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MTE 3 Results Phy208 Exam 3 50 Get prepared for the Final! 40 # SCORES Final Mon. May 12, 12:25-2:25, Ingraham B10 Average 79.75/100 std 12.30/100 A 19.9% AB 20.8% B 26.3% BC 17.4% C 13.1% D 2.1% F 0.4% 30 20 10 0 30 40 50 60 70 80 SCORE 90 Remember Final counts 25% of final grade! It will contain new material and MTE1-3 material (no alternate exams!!! but notify SOON any potential and VERY serious problem you have with this time) 100 Atomic Physics HONOR LECTURE ! Previous Lecture: Particle in a Box, wave functions and energy levels Quantum-mechanical tunneling and the scanning tunneling microscope Start Particle in 2D,3D boxes ! This Lecture: More on Particle in 2D,3D boxes Other quantum numbers than n: angular momentum H-atom wave functions Pauli exclusion principle PROF. R. Wakai (Medical Physics) Biomagnetism Biomagnetism deals with the registration and analysis of magnetic fields which are produced by organ systems in the body. 3 From last week: particle in a box Summary of quantum information Energy is quantized Sometimes velocity is to left, sometimes to right En = Quantum mechanics: Particle is a wave: p = mv = h/! ! standing wave: superposition of waves traveling left and right => integer number of wavelengths in the tube " h2 % p2 = n 2$ 2' 2m # 8mL & the larger the box the lower the energy of the particle in the box A quantum particle in a box cannot be at rest! Fundamental state energy is not zero: En=1 = 0.38 eV for an electron in a quantum well of L = 1 nm Consequence of uncertainty principle: "x = L # "px $ 1/L % 0! ! Energy Classical: particle bounces back and forth. n=5 n=4 n=3 n=2 n=1 Classical/Quantum Probability n=3 Probability (2D) x Ground state: same wavelength (longest) in both x and y Need two quantum #’s, one for x-motion one for y-motion Use a pair (nx, ny) Ground state: (1,1) n=2 Tunneling: nonzero probability of escaping the box. Tunneling Microscope: tunneling electron current from sample to probe sensitive to surface variations n=1 y Particle in a 2D box Same energy but different probability in space Similar when n $% (nx, ny) = (2,1) Particle in 3D box (nx, ny) = (1,2) With increasing energy... (222) (221) ! ! Ground state surface of constant probability (nx, ny, nz)=(1,1,1) px = h h = nx "n x 2L same for y,z ! (211) (121) (112) E= p2 p2 px2 + y + z = E o (n x2 + n y2 + n z2 ) quantum states with 2m 2m 2m same nx, ny, nz have same E ( 2 manyo 3Dx Eg: howE = E n + n + n All these states have the same energy, but different probabilities ! ) 2 2 particle y zstates have 18E0? (n ,n ,n ) = (4,1,1), (1,4,1), (1,1,4) x y z ! Other quantum numbers? H-atom Quantization of angular momentum !Bohr model fails describing atoms heavier than H !Does it violate the Heisenberg uncertainty principle? A) YES B) No 13.6 eV rn = n 2 ao Bohr n2 radius and energy of electron cannot be exactly known at the same time! ! Hydrogen atom is 3D structure. ! Schrödinger: L = h l(l + 1) En = " ! " is the orbital quantum number States with same n, have same energy and can have " = 0,1,2,...,n-1 orbital quantum number " =0 orbits are most elliptical " =n-1 most circular The z component of the angular momentum must also be quantized ! Should have 3 quantum numbers. !Coulomb potential (electron-proton interaction) is spherically symmetric. x, y, z not as useful as r, !, # ! Modified H-atom should have 3 quantum numbers Lz = m l h m ℓ ranges from - ℓ, to ℓ integer values=> (2ℓ+1) different values ! The experiment: Stern and Gerlach It is possible to measure the number of possible values of Lz respect to the axis of the B-field produced by the electron r current Orbital magnetic dipole For a quantum state with " = 2, how many different orientations of the orbital angular momentum respect to the z-axis are there? ! A. 1 B. 2 C. 3 D. 4 E. 5 µ electron Current e- ! ! ! ! ! ! s: ℓ=0 p: ℓ=1 d: ℓ=2 f: ℓ=3 g: ℓ=4 The electron moving on the orbit is like a current that produces a magnetic momentum µ=IA r e ev 2 e e µ = "r 2 = "r = (mvr) = L µ = µB l(l + 1) T 2"r 2m “atomic shells” 2m ! ! Summary of quantum numbers 3D Surfaces of constant prob. for H-atom Electron cloud: probability density in 3D of electron around the nucleus 2 P(r," ,# )dV = $(r," ,# ) dV ! ! For hydrogen atom: ! ! ! En = " 13.6 ! eV n : describes energy of orbit n2 !"describes the magnitude of orbital angular momentum m ! describes the angle of the orbital angular momentum ! L = h l(l + 1) Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability Lz = m l h n = 1, l = 0, ml = 0 ! ! ! ! Next highest energy: n = 2 2s-state n = 2, l = 0, ml = 0 2p-state 3p-state 3s-state 2p-state n = 2, l = 1, ml = 0 ! n = 3: 2 s-states, 6 p-states and... 3p-state n = 3, l = 0, ml = 0 n = 2, l = 1, ml = ±1 n = 3, l = 1, m l = 0 Same energy, but different probabilities ! ! ! ! ! n = 3, l = 1, ml = ±1 ...10 d-states Radial probability "n,l,m (r,# ,$ ) = Rn,l (r)Yl,m (# ,$ ) Angular Radial ! ! 3d-state 3d-state 3d-state n = 3, l = 2, ml = 0 n = 3, l = 2, ml = ±1 n = 3, l = 2, ml = ±2 ! rn = n 2 ao ! ! ! ! ! For 1s, 2p, 3d, rpeak = a0, 4a0, 9a0 These are the Bohr orbit radii! ! most probable distance of electron from nucleus! They behave like Bohr orbits because for states with same E, larger angular momentum corresponds to more spherical orbits, orbits are elliptical for small " All quantum numbers of electrons in atoms Electron spin ! New electron property: Electron acts like a bar magnet with N and S pole. Magnetic moment fixed… Quantum state specified by four quantum numbers: (n, l, ml , ms ) ! Three spatial quantum numbers (3-dimensional) ! One spin quantum number ! …but 2 possible orientations of magnet: up and down How many different quantum states exist with n=2? A. 1 B. 2 C. 4 D. 8 Described by spin quantum number ms z-component of spin angular momentum Sz = msh ! = 0 : 2s2 ml = 0 : ms = 1/2 , -1/2 ! = 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 2 states ! Pauli exclusion principle ! ! Building Atoms Electrons obey Pauli exclusion principle Only one electron per quantum state (n, !, m!, ms) unoccupied Atom Configuration H 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 1s shell filled occupied Hydrogen: 1 electron n=1 states one quantum state occupied (n = 1,l = 0,ml = 0,ms = +1/2) ! Helium: 2 electrons two quantum states occupied (n = 1,l = 0,ml = 0,ms = +1/2) (n = 1,l = 0,ml = 0,ms = "1/2) ! ! n=1 states Ne etc (n=1 shell filled noble gas) 2s shell filled 1s22s22p6 2p shell filled (n=2 shell filled noble gas) 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 Sc 3d1 Y 3d2 8 more transition metals B C N O F 2p1 2p2 2p3 2p4 2p5 Al Si P S Cl 3p1 3p2 3p3 3p4 3p5 Ga Ge As Se Br 4p1 4p2 4p3 4p4 4p5 He 1s2 Ne 2p6 Ar 3p6 Kr 4p6