Chapter07 3-Dim Problems aka the Hydrogen Atom

1. Chapter073-Dim Problemsakathe Hydrogen Atom

2. Outline • 7.1 Application of Schro Eqn to H-atom • 7.2 Solution of Schro Eqn • Angular Shapes of Wave Functions • Radial Dependence of Wave Functions • 7.3 Orbital Angular Momentum & Quantum Numbers • 7.4 Magnetic Effects upon the H-like atoms • 7.5 Intrinsic Spin • 7.6 Energy Levels

3. 3-D Problems • Separation of Variables • Y(rqf) = R(r) Q(q) F(f) • F(f) solution • Q(q) solution • R(r) equation • Effective Potentials

4. Separation of Variables

5. Coordinates & Normalization r

6. Note derivatives are cleanly separated

7. {a} {b} {c} {d}

8. {d} {e} LHS = const = RHS m2 {f}

9. Azimuthal Behavior …, -2, -1, 0, 1, 2, … Note: 1) EVP 2) Since no V involved only have to do this once forevermore

10. Other Piece {f} {g} {h} {i} {j}

11. {j} LHS = const = RHS {k} {l}

12. Other Angular Piece (co-lattitude) {l} Note: 1) EVP 2) Since no V involved only have to do this once forevermore Solns depend on choice of both l and m Associated Legendre Polynomials defer solving til later when we have nicer techniques

13. Summarizing the Angular Parts So Far Since the angular basis functions are the same regardless of the potential chosen. John Day @ http://www.cloudman.com/gallery1/gallery1_2.html Define the “spherical harmonics” http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi

14. (0,0) http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi (1,±1) (1,0) (2,±2) (2,0) (2,±1)

15. (0,0) (1,±1) (1,0) (2,±2) (2,0) (2,±1)

18. http://www2.physics.umd.edu/~gcchang/courses/phys402/common/notebooks.htmlhttp://www2.physics.umd.edu/~gcchang/courses/phys402/common/notebooks.html

19. Radial Piece {k} effective potential Note: 1) EVP 2) This has to be solved for every different choice of V(r) 3) Will determine the allowed Etot ‘s

21. http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgihttp://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi Summary So Far

22. Summary So Far (0,0) (1,0) (1,±1) (2,0) (2,±2) (2,±1)

24. Effective Potential Depends on the forces involved Atomic motion? Nuclear motion? … Centripetal Term

25. Bare Coulomb Potential He Li Be B C * * * H-atom positronium atom

26. Effective Potential: H atom Free States Etot > 0 l = 0 Etot Bound States Etot < 0 Atomic Potential Example Vcoul := -14.42/r 1 := 1 Vorbital := 3.818 * 1* (1+1) /r^2 Veff := Vcoul + Vorbital Plot[ {Vcoul, Vorbital, Veff}, {r, 0.3, 8}, PlotStyle ~ {{RGBColor[0, 0,1]}, {RGBColor[0, 1,0]}, {RGBColor[l, 0,0]}}, AxesLabel ~ {"r (A)", "Energy (eV)"}]

27. Effective Potential: H atom l = 1 Bound States Etot < 0 Atomic Potential Example Vcoul := -14.42/r 1 := 1 Vorbital := 3.818 * 1* (1+1) /r^2 Veff := Vcoul + Vorbital Plot[ {Vcoul, Vorbital, Veff}, {r, 0.3, 8}, PlotStyle ~ {{RGBColor[0, 0,1]}, {RGBColor[0, 1,0]}, {RGBColor[l, 0,0]}}, AxesLabel ~ {"r (A)", "Energy (eV)"}]

28. Bound States Etot < 0 l = 1 l = 2

30. Probability Distribution Functions

31. Electron Clouds – dot plots http://www.uark.edu/misc/julio/orbitals/ Scatter plots of hydrogen-atom wavefunctions This is a tentative project. The figures that you can link to from this page are made by choosing 2000 points at random, with a probability given by one of the hydrogen atom's wavefunctions. The resulting plots give an idea of the "shape" of the atomic wavefunctions. You can rotate them by clicking and dragging with the mouse; you can also magnify the figure by clicking and dragging vertically while holding down the "shift" key. The points were generated in Mathematica and the interactive figures were generated using LiveGraphics3D. LiveGraphics3D is an applet (not written by me); for it to work, you need to have java enabled in your browser. 31

33. What We Know So Far En independent of l, m 33

34. Electron Clouds – dot plots http://www.uark.edu/misc/julio/orbitals/ Scatter plots of hydrogen-atom wavefunctions This is a tentative project. The figures that you can link to from this page are made by choosing 2000 points at random, with a probability given by one of the hydrogen atom's wavefunctions. The resulting plots give an idea of the "shape" of the atomic wavefunctions. You can rotate them by clicking and dragging with the mouse; you can also magnify the figure by clicking and dragging vertically while holding down the "shift" key. The points were generated in Mathematica and the interactive figures were generated using LiveGraphics3D. LiveGraphics3D is an applet (not written by me); for it to work, you need to have java enabled in your browser.

35. 7.3 Orbital Angular Momentum

36. Orbital Angular Momentum

37. Vector Model Picture Vector Model ≠ Quantum Mechanical Ang Mom.

38. What We Know So Far 38

40. 7.4 Magnetic Effects external applied Bz

41. 7.4 Magnetic Effects external applied Bz Bohr magneton

42. 7.4 Magnetic Effects

43. 7.4 Magnetic Effects

45. 7.5 Intrinsic Spin

46. external applied Bz gyromagnetic ratios

48. The 21-cm Line http://intro.chem.okstate.edu/1314f00/Lecture/Chapter7/Lec11300.html Locates hot H in stars Locates cool H in clouds http://physics.gmu.edu/~lhorne/research1.html

49. http://www.cv.nrao.edu/course/astr534/HILine.html

50. http://www.naoj.org/Pressrelease/2009/09/08/index.html