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Last Time…

Last Time…. Decreasing particle size. 3-dimensional wave functions. Quantum dots (particle in box). Quantum tunneling. This week’s honors lecture: Prof. Brad Christian, “Positron Emission Tomography”. Exam 3 results. Average. Exam average = 76% Average is at B/BC boundary.

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Last Time…

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  1. Last Time… Decreasing particle size 3-dimensional wave functions Quantum dots (particle in box) Quantum tunneling This week’s honors lecture:Prof. Brad Christian, “Positron Emission Tomography” Physics 208, Lecture 26

  2. Exam 3 results Average • Exam average = 76% • Average is at B/BC boundary Course evaluations: Rzchowski: Thu, Dec. 6 Montaruli: Tues, Dec. 11 Physics 208, Lecture 26

  3. 3-D particle in box: summary • Three quantum numbers (nx,ny,nz)label each state • nx,y,z=1, 2, 3 … (integers starting at 1) • Each state has different motion in x, y, z • Quantum numbersdetermine • Momentum in each direction: e.g. • Energy: • Some quantum states have same energy Physics 208, Lecture 26

  4. Question How many 3-D particle in box spatial quantum states have energy E=18Eo? A. 1 B. 2 C. 3 D. 5 E. 6 Physics 208, Lecture 26

  5. 3-D Hydrogen atom • Bohr model: • Restricted to circular orbits • Found 1 quantum number n • Energy , orbit radius • From 3-D particle in box, expect that • H atom should have more quantum numbers • Expect different types of motion w/ same energy Physics 208, Lecture 26

  6. Modified Bohr model C B A Big angular momentum Small angular momentum • Different orbit shapes A, B, C C, B, A B, C, A B, A, C C, A, B These orbits have same energy, but different angular momenta: Rank the angular momenta from largest to smallest: Physics 208, Lecture 26

  7. Angular momentum is quantized orbital quantum number ℓ Angular momentum quantized , ℓ is the orbital quantum number For a particular n, ℓ has values 0, 1, 2, … n-1 ℓ=0, most elliptical ℓ=n-1, most circular For hydrogen atom, all have same energy Physics 208, Lecture 26

  8. Orbital mag. moment Orbital magnetic dipole electron Current • Each orbit has • Same energy: • Different orbit shape (angular momentum): • Different magnetic moment: • Orbital charge motion produces magnetic dipole • Proportional to angular momentum Physics 208, Lecture 26

  9. Orbital mag. quantum number mℓ • Directions of ‘orbital bar magnet’ quantized. • Orbital magnetic quantum number • mℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values • Determines z-component of L: • This is also angle of L For example: ℓ=1 gives 3 states: Physics 208, Lecture 26

  10. Question • For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there?A. 1B. 2C. 3D. 4E. 5 Physics 208, Lecture 26

  11. Summary of quantum numbers • n : describes energy of orbit • ℓ describes the magnitude of orbital angular momentum • mℓ describes the angle of the orbital angular momentum For hydrogen atom: Physics 208, Lecture 26

  12. Hydrogen wavefunctions • Radial probability • Angular not shown • For given n, probability peaks at ~ same place • Idea of “atomic shell” • Notation: • s: ℓ=0 • p: ℓ=1 • d: ℓ=2 • f: ℓ=3 • g: ℓ=4 Physics 208, Lecture 26

  13. Full hydrogen wave functions: Surface of constant probability • Spherically symmetric. • Probability decreases exponentially with radius. • Shown here is a surface of constant probability 1s-state Physics 208, Lecture 26

  14. n=2: next highest energy 2s-state 2p-state 2p-state Same energy, but different probabilities Physics 208, Lecture 26

  15. n=3: two s-states, six p-states and… 3p-state 3s-state 3p-state Physics 208, Lecture 26

  16. …ten d-states 3d-state 3d-state 3d-state Physics 208, Lecture 26

  17. Electron spin N S • Spin up S N • Spin down z-component of spin angular momentum New electron property:Electron acts like a bar magnet with N and S pole. Magnetic moment fixed… …but 2 possible orientations of magnet: up and down Described by spin quantum number ms Physics 208, Lecture 26

  18. Include spin • Quantum state specified by four quantum numbers: • Three spatial quantum numbers (3-dimensional) • One spin quantum number Physics 208, Lecture 26

  19. ℓ = 0 : ml = 0 : ms = 1/2 , -1/2 2 states 2s2 ℓ = 1 : ml = +1: ms = 1/2 , -1/2 2 states ml = 0: ms = 1/2 , -1/2 2 states ml = -1: ms = 1/2 , -1/2 2 states 2p6 There are a total of 8 states with n=2 Quantum Number Question How many different quantum states exist with n=2? A. 1 B. 2 C. 4 D. 8 Physics 208, Lecture 26

  20. Question How many different quantum states are in a 5g (n=5, ℓ=4) sub-shell of an atom? A. 22 B. 20 C. 18 D. 16 E. 14 ℓ =4, so 2(2 ℓ +1)=18. In detail, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4and ms=+1/2 or -1/2 for each. 18 available quantum states for electrons Physics 208, Lecture 26

  21. Putting electrons on atom unoccupied occupied n=1 states Hydrogen: 1 electronone quantum state occupied Helium: 2 electronstwo quantum states occupied n=1 states • Electrons obey Pauli exclusion principle • Only one electron per quantum state (n, ℓ, mℓ, ms) Physics 208, Lecture 26

  22. Atoms with more than one electron • Electrons interact with nucleus (like hydrogen) • Also with other electrons • Causes energy to depend on ℓ Physics 208, Lecture 26

  23. Other elements: Li has 3 electrons n=2 states, 8 total, 1 occupied n=1 states, 2 total, 2 occupiedone spin up, one spin down Physics 208, Lecture 26

  24. Electron Configurations Atom Configuration H 1s1 He 1s2 1s shell filled (n=1 shell filled - noble gas) Li 1s22s1 Be 1s22s2 2s shell filled B 1s22s22p1 etc (n=2 shell filled - noble gas) Ne 1s22s22p6 2p shell filled Physics 208, Lecture 26

  25. The periodic table H1s1 He1s2 Li2s1 Be2s2 B2p1 C2p2 N2p3 O2p4 F2p5 Ne 2p6 Na3s1 Mg3s2 Al3p1 Si3p2 P3p3 S3p4 Cl3p5 Ar3p6 K K4s1 Ca Ca4s2 Sc3d1 Y3d2 Ga4p1 Ge4p2 As4p3 Se4p4 Br4p5 Kr4p6 8 moretransition metals • Atoms in same column have ‘similar’ chemical properties. • Quantum mechanical explanation: similar ‘outer’ electron configurations. Na3s1 Physics 208, Lecture 26

  26. Excited states of Sodium • Na level structure • 11 electrons • Ne core = 1s2 2s2 2p6(closed shell) • 1 electron outside closed shell Na = [Ne]3s1 • Outside (11th) electron easily excited to other states. Physics 208, Lecture 26

  27. Emitting and absorbing light Zero energy Photon is emitted when electron drops from one quantum state to another n=4 n=4 n=3 n=3 n=2 n=2 Photon emittedhf=E2-E1 Photon absorbedhf=E2-E1 n=1 n=1 Absorbing a photon of correct energy makes electron jump to higher quantum state. Physics 208, Lecture 26

  28. Optical spectrum • Optical spectrum of sodium • Transitions from high to low energystates • Relatively simple • 1 electronoutside closed shell Na 589 nm, 3p -> 3s Physics 208, Lecture 26

  29. How do atomic transitions occur? • How does electron in excited state decide to make a transition? • One possibility: spontaneous emission • Electron ‘spontaneously’ drops from excited state • Photon is emitted ‘lifetime’ characterizes average time for emitting photon. Physics 208, Lecture 26

  30. Another possibility: Stimulated emission • Atom in excited state. • Photon of energy hf=E ‘stimulates’ electron to drop. Additional photon is emitted, Same frequency, in-phase with stimulating photon One photon in,two photons out: light has been amplified E hf=E Before After If excited state is ‘metastable’ (long lifetime for spontaneous emission) stimulated emission dominates Physics 208, Lecture 26

  31. LASER : Light Amplification by Stimulated Emission of Radiation Atoms ‘prepared’ in metastable excited states …waiting for stimulated emission Called ‘population inversion’ (atoms normally in ground state) Excited states stimulated to emit photon from a spontaneous emission. Two photons out, these stimulate other atoms to emit. Physics 208, Lecture 26

  32. Ruby Laser • Ruby crystal has the atoms which will emit photons • Flashtube provides energy to put atoms in excited state. • Spontaneous emission creates photon of correct frequency, amplified by stimulated emission of excited atoms. Physics 208, Lecture 26

  33. Ruby laser operation Relaxation to metastable state(no photon emission) Transition by stimulated emission of photon PUMP 3 eV 2 eV Metastable state 1 eV Ground state Physics 208, Lecture 26

  34. The wavefunction • Wavefunction =  = |moving to right> + |moving to left> • The wavefunction is an equal ‘superposition’ of the two states of precise momentum. • When we measure the momentum (speed), we find one of these two possibilities. • Because they are equally weighted, we measure them with equal probability. Physics 208, Lecture 26

  35. Silicon • 7x7 surface reconstruction • These 10 nm scans show the individual atomic positions Physics 208, Lecture 26

  36. Particle in box wavefunction x=0 x=L Prob. Of finding particle in region dx about x Particle is never here Particle is never here Physics 208, Lecture 26

  37. Making a measurement Suppose you measure the speed (hence, momentum) of the quantum particle in a tube. How likely are you to measure the particle moving to the left? A. 0% (never) B. 33% (1/3 of the time) C. 50% (1/2 of the time) Physics 208, Lecture 26

  38. Interaction with applied B-field • Like a compass needle, it interacts with an external magnetic field depending on its direction. • Low energy when aligned with field, high energy when anti-aligned • Total energy is then This means that spectral lines will splitin a magnetic field Physics 208, Lecture 26

  39. Physics 208, Lecture 26

  40. Orbital magnetic dipole moment Current = Area = In quantum mechanics, magnitude of orb. mag. dipole moment Can calculate dipole moment for circular orbit Dipole moment µ=IA Physics 208, Lecture 26

  41. Physics 208, Lecture 26

  42. Electron magnetic moment • Why does it have a magnetic moment? • It is a property of the electron in the same way that charge is a property. • But there are some differences • Magnetic moment has a size and a direction • It’s size is intrinsic to the electron, but the direction is variable. • The ‘bar magnet’ can point in different directions. Physics 208, Lecture 26

  43. Additional electron properties N S • Free electron, by itself in space, not only has a charge, but also acts like a bar magnet with a N and S pole. • Since electron has charge, could explain this if the electron is spinning. • Then resulting current loops would produce magnetic field just like a bar magnet. • But… • Electron in NOT spinning. • As far as we know, electron is a point particle. Physics 208, Lecture 26

  44. Spin: another quantum number • There is a quantum # associated with this property of the electron. • Even though the electron is not spinning, the magnitude of this property is the spin. • The quantum numbers for the two states are +1/2 for the up-spin state -1/2 for the down-spin state • The proton is also a spin 1/2 particle. • The photon is a spin 1 particle. Physics 208, Lecture 26

  45. Orbital mag. moment Orbital magnetic moment electron Current Make a question out of this • Since • Electron has an electric charge, • And is moving in an orbit around nucleus… • produces a loop of current,and a magnetic dipole moment , • Proportional to angular momentum magnitude of orb. mag. dipole moment Physics 208, Lecture 26

  46. Orbital mag. quantum number mℓ N S S N S N mℓ = +1 mℓ = -1 mℓ = 0 • Possible directions of the ‘orbital bar magnet’ are quantized just like everything else! • Orbital magnetic quantum number • mℓ ranges from - ℓ, to ℓ in integer steps • Number of different directions = 2ℓ+1 For example: ℓ=1 gives 3 states: Physics 208, Lecture 26

  47. Particle in box quantum states L n=1 n p E Wavefunction Probability n=3 n=2 Physics 208, Lecture 26

  48. Particle in box energy levels n=5 Energy n=4 n=3 n=2 n=1 • Quantized momentum • Energy = kinetic • Or Quantized Energy n=quantum number Physics 208, Lecture 26

  49. Hydrogen atom energies Zero energy n=4 n=3 n=2 Energy n=1 • Quantized energy levels: • Each corresponds to different • Orbit radius • Velocity • Particle wavefunction • Energy • Each described by a quantum number n Physics 208, Lecture 26

  50. Physics 208, Lecture 26

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