 Unperturbed states Building the true states and true energies to some order H E 0 0 n 0 n 0 n n ... 0 n 1 n 2 n En E E E ... 0 n 1 n zeroorder firstorder 2 n secondorder Phys 452 Non-degenerate Perturbation theory First-order correction Energy State En1 n0 H ' n0 1 n nm H' 0 m E 0 n E 0 m 0 n 0 m Phys 452 Non-degenerate Perturbation theory Second-order correction Energy E 2 n 0 n E 2 n m n H' 0 m 1 n H' 0 n 2 En0 Em0 Only works if the energies are non-degenerate Phys 452 Degenerate perturbation theory General method • Start with an ortho-normal basis of the unperturbed states 0 1 , 20 , 30 ,... • If the state is non-degenerate: En1 n0 H ' n0 • If the state is degenerate: build W Wij i0 H ' 0j • Diagonalize W : the eigenvalues are En1,4 En1,3 1 n,k E 0 n E En1,2 En1,1 l Phys 452 The fine structure of hydrogen 2 p H V (r ) 2m 0 Motion of the electron Bohr’s energies Coulomb interaction between e- and nucleus 2 mc 2 e2 2 1 2 1 mc 2 1 0 En 2 2 2 2 n2 2ma n 2 4 0 c n Phys 452 Quiz 32 b What kind of interaction is at the origin of the spin-orbit coupling effect? A. An interaction between the spins of two electrons located at different orbits B. The spin of an electron interacting with the spin of the nucleus C. The spin of an electron interacting with field created by its angular momentum D. The spin of an electron interacting with the field created by another electron’s angular momentum E. An interaction between the spins of two electrons located in the same orbit Phys 452 The fine structure of hydrogen Fine structure Bohr’s energy E= E 0 n + Relativistic correction 2 2 coupling B mc 1 2 + Spin-orbit H ' Trel Tclassic n2 e S - p4 E H' 3 2 8m c 1 r “Classical e + view” Emag M S .BL Phys 452 The fine structure of hydrogen Fine structure E= E 0 n Bohr’s energy + Relativistic correction mc 1 2 E 2 2 n + Spin-orbit 2 0 n coupling 2 4n 3 2 2mc l 1/ 2 1 n E 1 n E E n j ( j 1) l (l 1) s(s 1) 0 n mc 2 2 l (l 1/ 2)(l 1) Phys 452 The fine structure of hydrogen E= En0 Bohr’s energy 2 mc 2 1 2 n2 + Fine structure + En0 4n E fs E 1 3 2 j 1/ 2 2mc 0 n New relevant quantum numbers: n, l, s, j and mj Zeeman effect ? Phys 452 Zeeman effect Bext S L e- “Classical view” Weak-field Fine structure dominates e H ( L 2S ).Bext 2m ' Z • Comparing: Bext Intermediate field ' H ' H SO H Z' and Bint Strong field Zeeman effect dominates Phys 452 Zeeman effect Bext S Weak -field L Good eigenstates: e- Bext Bint n, l , j , m j E B g j m j Bext 1 Z j ( j 1) l (l 1) 3 / 4 with Lande factor: g j 1 2 j ( j 1) 2 0 Etot En 1 2 cnj B g j m j Bext n Phys 452 Zeeman effect Bext S Strong -field L e- Bext Bint Good eigenstates: n, l , ml , s, ms E B ml 2mS Bext 1 Z 0 l (l 1) ml ms E 1 2 3 1 E fs 3 n 4n l (l 1/ 2)(l 1) Phys 452 Variational principle ??? Hamiltonian Schrödinger Equation… … very hard to solve! Many particles Egs H Ground state Expectation value on any normalized function Phys 452 Quiz 32 c / d With the variational principle, we are guaranteed to find out the ground state A. TRUE B. FALSE What are we basically adjusting in the variational principle? A. The Hamiltonian B. The wave function C. Both Phys 452 Variational principle The method: • Define your system, and the Hamiltonian H • Pick a normalized wave function • Calculate H • Minimize H • You get an estimate of ground state energy Egs H min Phys 452 The ground state of Helium 2 particles system r1 p12 p22 2e2 1 1 e2 1 H 2m 2m 4 0 r1 r2 4 0 r1 r2 r2 Kinetic energy Interaction with proton He atom Electron- electron interaction Zero-order Hamiltonian H0 Perturbation Vee Phys 452 The ground state of Helium • Use the variational principle to account for screening effects Z3 Z r1 r2 / a r1 , r2 3 Ae a r1 r2 Z 2 Vee e2 2 3 1 r1 , r2 d r2 d 3r1 4 0 r1 r2 He atom Vee 5Z E1 4 Phys 452 The ground state of Helium • Use the variational principle to account for screening effects r1 r2 Z 2 p12 p22 2e2 1 1 e2 1 H 2m 2m 4 0 r1 r2 4 0 r1 r2 He atom p12 p22 Ze 2 1 1 e 2 Z 2 Z 2 1 H 2m 2m 4 0 r1 r2 4 0 r1 r2 r1 r2 H 2 1 e 2 2Z E1 2( Z 2) 4 r 0 Vee Phys 452 Hydrogen molecule ion H2+ 0 r2 0 r1 Energy Minimization H E1 R Equilibrium separation distance: Req 2.4a 1.3 x Presence of a minimum: Evidence of bonding R a Phys 452 The WKB approximation V(x) Turning points E Non-classical region (E<V) Non-classical region (E<V) Classical region (E>V) Phys 452 The WKB approximation Excluding the turning points: E V x E V x i WKB x p ( x ) dx C e p( x) WKB x p ( x ) dx e C 1 p( x) where p x 2m( E V x ) Phys 452 Tunneling trough a barrier V(x) V0 A F B -a x +a 2 Transmission coefficient F 2 T e A 1 a p x dx a Phys 452 Quiz 34 e A particle with an incident energy E is approaching a barrier of potential V and width a. Which one of these statements does NOT apply? A. The transmission coefficient through the barrier depends on E, V and a B. The transmission coefficient increases when a decreases for a given E and V C. The transmission coefficient increases when V decreases for a given E and a D. The transmission coefficient increases when E decreases for a given V and a E. The particle has some chances to be reflected by the barrier if V>E Phys 452 The WKB approximation Patching – upward slope V(x) Patching region Linear approximation Overlap 1 Overlap 2 E Classical region (E>V) X=0 Non-classical region (E<V) Phys 452 The WKB approximation Patching – upward slope • General expression for the wave function 2D 1 x2 sin p x ' dx ' x 4 p x x x D 1 exp p x ' dx ' x2 p x x x2 x x2 Phys 452 The WKB approximation Connection formulas • Potential with 2 walls a p x dx n 0 x2 • Potential with 1 wall 0 x2 • Potential with no walls x2 x1 1 p x dx n 4 1 p x dx n 2 Phys 452 Dynamical systems d H i dt V depends on time General solution: cn t n eiEnt / n Probability to measure the energy En: cn t 2 Phys 452 Two- level systems Time- dependent perturbation Sinusoidal perturbation Eb Ea H ' V cos t 2 Vab sin 0 t / 2 2 2 Probability of transition: Pa b t 0 2 P() for a given time t Resonance effect 0 Phys 452 Emission and absorption of a radiation z Eb x Ea y E E0 cos t nz Hab ' pab E0 cos t Transition rate Rba t with pab 3 0 2 pab q a r b 2 0 B 0 Phys 452 Quiz 32f What can we say about the transition rate of a stimulated emission ? A. It is the same as the transition rate for absorption B. It is opposite to the transition rate for absorption C. It is inverse of the transition rate for absorption D. It adds up with the spontaneous emission rate to cancel the absorption’s one Phys 452 Emission and absorption: Einstein coefficients Eb Nb Ea Na Bab : stimulated absorption rate Bab : stimulated emission rate A : spontaneous emission rate Bab Bba A Thermal equilibrium 0 dNb Nb A Nb Bba 0 N a Bab 0 0 dt A Na / Nb Bab Bba 03 2 A p 3 3 0 c Boltzman distribution of particles Analogy with Planck’s blackbody formula Excited state lifetime 1/ A Phys 452 Emission and absorption Selection rules E l 0 n4 n3 n2 n 1 l 1 l 2 l 3 Electric Dipole transitions l 1 m 0, 1 Phys 452 Adiabatic approximation t cn 0 ei t ei t n t General solution Adiabatic approx with n m t i t 0 m m dt ' t m R m n t 1 t E t 'dt ' n 0 Geometric phase m t i n dR Dynamic phase Berry’s phase Phys 452 Scattering Develop the solution in terms of spherical harmonics, Solution to Coulomb potential r, , R r Yl m , 2 d 2u l (l 1) V r u Eu 2 2 2m dr 2m r 2 kr 1 V 0 Radiation zone V 0 V 0 intermediate zone Scattering zone Phys 452 Scattering Partial wave analysis kr 1 Connecting all three regions and expressing the Global wave function in spherical coordinates ikz eikr r , A e f r V 0 V 0 Rayleigh’s formula Scattered waves r , A i l 2l 1 jl kr ikal hl1 kr Pl cos l Total cross-section D d 4 2l 1 al l 2 To be determined by the type of potential + boundary conditions Phys 452 Scattering Phase - shifts al Scattering factor Scattering Cross-section 1 2il 1 e 1 eil sin l 2ik k 1 f 2l 1eil sin l Pl cos k l 0 4 2 k 2 2 l 1 sin l l 0 Phys 452 Born approximation k' Scattering vector q k Spherical wave Plane wave f , m 2 2 iq .r 3 e V r d r Phys 452 Born approximation • Low energy approximation f , q.r m 2 2 1 3 V r d r • Case of spherical potential 2m f , 2 rV r sin qr dr q0 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 q2 k' d 2 . ' d k ' 4 0 mc 2 k 2