1 / 54

Total Angular Momentum

Total Angular Momentum. Orbital angular momentum. Spin angular momentum. Total angular momentum. L , L z , S , S z J and J z are quantized. Total Angular Momentum. If j and m j are quantum numbers for the single electron (hydrogen atom) Quantization of the magnitudes

juanitamiller
Download Presentation

Total Angular Momentum

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Total Angular Momentum Orbital angular momentum Spin angular momentum Total angular momentum L, Lz, S, SzJ and Jz are quantized

  2. Total Angular Momentum • If j and mj are quantum numbers for the single electron (hydrogen atom) • Quantization of the magnitudes • The total angular momentum quantum number for the single electron can only have the values

  3. The Total Angular Momentum Diagram Figure 8.5 When forming the total angular momentum from the orbital and spin angular momenta, the addition must be done vectorially, .

  4. Spin-Orbit Coupling • The dipole potential energy • The spin magnetic moment µ . • . An effect of the spins of the electron and the orbital angular momentum interaction is called spin-orbit coupling. is th·e magnetic field due to the proton where cos a is the angle between

  5. Total Angular Momentum No external magnetic field: J can have a fixed valuein only one direction. This direction can be chosen randomly • Only Jz can be known because the uncertainty principle forbids Jx or Jy from being known at the same time as Jz

  6. Total Angular Momentum With an external magnetic field: • will precess about

  7. Total Angular Momentum • Now the selection rules for a single-electron atom become • Δn = anything Δℓ = ±1 • Δmj = 0, ±1 Δj = 0, ±1 • Hydrogen energy-level diagram for n = 2 and n = 3 with the spin-orbit splitting

  8. The Energy-Level Diagram of Sodium Fine structure splitting is too small on this scale(not shown)

  9. Many-Electron Atoms Hund’s rules: The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as rule 1 is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized. We label with 1 and 2 the two-electron atom There are LS coupling and jj coupling to combine four angular momenta J.

  10. LS Coupling This is used for most atoms when the magnetic field is weak. If two electrons are in a single subshell, S = 0 or 1 depending on whether the spins are antiparallel or parallel. For given L, there are 2S + 1 values of J For L > S, J goes from L −S to L + S For L < S, there are fewer than 2S + 1 possible J values The value of 2S + 1 is the multiplicity of the state

  11. Table 8-2 p287

  12. LS Coupling • The notation for a single-electron atom becomes n2S+1LJ • The letters and numbers are called spectroscopic symbols. • There are singlet states (S = 0) and triplet states (S = 1) for two electrons.

  13. LS Coupling Magnesium(0ne electron in 3s subshell and the other excited to the nl subshell) • There are separated energy levels according to whether they are S = 0 or 1 • Allowed transitions must have ΔS = 0 • No allowed (forbidden) transitions are possible between singlet and triplet states with much lower probability

  14. LS Coupling • The allowed transitions for the LS coupling scheme are • ΔL = ±1 ΔS = 0 • ΔJ = 0, ±1 (J = 0 →J = 0 is forbidden) • A magnesium atom excited to the 3s3p triplet state has no lower triplet state to which it can decay. • It is called metastable, because it lives for such a long time on the atomic scale.

  15. jj Coupling It is for the heavier elements, where the nuclear charge causes the spin-orbit interactions to be as strong as the force between the individual and .

  16. Mercury

  17. Figure 8-11 p290

  18. Figure 8-12 p290

  19. Clicker - Questions

  20. 8.3: Anomalous Zeeman Effect Orbital contribution and Spin magnetic moment More than three closely spaced optical lines were observed. The interaction that splits the energy levels in an external magnetic field is caused by interaction. The magnetic moment depends on The 2J + 1 degeneracy for a given total angular momentum state J is removed by the effect of the . If the is small compared to internal magnetic field, then and precess about while precesses slowly about .

  21. Total Angular Momentum With an external magnetic field: • will precess about

  22. The total magnetic moment is • The magnetic total angular momentum numbers mJ from −J to J in integral steps. • splits each state J into 2J + 1 equally spaced levels separated ΔE = V. • For photon transitions between energy levels ΔmJ = ±1, 0 but is forbidden when ΔJ = 0. Anomalous Zeeman Effect μB is the Bohr magneton and it is called the Landég factor

  23. Figure 8.15 Examples of transitions for the normal Zeeman effect. The nine possible transitions are labeled, but there are only three distinctly different energies because the split energy levels are equally spaced (∆E) for both the 1D2 and 1P1 states Figure 8-15 p293

  24. AnomalousZeeman effect for Na Figure 8-16 p294

  25. Rydberg Atom An atom that is highly excited with the outermost electron in a high energy level near ionization Properties of Rydberg atoms:

  26. Useful Combinations of Constants

  27. Solution: In the first excited state, go to the next higher level. In neon one of the 2p electrons is promoted to 3s, so the configuration is 2p^5 3s^1 . By the same reasoning the first excited state of xenon is 5p^5 6s^1.

  28. The 3s state of Na has energy of -5.14eV.Determine the effective nuclear charge. From Figure 8.4 we see that the radius of Na is about 0.16 nm. We know that for single-electron atoms it holds

  29. Problem 8.21

  30. Problem 8.31

  31. Problem 8.33

  32. Problem 8.15

  33. Problem 8.17

  34. Problem 8.19

  35. Problem 10. 18

  36. Problem 10.19

  37. Problem 10.20

  38. Problem 10.22

  39. Why does Bose-Einstein Condensation of Atoms Occur? Rb atom Eric Cornell andCarl Wieman Na atom Wolfgang Ketterle______ Nobel Price 2001 Consider boson and fermion wave functions of two identical particles labeled “1” and “2”. For now they can be either fermions or bosons: Solutions: Identical probability density the same. The solutions to this equation are + symmetric =boson - antisymmetric=fermion Proof: = nonzero probability occupying the same state favors to be in the lower states for Bose-Einstein Condensation Two bosons can occupy the same state this can be generalized to many bosons and under favorable conditions Bose- Einstein condensation occurs. Bosons have integer spins and Fermions half integer spins.

  40. Bose –Einstein condensation in Gases

  41. Clicker - Questions

  42. Stimulated Emission and Lasers Tunable laser: The emitted radiation wavelength can be adjusted as wide as 200 nm. Semi conductor lasers are replacing dye lasers. Free-electron laser:

  43. From: Demonstration of electron acceleration in a laser-driven dielectric microstructure (Byer et al.Vol 503,nature,2013) a b DLA structure and experimental set-up. a, Scanning electron microscope image of the longitudinal cross-section of a DLA structure fabricated as depicted in Extended Data Fig. 1a. Scale bar, 2 mm.b, Experimental set-up. Inset, a diagram of the DLA structure indicating the field polarization direction and the effective periodic phase reset, depicted as alternating red (acceleration) and black (deceleration) arrows. A snapshot of the simulated fields in the structure shows the corresponding spatial modulation in the vacuum channel.

  44. Stimulated Emission and Lasers This laser relies on charged particles. A series of magnets called wigglers is used to accelerate a beam of electrons. Free electrons are not tied to atoms; they aren’t dependent upon atomic energy levels and can be tuned to wavelengths well into the UV part of the spectrum.

  45. Scientific Applications of Lasers • An extremely coherent and nondivergent beam is used in making precise determination of large and small distances. The speed of light in a vacuum is defined. c = 299,792,458 m/s. • Pulsed lasers are used in thin-film deposition to study the electronic properties of different materials. • The use of lasers in fusion research • Inertial confinement: A pellet of deuterium and tritium would be induced into fusion by an intense burst of laser light coming simultaneously from many directions.

  46. Holography • Consider laser light emitted by a reference source R. • The light through a combination of mirrors and lenses can be made to strike both a photographic plate and an object O. • The laser light is coherent; the image on the film will be an interference pattern.

  47. Holography After exposure this interference pattern is a hologram, and when the hologram is illuminated from the other side, a real image of O is formed. If the lenses and mirrors are properly situated, light from virtually every part of the object will strike every part of the film. Each portion of the film contains enough information to reproduce the whole object!

More Related