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Last Time… Decreasing particle size Quantum dots (particle in box) Quantum tunneling 3-dimensional wave functions This week’s honors lecture: Prof. Brad Christian, “Positron Emission Tomography” Tue. Dec. 4 2007 Physics 208, Lecture 26 1 Exam 3 results Phy208 Exam 3 Exam average = 76% Average is at B/BC boundary Course evaluations: Rzchowski: Thu, Dec. 6 Montaruli: Tues, Dec. 11 Average 25 20 AB Count 30 15 B 10 C A D 5 F 0 10 Tue. Dec. 4 2007 BC 20 30 Physics 208, Lecture 26 40 50 60 70 Range 80 90 100 2 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 numbers determine px h n nx x h 2L Momentum in each direction: e.g. 2 2 p p p Energy: E y z E o nx2 ny2 nz2 2m 2m 2m 2 x Some quantum states have same energy Tue. Dec. 4 2007 Physics 208, Lecture 26 3 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 Tue. Dec. 4 2007 E E o n x2 n y2 n z2 n ,n ,n 4,1,1, 1,4,1, (1,1,4) x y z Physics 208, Lecture 26 4 3-D Hydrogen atom Bohr model: Restricted to circular orbits Found 1 quantum number n Energy 13.6 E n 2 eV , orbit radius n rn n ao 2 From 3-D particle in box, expect that H atom should have more quantum numbers Expect different types of motion w/ same energy Tue. Dec. 4 2007 Physics 208, Lecture 26 5 Modified Bohr model Different orbit shapes A B C Big angular momentum Small angular momentum These orbits have same energy, a) A, B, but different angular momenta: b) C, B, Lrp c) B, C, Rank the angular momenta d) B, A, from largest to smallest: A A C e) C, A, B Tue. Dec. 4 2007 C Physics 208, Lecture 26 6 Angular momentum is quantized orbital quantum number ℓ Angular momentum quantized L ℓ is the orbital quantum number 1, 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 Tue. Dec. 4 2007 Physics 208, Lecture 26 7 Orbital mag. moment Orbital magnetic dipole Orbital charge motion produces magnetic dipole Proportional to angular momentum e B 0.927 1023 A m 2 B L / B 1 electron Current 2m Each orbit n, has 2 eV Same energy: E n 13.6 /n Different orbit shape 1 (angular momentum): L Different magnetic moment: B L / Tue. Dec. 4 2007 Physics 208, Lecture 26 8 Orbital mag. quantum number mℓ Directions of ‘orbital bar magnet’ quantized. Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values Determines z-component of L: Lz m This is also angle of L For example: ℓ=1 gives 3 states: Tue. Dec. 4 2007 Physics 208, Lecture 26 9 Question For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there? A. 1 B. 2 C. 3 D. 4 E. 5 Tue. Dec. 4 2007 Physics 208, Lecture 26 10 Summary of quantum numbers For hydrogen atom: n : describes energy of orbit ℓ describes the magnitude of orbital angular momentum m ℓ describes the angle of the orbital angular momentum Tue. Dec. 4 2007 Physics 208, Lecture 26 11 Hydrogen wavefunctions Radial probability Angular not shown For given n, probability peaks at ~ same place Idea of “atomic shell” Notation: s: ℓ=0 p: d: f: g: ℓ=1 ℓ=2 ℓ=3 ℓ=4 Tue. Dec. 4 2007 Physics 208, Lecture 26 12 Full hydrogen wave functions: Surface of constant probability 1s-state n 1, Tue. Dec. 4 2007 Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability 0, m 0 Physics 208, Lecture 26 13 n=2: next highest energy 2s-state 2p-state n 2, 0, m 0 n 2, 1, m 0 2p-state n 2, 1, m 1 Same energy, but different probabilities Tue. Dec. 4 2007 26 Physics 208, Lecture 14 n=3: two s-states, six p-states and… 3p-state 3s-state 3p-state n 3, 0, m 0 Tue. Dec. 4 2007 n 3, 1, m 0 Physics 208, Lecture 26 n 3, 1, m 1 15 …ten d-states 3d-state 3d-state n 3, 3d-state 2, m 0 Tue. Dec. 4 2007 n 3, 2, m 1 Physics 208, Lecture 26 n 3, 2, m 2 16 Electron spin 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 Spin up ms 1/2 Spin down ms 1/2 z-component of spin angular momentum Sz ms Tue. Dec. 4 2007 Physics 208, Lecture 26 17 Include spin Quantum state specified by four quantum numbers: n, , m , ms Three spatial quantum numbers (3-dimensional) One spin quantum number Tue. Dec. 4 2007 Physics 208, Lecture 26 18 Quantum Number Question How many different quantum states exist with n=2? A. 1 B. 2 C. 4 D. 8 ℓ = 0 : 2s2 ml = 0 : ms = 1/2 , -1/2 2 states ℓ = 1 : 2p6 ml = +1: ms = 1/2 , -1/2 ml = 0: ms = 1/2 , -1/2 ml = -1: ms = 1/2 , -1/2 2 states 2 states 2 states There are a total of 8 states with n=2 Tue. Dec. 4 2007 Physics 208, Lecture 26 19 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 ℓ =4, so 2(2 ℓ +1)=18. E. 14 In detail, m = -4, -3, -2, -1, 0, 1, 2, 3, 4 l and ms=+1/2 or -1/2 for each. 18 available quantum states for electrons Tue. Dec. 4 2007 Physics 208, Lecture 26 20 Putting electrons on atom Electrons obey Pauli exclusion principle Only one electron per quantum state (n, ℓ, mℓ, ms) unoccupied occupied n=1 states Hydrogen: 1 electron one quantum state occupied n 1, 0,m 0,ms 1/2 Helium: 2 electrons n=1 states two quantum states occupied n 1, 0,m 0,ms 1/2 n 1, 0,m 0,ms 1/2 Tue. Dec. 4 2007 Physics 208, Lecture 26 21 Atoms with more than one electron Electrons interact with nucleus (like hydrogen) Also with other electrons Causes energy to depend on ℓ Tue. Dec. 4 2007 Physics 208, Lecture 26 22 Other elements: Li has 3 electrons n 2 0 m 0 1 ms 2 n 2 0 m 0 1 ms 2 n 2 1 m 0 1 ms 2 n 2 1 m 0 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n=2 states, 8 total, 1 occupied n=1 states, 2 total, 2 occupied n 1 n 1 0 0 m 0 m 0 m s 1/2 m s 1/2 Tue. Dec. 4 2007 one spin up, one spin down Physics 208, Lecture 26 23 Electron Configurations Atom Configuration H 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 Ne Tue. Dec. 4 2007 etc 1s shell filled 1s22s22p6 (n=1 shell filled noble gas) 2s shell filled 2p shell filled Physics 208, Lecture 26 (n=2 shell filled noble gas) 24 The periodic table Atoms in same column have ‘similar’ chemical properties. Quantum mechanical explanation: similar ‘outer’ electron configurations. H 1s1 Li 2s1 Na 3s1 K 4s1 Be 2s2 Mg 3s2 Ca 4s2 Tue. Dec. 4 2007 Sc 3d1 Y 3d2 8 more transition metals B 2p1 Al 3p1 Ga 4p1 C 2p2 Si 3p2 Ge 4p2 Physics 208, Lecture 26 N 2p3 P 3p3 As 4p3 O 2p4 S 3p4 Se 4p4 F 2p5 Cl 3p5 Br 4p5 He 1s2 Ne 2p6 Ar 3p6 Kr 4p6 25 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 26 Emitting and absorbing light Zero energy n=4 n=3 13.6 E 3 2 eV 3 n=2 13.6 E 2 2 eV 2 Photon emitted hf=E2-E1 n=1 E3 13.6 eV 32 n=2 E2 13.6 eV 22 E1 13.6 eV 12 Photon absorbed hf=E2-E1 E1 13.6 eV 12 Photon is emitted when electron drops fromone quantum state to another Tue. Dec. 4 2007 n=4 n=3 n=1 Absorbing a photon of correct energy makeselectron jump to higher quantum state. Physics 208, Lecture 26 27 Optical spectrum of sodium Transitions from high to low energy states Relatively simple 1 electron outside closed shell Tue. Dec. 4 2007 Physics 208, Lecture 26 589 nm, 3p -> 3s Optical spectrum Na 28 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 29 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 Tue. Dec. 4 2007 Physics 208, Lecture 26 30 : Light Amplification by Stimulated Emission of Radiation LASER 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 31 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 32 Ruby laser operation Relaxation to metastable state (no photon emission) 3 eV 2 eV 1 eV Metastable state PUMP Transition by stimulated emission of photon Ground state Tue. Dec. 4 2007 Physics 208, Lecture 26 33 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 34 Silicon Tue. Dec. 4 2007 Physics 208, Lecture 26 7x7 surface reconstruction These 10 nm scans show the individual atomic positions 35 Particle in box wavefunction x dx Prob. Of finding particle in region dx about x 2 x L ? Particle is never here x=0 x=L Particle is never here x 0 ? Tue. Dec. 4 2007 Physics 208, Lecture 26 36 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) Tue. Dec. 4 2007 Physics 208, Lecture 26 37 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 13.6 E eV B Total energy is then 2 This means that spectral lines will split in a magnetic field Tue. Dec. 4 2007 n 13.6 2 eV z B n 13.6 2 eV m B B n Physics 208, Lecture 26 38 Tue. Dec. 4 2007 Physics 208, Lecture 26 39 Orbital magnetic dipole moment Can calculate dipole moment for circular orbit charge e ev Current = period 2r /v 2r Dipole moment µ=IA Area = r evr e mvr/ 2 2m B L / In quantum mechanics, L B Tue. Dec. 4 2007 2 B e 0.927 1023 A m 2 2m 5.79 105 eV /Tesla 1 1 magnitude of orb. mag. dipole moment Physics 208, Lecture 26 40 Tue. Dec. 4 2007 Physics 208, Lecture 26 41 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 42 Additional electron properties 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 43 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. Tue. Dec. 4 2007 Physics 208, Lecture 26 44 Orbital magnetic moment Orbital mag. moment Since Make a question out of this electron Current 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 B L / magnitude of mag. dipole moment orb. Tue. Dec. 4 2007 B e 0.927 1023 A m 2 2m 5.79 105 eV /Tesla B Physics 208, Lecture 26 1 45 Orbital mag. quantum number mℓ Possible directions of the ‘orbital bar magnet’ are quantized just like everything else! Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps Number of different directions = 2ℓ+1 For example: ℓ=1 gives 3 states: m ℓ = +1 Tue. Dec. 4 2007 S N mℓ = 0 Physics 208, Lecture 26 m ℓ = -1 46 Particle in box quantum states L n p n=3 h 3 2L E h2 2 8mL2 h 2L h2 8mL2 Tue. Dec. 4 2007 Probability 2 h 32 8mL2 n=2 2 h 2L n=1 Wavefunction 2 Physics 208, Lecture 26 47 Particle in box energy levels Quantized momentum h h p n npo 2L Energy = kinetic 2 2 npo p E n2Eo 2m 2m Or Quantized Energy Energy n=5 n=4 En n2Eo n=3 n=quantum number n=2 n=1 Tue. Dec. 4 2007 Physics 208, Lecture 26 48 Hydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n Zero energy n=4 n=3 E3 13.6 eV 32 n=2 E2 13.6 eV 22 E1 13.6 eV 12 n=1 13.6 E n 2 eV n Tue. Dec. 4 2007 Energy Physics 208, Lecture 26 49 Tue. Dec. 4 2007 Physics 208, Lecture 26 50 Quantum numbers Two quantum numbers Tue. Dec. 4 2007 Physics 208, Lecture 26 51 Pauli Exclusion Principle Where do the electrons go? In an atom with many electrons, only one electron is allowed in each quantum state (n, ℓ, mℓ, ms). Atoms with many electrons have many atomic orbitals filled. Chemical properties are determined by the configuration of the ‘outer’ electrons. Tue. Dec. 4 2007 Physics 208, Lecture 26 52 Atomic sub-shells Each Tue. Dec. 4 2007 Physics 208, Lecture 26 53