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Total Angular Momentum Orbital angular momentum Spin angular momentum Total angular momentum L, Lz, S, SzJ and Jz are quantized 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 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, . Spin-Orbit Coupling An effect of the spins of the electron and the orbital angular momentum interaction is called spin-orbit coupling. • The dipole potential energy • The spin magnetic moment • . is th·e magnetic field due to the proton where cos a is the angle between 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 Total Angular Momentum With an external magnetic field: will precess about Total Angular Momentum Now the selection rules for a single-electron atom become Δn = anything Δmj = 0, ±1 Δℓ = ±1 Δj = 0, ±1 Hydrogen energy-level diagram for n = 2 and n = 3 with the spinorbit splitting The Energy-Level Diagram of Sodium Fine structure splitting is too small on this scale(not shown) Many-Electron Atoms Hund’s rules: 1) The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. 2) Insofar as rule 1 is not violated, L should also be maximized. 3) 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. 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 Table 8-2 p287 LS Coupling The notation for a single-electron atom becomes n2S+1 LJ The letters and numbers are called spectroscopic symbols. There are singlet states (S = 0) and triplet states (S = 1) for two electrons. 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 LS Coupling The allowed transitions for the LS coupling scheme are ΔL = ±1 ΔJ = 0, ±1 ΔS = 0 (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. 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 . Mercury Figure 8-11 p290 Figure 8-12 p290 Clicker - Questions 8.3: Anomalous Zeeman Effect 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. Orbital contribution and Spin magnetic moment 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 . Total Angular Momentum With an external magnetic field: will precess about Anomalous Zeeman Effect The total magnetic moment is μB is the Bohr magneton and it is called the Landé g factor 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. 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 Anomalous Zeeman effect for Na Figure 8-16 p294 Rydberg Atom An atom that is highly excited with the outermost electron in a high energy level near ionization Properties of Rydberg atoms: Useful Combinations of Constants 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. 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 Problem 8.21 1 1 3 1240eV nm 7.334 10 eV. 1 2 766.41 nm 769.90 nm As in Example 8.8 the internal magnetic field is 1. E hc hc 9.109 1031 kg 7.334 103 eV mE B 63.4 T 19 16 e 1.602 10 C 6.582 10 eV s Problem 8.31 1. av B B L 2S and J L S L 2S L S J Using the equation in the problem, we have J2 Now L 2S L S L2 2S 2 3L S and using J L S we have J J L2 S 2 2L S so J 2 L2 S 2 and L·S 2 B 3J 2 S 2 L2 3J 2 S 2 L2 . Therefore av L 2S L S J 2 2J 2 . With B defined to be in the +z-direction, V av B B B 3J 2 S 2 L2 2J 2 J z . As usual 3J 2 S 2 L2 J z mJ , so V B BmJ . Now the vector magnitudes are J 2 j ( j 1) 2 2J S 2 s( s 1) 2 , and L2 ( 1) 2 so we obtain V B BmJ g where g 3 j ( j 1) s( s 1) ( 1) j ( j 1) s( s 1) ( 1) . 1 2 j ( j 1) 2 j ( j 1) 2 , Problem 8.33 1/ 2 3 / 2 1/ 2 3 / 2 1 1 2 2 1/ 2 3 / 2 3 / 2 5 / 2 1/ 2 3 / 2 1(2) 1 1 4 g 1 2 3 / 2 5 / 2 3 3 1. a) L 0 , S J 1/ 2 ; g 1 (b) L 1 , S 1/ 2 , and J 3 / 2 ; (c) L 2 , S 1/ 2 , and J 5 / 2 ; g 1 5 / 2 7 / 2 1/ 2 3 / 2 2(3) 1 1 6 2 5 / 2 7 / 2 5 5 Problem 8.15 Problem 8.17 Problem 8.19 Problem 10. 18 1. (a) Using dimensional analysis and the fact that the energy of each photon is hc / 3.14 1019 J , then N 5 103 J/s 1.59 1016 photon/s . 19 t 3.14 10 J/photon (b) 0.02 mole is equivalent to 0.02 N A 1.20 1022 atoms . The fraction participating is 1.59 1016 1.33 106 . 22 1.20 10 (c) The transitions involved have a fairly low probability, even with stimulated emission. We are saved by the large number of atoms available. Problem 10.19 1.15 eV 1.602 1019 J 1. (a) The energy of each photon is E2 E1 · 1.84 1019 J/photon . photon 1eV Since P E / t then we have P 1.84 1019 J/photon 5.50 1018 photons/s 1.00W . hc 1.986 1025 J m 1.08 106 m 1.08 μm (b) 19 E 1.84 10 J Problem 10.20 1. (a) With the given wavelength, we know that each photon has an energy of 1.986 1025 J m E 5.658 1019 J . 9 35110 m hc Therefore the number of transitions required is N 40 103 J 7.07 1022 photons . 19 5.658 10 J/photon 40 103 J (b) Average power is energy divided by time: P 1.14 1013 W 9 3.5 10 s or about 11 TW Problem 10.22 2 1 m 6.67 109 s 8 2.998 10 m/s 16 103 m 16 103 m (b) For the 16-km round trip t 1.6 108 s . 8 4 8 2.998 10 m/s 1 3 10 2.998 10 m/s 1. (a) t Because this result is larger than the desired uncertainty in timing, it is important to take atmospheric effects into account. Why does Bose-Einstein Condensation of Atoms Occur? Rb atom Wieman Na atom Ketterle______ Eric Cornell and Carl Wolfgang 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. Bose –Einstein condensation in Gases Clicker - Questions 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: 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. 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. 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. 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. 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! Holography Transmission hologram: • The reference beam is on the same side of the film as the object and the illuminating beam is on the opposite side. Reflection hologram: • Reverse the positions of the reference and illuminating beam. The result will be a white light hologram in which the different colors contained in white light provide the colors seen in the image. Interferometry: • Two holograms of the same object produced at different times can be used to detect motion or growth that could not otherwise be seen. Quantum Entanglement, Teleportation, and Information • Schrödinger used the term “quantum entanglement” to describe a strange correlation between two quantum systems. He considered entanglement for quantum states acting across large distances, which Einstein referred to as “spooky action at a distance.” Quantum teleportation: • No information can be transmitted through only quantum entanglement, but transmitting information using entangled systems in conjunction with classical information is possible. Quantum Entanglement, Teleportation, and Information Alice, who does not know the property of the photon, is spacially separated from Bob and tries to transfer information about photons. 1) A beam splitter can be used to produce two additional photons that can be used to trigger a detector. 2) Alice can manipulate her quantum system and send that information over a classical information channel to Bob. 3) Bob then arranges his part of the quantum system to detect information. Other Laser Applications • Used in surgery to make precise incisions Ex: eye operations • We see in everyday life such as the scanning devices used by supermarkets and other retailers Ex. Bar code of packaged product CD and DVD players • Laser light is directed toward disk tracks that contain encoded information. The reflected light is then sampled and turned into electronic signals that produce a digital output.