Quantum Mechanics II: Final Test Preparation Tips

1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

2. Phys 452 Quiz 35 on Wed April 13 5 points Class- evaluation Please take some time to submit your class evaluation on line by April 15 http://studentratings.byu.edu

3. Phys 452 Class schedule • Today, April 11 : • Finish the presentations • Final Review Wed, April 13 : Practice final test • Review the chosen problem with your group • Type the solution • Decide on who is going to present Quiz 36: 10 pts

4. Phys 452 Tentative grade scale A 93 A- 89 B+ 85 B 80 B- 75 C+ 70 C 60 C- 55 D+ 50 D 45 D- 40 Class scores W 2011 90.1 90.8 80.8 Homework (40%) Quizzes (10%) Midterms (30%) Final (20%) Class average 86.7 Prepare well for the final!!

5. Phys 452 Final test Tuesday, April 19 2pm - 5 pm Where: classroom C247 Time – limited: 3 hours Comprehensive Closed book, closed notes Bring a calculator

6. Phys 452 Final test Choose 5 out of 6 problems Similar to • Time-independent perturbation theory • 2. Hydrogen fine structure, Zeeman effect • 3. Variational principle • 4. WKB approximation • 5. Emission, absorption • 6. Scattering

7. Phys 452 • Quiz 34 a • For which of these techniques • you need to have a first • estimate of the eigenstates? • 1 • 2 • 3 • 1 & 2 • All of them Techniques to solvefor the allowed energies 1. The perturbation theory 2. The variational principle 3. The WKB approximation

8. Phys 452 Unperturbed states Building the true states and true energies to some order first- order second- order zero- order Perturbation theory

9. Phys 452 Energy State Non-degenerate Perturbation theory First-order correction

10. Phys 452 Energy Non-degenerate Perturbation theory Second-order correction Only works if the energies are non-degenerate

11. Phys 452 • Start with an ortho-normal basis of the unperturbed states • If the state is non-degenerate: • If the state is degenerate: build • Diagonalize W : the eigenvalues are l Degenerate perturbation theory General method

12. Phys 452 Coulomb interaction between e- and nucleus Motion of the electron Bohr’s energies The fine structure of hydrogen

13. Phys 452 Quiz 34 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

14. Phys 452 Fine structure B Relativistic correction Spin-orbit coupling + + S e- e+ “Classical view” The fine structure of hydrogen Bohr’s energy E =

15. Phys 452 Fine structure Relativistic correction Spin-orbit coupling + + The fine structure of hydrogen Bohr’s energy E =

16. Phys 452 + Fine structure ? New relevant quantum numbers: n, l, s, j and mj The fine structure of hydrogen Bohr’s energy E = + Zeeman effect

17. Phys 452 Zeeman effect Bext S L • Comparing: and e- “Classical view” Intermediate field Strong field Weak-field Fine structure dominates Zeeman effect dominates

18. Phys 452 Good eigenstates: with Lande factor: Zeeman effect Bext S Weak -field L e-

19. Phys 452 Good eigenstates: Bext S L e- Zeeman effect Strong -field

20. Phys 452 Hamiltonian ??? Ground state Expectation value on any normalized function y Schrödinger Equation… … very hard to solve! Many particles Variational principle

21. Phys 452 Quiz 34 c / d What are we basically adjusting in the variational principle? A. The Hamiltonian B. The wave function C. Both With the variational principle, we are guaranteed to find out the ground state A. TRUE B. FALSE

22. Phys 452 • Calculate • Minimize • You get an estimate • of ground state energy Variational principleThe method: • Define your system, and the Hamiltonian H • Pick a normalized wave function y

23. Phys 452 The ground state of Helium Kinetic energy Interaction with proton Electron- electron interaction Zero-order Hamiltonian H0 Perturbation 2 particles system He atom

24. Phys 452 The ground state of Helium • Use the variational principle to account for screening effects He atom

25. Phys 452 He atom The ground state of Helium • Use the variational principle to account for screening effects

26. Hydrogen molecule ion H2+ Phys 452 Equilibrium separation distance: Presence of a minimum: Evidence of bonding Energy Minimization

27. Phys 452 Turning points E Non-classical region (E<V) Non-classical region (E<V) Classical region (E>V) The WKB approximation V(x)

28. Phys 452 The WKB approximation Excluding the turning points: where

29. Phys 452 Tunneling trough a barrier V(x) V0 A F B x -a +a Transmission coefficient

30. 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 adecreases for a given E and V C. The transmission coefficient increases when V decreases for a given E anda D. The transmission coefficient increases when E decreases for a given V anda E. The particle has some chances to be reflected by the barrier if V>E

31. The WKB approximation Phys 452 Patching region Overlap 1 Overlap 2 E Patching – upward slope V(x) Linear approximation X=0 Classical region (E>V) Non-classical region (E<V)

32. The WKB approximation Phys 452 Patching – upward slope • General expression for the wave function

33. The WKB approximation Phys 452 • Potential with 2 walls • Potential with 1 wall • Potential with no walls Connection formulas

34. Phys 452 General solution: Probability to measure the energy En: Dynamical systems V depends on time

35. Phys 452 Two- level systemsTime- dependent perturbation P(w) for a given time t Resonance effect w Sinusoidal perturbation Probability of transition:

36. Phys 452 Emission and absorption of a radiation with z x y Transition rate

37. Phys 452 Quiz 36e 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

38. Phys 452 Emission and absorption: Einstein coefficients : stimulated absorption rate : stimulated emission rate : spontaneous emission rate Thermal equilibrium Excited state lifetime Boltzman distribution of particles Analogy with Planck’s blackbody formula

39. Phys 452 Emission and absorption Selection rules E Electric Dipole transitions

40. Phys 452 Dynamic phase Geometric phase Berry’s phase Adiabatic approximation General solution Adiabatic approx with

41. Phys 452 Scattering Develop the solution in terms of spherical harmonics, Solution to Coulomb potential Radiation zone intermediate zone Scattering zone

42. Phys 452 Scattering To be determined by the type of potential + boundary conditions Total cross-section Partial wave analysis Connecting all three regions and expressing the Global wave function in spherical coordinates Rayleigh’s formula Scattered waves

43. Phys 452 Scattering Scattering factor Scattering Cross-section Phase - shifts

44. Phys 452 Born approximation Scattering vector Spherical wave Plane wave

45. Phys 452 Born approximation • Low energy approximation • Case of spherical potential

46. Phys 452 Compton scattering Quantum theory • We retrieve the conservation laws: • Furthermore, we can evaluate • the cross-section: